diff --git a/HGD/fluid.py b/HGD/fluid.py
new file mode 100644
index 0000000..2b48b6a
--- /dev/null
+++ b/HGD/fluid.py
@@ -0,0 +1,440 @@
+"""Two-way fluid coupling utilities for heterarchical granular dynamics.
+
+The layer drag, particle relaxation, aggregation, and coupling order follow
+Li et al.'s HGFD formulation.  The bundled fluid participant is a compact
+Python finite-volume projection solver; the paper used an extended OpenFOAM
+``pisoFoam`` participant coupled through preCICE.
+"""
+
+from dataclasses import dataclass
+
+import numpy as np
+from scipy import sparse
+from scipy.sparse.linalg import spsolve
+
+
+@dataclass
+class FluidState:
+    """Eulerian fluid fields and coupling diagnostics on the HGD grid."""
+
+    u: np.ndarray
+    v: np.ndarray
+    pressure: np.ndarray
+    beta: np.ndarray
+    solid_u: np.ndarray
+    solid_v: np.ndarray
+    momentum_source_u: np.ndarray
+    momentum_source_v: np.ndarray
+    divergence: np.ndarray
+    fluidized: np.ndarray
+    void_fraction: np.ndarray
+    pressure_drop: float = 0.0
+    fluidization_ratio: float = 0.0
+
+
+def _void_fraction(solid_fraction, p):
+    minimum = getattr(p, "fluid_min_void_fraction", 0.05)
+    return np.clip(1.0 - solid_fraction, minimum, 1.0)
+
+
+def initialize(p, s):
+    """Initialise a co-located gas field from a prescribed superficial inlet velocity."""
+
+    solid_fraction = 1.0 - np.mean(np.isnan(s), axis=2)
+    void_fraction = _void_fraction(solid_fraction, p)
+    superficial_velocity = getattr(p, "fluid_inlet_velocity", 0.0)
+    shape = solid_fraction.shape
+
+    # n * u_f is the superficial velocity.  This initial field therefore
+    # satisfies the one-dimensional variable-porosity continuity equation.
+    u = np.zeros(shape)
+    v = superficial_velocity / void_fraction
+    pressure = np.zeros(shape)
+    zeros = np.zeros(shape)
+    return FluidState(
+        u=u,
+        v=v,
+        pressure=pressure,
+        beta=zeros.copy(),
+        solid_u=zeros.copy(),
+        solid_v=zeros.copy(),
+        momentum_source_u=zeros.copy(),
+        momentum_source_v=zeros.copy(),
+        divergence=zeros.copy(),
+        fluidized=np.zeros(shape, dtype=bool),
+        void_fraction=void_fraction.copy(),
+    )
+
+
+def drag_coefficients(s, particle_u, particle_v, fluid_u, fluid_v, p):
+    """Return layer and cell Gidaspow momentum-exchange coefficients.
+
+    ``layer_beta`` includes the paper's ``1 / M_s`` distribution over the
+    occupied internal coordinates.  Summing it over ``k`` gives the
+    cell-level coefficient supplied to the fluid momentum equation.
+    """
+
+    occupied = np.isfinite(s)
+    solid_fraction = np.mean(occupied, axis=2)
+    void_fraction = _void_fraction(solid_fraction, p)
+    phi = solid_fraction[:, :, np.newaxis]
+    eps = void_fraction[:, :, np.newaxis]
+
+    fluid_u_3d = fluid_u[:, :, np.newaxis]
+    fluid_v_3d = fluid_v[:, :, np.newaxis]
+    rel_u = fluid_u_3d - particle_u
+    rel_v = fluid_v_3d - particle_v
+    relative_speed = np.sqrt(rel_u**2 + rel_v**2)
+
+    diameter_floor = getattr(p, "fluid_drag_min_particle_size", 1e-12)
+    diameter = np.where(occupied, np.maximum(s, diameter_floor), 1.0)
+    rho_f = p.gas_density
+    mu_f = p.gas_viscosity
+    reynolds = rho_f * relative_speed * diameter / mu_f
+
+    # Written as Cd*|ur| to remain finite in the Stokes limit Re -> 0.
+    beta_wen_yu = 18.0 * mu_f / diameter**2 * (1.0 + 0.15 * reynolds**0.687) * phi * eps ** (-2.65)
+    beta_ergun = 150.0 * mu_f * phi**2 / (eps * diameter**2) + 1.75 * rho_f * phi * relative_speed / diameter
+    cell_beta_per_size = np.where(phi < 0.2, beta_wen_yu, beta_ergun)
+    solid_count = np.sum(occupied, axis=2, keepdims=True)
+    layer_weight = np.divide(
+        1.0,
+        solid_count,
+        out=np.zeros_like(solid_count, dtype=float),
+        where=solid_count > 0,
+    )
+    layer_beta = cell_beta_per_size * layer_weight
+    layer_beta = np.where(occupied, layer_beta, 0.0)
+    cell_beta = np.sum(layer_beta, axis=2)
+    return layer_beta, cell_beta
+
+
+def update_particle_velocities(particle_u, particle_v, s, state, p):
+    """Advance layer-wise particle velocity using the analytic drag relaxation."""
+
+    layer_beta, _ = drag_coefficients(s, particle_u, particle_v, state.u, state.v, p)
+    occupied = np.isfinite(s)
+    # Each occupied internal coordinate represents 1/M of the cell volume.
+    # Its phase mass density is therefore rho_p/M.  This M factor is required
+    # to reproduce the three single-particle terminal velocities in Fig. 4 of
+    # the paper; Eq. (8) prints rho_p/beta_k without making this layer mass
+    # normalisation explicit.
+    relaxation_rate = p.nm * layer_beta / p.solid_density
+    decay = np.exp(-relaxation_rate * p.dt)
+
+    buoyancy = 1.0 - p.gas_density / p.solid_density
+    gravity_x = getattr(p, "fluid_gravity_x", 0.0) * buoyancy
+    gravity_y = -abs(p.g) * buoyancy
+    fluid_u = state.u[:, :, np.newaxis]
+    fluid_v = state.v[:, :, np.newaxis]
+
+    with np.errstate(divide="ignore", invalid="ignore"):
+        terminal_u = fluid_u + np.divide(
+            gravity_x,
+            relaxation_rate,
+            out=np.zeros_like(relaxation_rate),
+            where=relaxation_rate > 0,
+        )
+        terminal_v = fluid_v + np.divide(
+            gravity_y,
+            relaxation_rate,
+            out=np.zeros_like(relaxation_rate),
+            where=relaxation_rate > 0,
+        )
+
+    new_u = terminal_u + (particle_u - terminal_u) * decay
+    new_v = terminal_v + (particle_v - terminal_v) * decay
+
+    # In cells where the drag coefficient is exactly zero, integrate gravity
+    # directly instead of using the singular terminal-velocity expression.
+    no_drag = relaxation_rate == 0
+    new_u = np.where(no_drag, particle_u + gravity_x * p.dt, new_u)
+    new_v = np.where(no_drag, particle_v + gravity_y * p.dt, new_v)
+    return np.where(occupied, new_u, 0.0), np.where(occupied, new_v, 0.0)
+
+
+def coupling_fields(s, particle_u, particle_v, state, p):
+    """Aggregate heterarchical drag fields for the fluid momentum equation."""
+
+    layer_beta, cell_beta = drag_coefficients(s, particle_u, particle_v, state.u, state.v, p)
+    virtual_displacement = getattr(p, "hgfd_virtual_displacement", None)
+    if virtual_displacement is None:
+        active = np.isfinite(s)
+    else:
+        active = np.isfinite(s) & (virtual_displacement < p.dx)
+    active_beta = np.where(active, layer_beta, 0.0)
+    active_beta_sum = np.sum(active_beta, axis=2)
+    with np.errstate(divide="ignore", invalid="ignore"):
+        solid_u = np.divide(
+            np.sum(active_beta * particle_u, axis=2),
+            active_beta_sum,
+            out=np.zeros_like(cell_beta),
+            where=active_beta_sum > 0,
+        )
+        solid_v = np.divide(
+            np.sum(active_beta * particle_v, axis=2),
+            active_beta_sum,
+            out=np.zeros_like(cell_beta),
+            where=active_beta_sum > 0,
+        )
+
+    # Equal and opposite to beta * (u_f - u_p), the drag on particles.  The
+    # paper prints S_m=beta*(u_f-u_p) with a positive fluid source, but also
+    # states that the source is equal-and-opposite; this physical sign is the
+    # one consistent with that statement and with momentum conservation.
+    source_u = cell_beta * (solid_u - state.u)
+    source_v = cell_beta * (solid_v - state.v)
+    return cell_beta, solid_u, solid_v, source_u, source_v
+
+
+def _lust_face_values(field, mass_flux, spacing, axis, lower_boundary, upper_boundary):
+    """Interpolate a transported field with OpenFOAM's 75/25 LUST blend."""
+
+    shape = list(field.shape)
+    shape[axis] += 1
+    faces = np.empty(shape, dtype=float)
+    gradient = np.gradient(field, spacing, axis=axis)
+
+    lower_cell = [slice(None)] * field.ndim
+    upper_cell = [slice(None)] * field.ndim
+    lower_cell[axis] = slice(0, -1)
+    upper_cell[axis] = slice(1, None)
+    internal_face = [slice(None)] * field.ndim
+    internal_face[axis] = slice(1, -1)
+
+    left = field[tuple(lower_cell)]
+    right = field[tuple(upper_cell)]
+    linear = 0.5 * (left + right)
+    linear_upwind = np.where(
+        mass_flux[tuple(internal_face)] >= 0,
+        left + 0.5 * spacing * gradient[tuple(lower_cell)],
+        right - 0.5 * spacing * gradient[tuple(upper_cell)],
+    )
+    faces[tuple(internal_face)] = 0.75 * linear + 0.25 * linear_upwind
+
+    lower_face = [slice(None)] * field.ndim
+    upper_face = [slice(None)] * field.ndim
+    lower_face[axis] = 0
+    upper_face[axis] = -1
+    faces[tuple(lower_face)] = lower_boundary
+    faces[tuple(upper_face)] = upper_boundary
+    return faces
+
+
+def _face_average(field, axis):
+    """Linear face interpolation with zero-gradient boundary extrapolation."""
+
+    shape = list(field.shape)
+    shape[axis] += 1
+    faces = np.empty(shape, dtype=float)
+    lower_cell = [slice(None)] * field.ndim
+    upper_cell = [slice(None)] * field.ndim
+    lower_cell[axis] = slice(0, -1)
+    upper_cell[axis] = slice(1, None)
+    internal_face = [slice(None)] * field.ndim
+    internal_face[axis] = slice(1, -1)
+    faces[tuple(internal_face)] = 0.5 * (field[tuple(lower_cell)] + field[tuple(upper_cell)])
+    lower_face = [slice(None)] * field.ndim
+    upper_face = [slice(None)] * field.ndim
+    lower_source = [slice(None)] * field.ndim
+    upper_source = [slice(None)] * field.ndim
+    lower_face[axis] = 0
+    upper_face[axis] = -1
+    lower_source[axis] = 0
+    upper_source[axis] = -1
+    faces[tuple(lower_face)] = field[tuple(lower_source)]
+    faces[tuple(upper_face)] = field[tuple(upper_source)]
+    return faces
+
+
+def _conservative_transport(state, void_fraction, p):
+    """Advance Eq. (19) without pressure using the paper's FVM operators.
+
+    The convective face value uses the LUST 75/25 linear/linear-upwind blend;
+    gradients and viscous stresses use linear interpolation on the orthogonal
+    grid.  Interphase drag is treated with backward Euler below, matching the
+    paper's first-order implicit temporal discretisation.
+    """
+
+    inlet = getattr(p, "fluid_inlet_velocity", 0.0)
+    qx, qy = _face_fluxes(state.u, state.v, void_fraction, inlet)
+    intrinsic_inlet = inlet / np.maximum(void_fraction[:, 0], 1e-30)
+
+    u_xface = _lust_face_values(state.u, qx, p.dx, axis=0, lower_boundary=0.0, upper_boundary=0.0)
+    u_yface = _lust_face_values(
+        state.u,
+        qy,
+        p.dy,
+        axis=1,
+        lower_boundary=0.0,
+        upper_boundary=state.u[:, -1],
+    )
+    v_xface = _lust_face_values(state.v, qx, p.dx, axis=0, lower_boundary=0.0, upper_boundary=0.0)
+    v_yface = _lust_face_values(
+        state.v,
+        qy,
+        p.dy,
+        axis=1,
+        lower_boundary=intrinsic_inlet,
+        upper_boundary=state.v[:, -1],
+    )
+    convection_u = _flux_divergence(qx * u_xface, qy * u_yface, p.dx, p.dy)
+    convection_v = _flux_divergence(qx * v_xface, qy * v_yface, p.dx, p.dy)
+
+    du_dx = np.gradient(state.u, p.dx, axis=0)
+    du_dy = np.gradient(state.u, p.dy, axis=1)
+    dv_dx = np.gradient(state.v, p.dx, axis=0)
+    dv_dy = np.gradient(state.v, p.dy, axis=1)
+    viscosity = p.gas_viscosity / p.gas_density
+    tau_xx = void_fraction * 2.0 * viscosity * du_dx
+    tau_xy = void_fraction * viscosity * (du_dy + dv_dx)
+    tau_yy = void_fraction * 2.0 * viscosity * dv_dy
+    viscous_u = _flux_divergence(_face_average(tau_xx, axis=0), _face_average(tau_xy, axis=1), p.dx, p.dy)
+    viscous_v = _flux_divergence(_face_average(tau_xy, axis=0), _face_average(tau_yy, axis=1), p.dx, p.dy)
+
+    old_void_fraction = state.void_fraction
+    momentum_u = old_void_fraction * state.u + p.dt * (-convection_u + viscous_u)
+    momentum_v = old_void_fraction * state.v + p.dt * (-convection_v + viscous_v)
+    return momentum_u, momentum_v
+
+
+def _face_fluxes(u, v, void_fraction, inlet_velocity=0.0):
+    """Interpolate superficial velocity to finite-volume cell faces."""
+
+    nx, ny = void_fraction.shape
+    qx = np.zeros((nx + 1, ny))
+    qy = np.zeros((nx, ny + 1))
+    qx[1:nx, :] = 0.5 * (void_fraction[:-1, :] * u[:-1, :] + void_fraction[1:, :] * u[1:, :])
+    qy[:, 1:ny] = 0.5 * (void_fraction[:, :-1] * v[:, :-1] + void_fraction[:, 1:] * v[:, 1:])
+    qy[:, 0] = inlet_velocity
+    qy[:, -1] = void_fraction[:, -1] * v[:, -1]
+    return qx, qy
+
+
+def _flux_divergence(qx, qy, dx, dy):
+    return (qx[1:, :] - qx[:-1, :]) / dx + (qy[:, 1:] - qy[:, :-1]) / dy
+
+
+def weighted_divergence(u, v, void_fraction, dx, dy, inlet_velocity=0.0):
+    """Calculate finite-volume ``div(n u_f)`` at cell centres."""
+
+    qx, qy = _face_fluxes(u, v, void_fraction, inlet_velocity)
+    return _flux_divergence(qx, qy, dx, dy)
+
+
+def _pressure_matrix(void_fraction, dx, dy):
+    """Build ``-div(n grad(.))`` with wall Neumann and open-top Dirichlet BCs."""
+
+    nx, ny = void_fraction.shape
+    matrix = sparse.lil_matrix((nx * ny, nx * ny))
+
+    def flat(i, j):
+        return i * ny + j
+
+    for i in range(nx):
+        for j in range(ny):
+            row = flat(i, j)
+            diagonal = 0.0
+            for di, dj, spacing in ((-1, 0, dx), (1, 0, dx), (0, -1, dy), (0, 1, dy)):
+                ii, jj = i + di, j + dj
+                if ii < 0 or ii >= nx or jj < 0:
+                    continue
+                if jj >= ny:
+                    continue
+                face_void = 0.5 * (void_fraction[i, j] + void_fraction[ii, jj])
+                coefficient = face_void / spacing**2
+                diagonal += coefficient
+                matrix[row, flat(ii, jj)] = -coefficient
+            if j == ny - 1:
+                # Fixed pressure at the open top face, half a cell away.
+                diagonal += 2.0 * void_fraction[i, j] / dy**2
+            matrix[row, row] = diagonal
+    return matrix.tocsr()
+
+
+def project_velocity(u, v, pressure, void_fraction, p):
+    """Project a tentative velocity onto ``div(n u_f) = 0``."""
+
+    del pressure
+    inlet = getattr(p, "fluid_inlet_velocity", 0.0)
+    qx, qy = _face_fluxes(u, v, void_fraction, inlet)
+    divergence = _flux_divergence(qx, qy, p.dx, p.dy)
+    rhs = -(p.gas_density / p.dt) * divergence.ravel()
+    correction = spsolve(_pressure_matrix(void_fraction, p.dx, p.dy), rhs).reshape(void_fraction.shape)
+
+    nx, ny = void_fraction.shape
+    void_x = 0.5 * (void_fraction[:-1, :] + void_fraction[1:, :])
+    void_y = 0.5 * (void_fraction[:, :-1] + void_fraction[:, 1:])
+    qx[1:nx, :] -= p.dt / p.gas_density * void_x * (correction[1:, :] - correction[:-1, :]) / p.dx
+    qy[:, 1:ny] -= p.dt / p.gas_density * void_y * (correction[:, 1:] - correction[:, :-1]) / p.dy
+    qy[:, -1] -= p.dt / p.gas_density * void_fraction[:, -1] * (0.0 - correction[:, -1]) / (0.5 * p.dy)
+
+    # Reconstruct intrinsic cell-centred velocities for HGD coupling.
+    u = 0.5 * (qx[:-1, :] + qx[1:, :]) / void_fraction
+    v = 0.5 * (qy[:, :-1] + qy[:, 1:]) / void_fraction
+    # The tentative step does not include an old pressure gradient, so this is
+    # the new pressure rather than an incremental pressure correction.
+    pressure = correction
+    return u, v, pressure, _flux_divergence(qx, qy, p.dx, p.dy)
+
+
+def minimum_fluidization_velocity(p, solid_fraction=None, particle_size=None):
+    """Estimate ``U_mf`` from the same dense Gidaspow/Ergun force balance."""
+
+    phi = p.nu_fill if solid_fraction is None else solid_fraction
+    diameter = p.s_m if particle_size is None else particle_size
+    eps = max(1.0 - phi, getattr(p, "fluid_min_void_fraction", 0.05))
+    weight = phi * (p.solid_density - p.gas_density) * p.g
+    linear = 150.0 * p.gas_viscosity * phi**2 / (eps**2 * diameter**2)
+    quadratic = 1.75 * p.gas_density * phi / (eps**2 * diameter)
+    if quadratic == 0:
+        return weight / linear
+    return (-linear + np.sqrt(linear**2 + 4.0 * quadratic * weight)) / (2.0 * quadratic)
+
+
+def _diagnostics(state, solid_fraction, p):
+    drag_on_solid_y = state.beta * (state.v - state.solid_v)
+    effective_weight = solid_fraction * (p.solid_density - p.gas_density) * p.g
+    occupied = solid_fraction > 0
+    state.fluidized = occupied & (drag_on_solid_y >= effective_weight)
+    state.pressure_drop = float(np.mean(state.pressure[:, 0]) - np.mean(state.pressure[:, -1]))
+    bed_weight = float(np.mean(np.sum(effective_weight, axis=1) * p.dy))
+    state.fluidization_ratio = state.pressure_drop / bed_weight if bed_weight > 0 else 0.0
+
+
+def advance(state, s, particle_u, particle_v, p):
+    """Advance the fluid one staggered explicit HGD--CFD coupling step."""
+
+    solid_fraction = 1.0 - np.mean(np.isnan(s), axis=2)
+    void_fraction = _void_fraction(solid_fraction, p)
+    beta, solid_u, solid_v, source_u, source_v = coupling_fields(s, particle_u, particle_v, state, p)
+
+    momentum_u, momentum_v = _conservative_transport(state, void_fraction, p)
+
+    # First-order implicit Euler treatment of the interphase source in
+    # Eq. (19): n U^{n+1} + dt beta/rho U^{n+1}
+    #              = m* + dt beta/rho U_s.
+    drag_weight = p.dt * beta / p.gas_density
+    denominator = void_fraction + drag_weight
+    tentative_u = (momentum_u + drag_weight * solid_u) / denominator
+    tentative_v = (momentum_v + drag_weight * solid_v) / denominator
+
+    state.u, state.v, state.pressure, state.divergence = project_velocity(
+        tentative_u, tentative_v, state.pressure, void_fraction, p
+    )
+    state.beta = beta
+    state.solid_u = solid_u
+    state.solid_v = solid_v
+    state.momentum_source_u = source_u
+    state.momentum_source_v = source_v
+    state.void_fraction = void_fraction.copy()
+    _diagnostics(state, solid_fraction, p)
+    courant = np.max(np.abs(state.u) * p.dt / p.dx + np.abs(state.v) * p.dt / p.dy)
+    p.fluid_courant = float(courant)
+    if not np.all(np.isfinite(state.u)) or not np.all(np.isfinite(state.v)):
+        raise FloatingPointError("Non-finite fluid velocity produced by the HGFD solver")
+    if courant > getattr(p, "fluid_cfl_limit", 0.8):
+        raise FloatingPointError(
+            f"Fluid CFL={courant:.3g} exceeds fluid_cfl_limit; reduce defined_time_step_size"
+        )
+    return state
diff --git a/HGD/initial.py b/HGD/initial.py
index 9202d78..c504a46 100644
--- a/HGD/initial.py
+++ b/HGD/initial.py
@@ -11,7 +11,7 @@ def IC(p):
     Returns:
         The array of grain sizes. Values of `NaN` are voids.
     """
-    rng = np.random.default_rng()
+    rng = np.random.default_rng(getattr(p, "random_seed", None))
 
     # First step: Generate BOTH the grain size distribution and the void distribution
 
@@ -137,6 +137,15 @@ def IC(p):
     if p.IC_mode == "random":  # voids everywhere randomly, this has been done above in the first step
         # mask = np.random.rand(p.nx, p.ny, p.nm) > p.nu_fill
         return s
+    elif p.IC_mode == "single_particle":
+        # Paper Case 1: one solid-occupied internal coordinate released from
+        # rest in an otherwise empty, fluid-filled domain.
+        s[:] = np.nan
+        i = getattr(p, "single_particle_i", p.nx // 2)
+        j = getattr(p, "single_particle_j", 3 * p.ny // 4)
+        k = getattr(p, "single_particle_k", 0)
+        s[i, j, k] = p.s_m
+        return s
     elif p.IC_mode == "top":  # material at the top
         mask = np.zeros([p.nx, p.ny, p.nm], dtype=bool)
         mask[:, : int((1 - p.fill_ratio) * p.ny), :] = True
diff --git a/HGD/main.py b/HGD/main.py
index 1dd39e3..e06294e 100644
--- a/HGD/main.py
+++ b/HGD/main.py
@@ -14,11 +14,15 @@
 from HGD import boundary
 from HGD import mask
 from HGD import operators
+from HGD import fluid
 
 # from HGD import cycles
 
 
 def init(p, cycles=None):
+    if p.random_seed is not None:
+        np.random.seed(p.random_seed)
+
     p.move_voids = importlib.import_module(f"HGD.motion.{p.motion_model}").move_voids
 
     if hasattr(importlib.import_module(f"HGD.motion.{p.motion_model}"), "stream"):
@@ -56,6 +60,18 @@ def init(p, cycles=None):
     # initial.set_boundary(s, p.X, p.Y, p)
     p = mask.update(p, s)
 
+    if p.fluid_coupling:
+        if not p.inertia:
+            raise ValueError("fluid_coupling requires inertia=true")
+        if p.motion_model != "hgfd":
+            raise ValueError("fluid_coupling requires motion_model='hgfd'")
+        p.fluid_state = fluid.initialize(p, s)
+        p.minimum_fluidization_velocity = fluid.minimum_fluidization_velocity(p)
+        print(
+            f"Estimated minimum fluidization velocity: {p.minimum_fluidization_velocity:.4g} m/s; "
+            f"inlet: {p.fluid_inlet_velocity:.4g} m/s"
+        )
+
     if hasattr(p, "temperature"):
         T = p.temperature["inlet_temperature"] * np.ones_like(s)
         T[np.isnan(s)] = np.nan
@@ -174,6 +190,9 @@ def time_step(p, state):
         u = u_new
         v = v_new
 
+    if p.fluid_coupling:
+        p.fluid_state = fluid.advance(p.fluid_state, s, u, v, p)
+
     state = (
         s,
         u,
diff --git a/HGD/motion/hgfd.py b/HGD/motion/hgfd.py
new file mode 100644
index 0000000..e0d1294
--- /dev/null
+++ b/HGD/motion/hgfd.py
@@ -0,0 +1,155 @@
+"""Inertial, four-direction HGD transport for two-way fluid coupling."""
+
+import numpy as np
+
+from HGD import fluid
+
+
+def concentration_dependent_alpha(solid_fraction, p):
+    """HGFD mixing coefficient that vanishes when dilute and saturates when dense."""
+
+    alpha_max = getattr(p, "fluid_alpha_max", p.alpha)
+    epsilon = getattr(p, "fluid_alpha_epsilon", 0.01)
+    phi_c = float(np.min(p.nu_cs)) if isinstance(p.nu_cs, np.ndarray) else p.nu_cs
+    phi_ref = phi_c - epsilon
+    clipped = np.minimum(solid_fraction, phi_ref)
+    numerator = np.maximum(phi_c - clipped, epsilon) ** -0.5 - phi_c**-0.5
+    denominator = epsilon**-0.5 - phi_c**-0.5
+    return np.clip(alpha_max * numerator / denominator, 0.0, alpha_max)
+
+
+def _destination_masks(s, solid_fraction, boundary, axis, step, p):
+    destination = np.roll(s, -step, axis=axis)
+    destination_void = np.isnan(destination)
+    destination_fraction = np.roll(solid_fraction, -step, axis=axis)[:, :, np.newaxis]
+    destination_boundary = np.roll(boundary, -step, axis=axis)[:, :, np.newaxis]
+    blocked = (destination_fraction >= p.nu_cs) | destination_boundary
+
+    edge = [slice(None)] * 3
+    edge[axis] = -1 if step > 0 else 0
+    blocked[tuple(edge)] = True
+    valid = destination_void & ~blocked
+    # An event exists either when the matching destination layer is void or
+    # when a wall/critically packed cell rejects the attempted event.  A
+    # matching occupied layer in an under-packed cell simply offers no swap.
+    candidate = destination_void | blocked
+    return candidate, valid
+
+
+def _conflict_free(sources, destinations):
+    if len(sources) == 0:
+        return sources, destinations
+    order = np.random.permutation(len(sources))
+    sources = sources[order]
+    destinations = destinations[order]
+    _, first = np.unique(destinations, axis=0, return_index=True)
+    keep = np.sort(first)
+    return sources[keep], destinations[keep]
+
+
+def move_voids(u, v, s, p, diag=0, c=None, T=None, chi=None, last_swap=None):
+    """Advance particles using force-derived velocities and stochastic swaps."""
+
+    del diag
+    if not getattr(p, "fluid_coupling", False):
+        raise ValueError("The 'hgfd' motion model requires fluid_coupling=true")
+    if last_swap is None:
+        last_swap = np.zeros_like(s)
+    if not hasattr(p, "hgfd_virtual_displacement") or p.hgfd_virtual_displacement.shape != s.shape:
+        p.hgfd_virtual_displacement = np.zeros_like(s)
+
+    u, v = fluid.update_particle_velocities(u, v, s, p.fluid_state, p)
+    occupied = np.isfinite(s)
+    virtual_displacement = p.hgfd_virtual_displacement
+    virtual_displacement[occupied] += np.hypot(u[occupied], v[occupied]) * p.dt
+    virtual_displacement[~occupied] = 0.0
+    solid_fraction = np.mean(occupied, axis=2)
+    boundary = getattr(p, "boundary_mask", np.zeros((p.nx, p.ny), dtype=bool))
+    alpha = concentration_dependent_alpha(solid_fraction, p)[:, :, np.newaxis]
+    diffusivity = alpha * np.nan_to_num(s) * np.abs(v)
+    horizontal_diffusion = diffusivity * p.dt / p.dx**2
+
+    probabilities = []
+    valid_destinations = []
+    directions = ((0, 1), (0, -1), (1, 1), (1, -1))
+    for axis, step in directions:
+        velocity = u if axis == 0 else v
+        directional = np.maximum(step * velocity, 0.0) * p.dt / (p.dx if axis == 0 else p.dy)
+        if axis == 0:
+            directional = directional + horizontal_diffusion
+        candidate, valid = _destination_masks(s, solid_fraction, boundary, axis, step, p)
+        probabilities.append(np.where(occupied & candidate, directional, 0.0))
+        valid_destinations.append(valid)
+
+    total = np.sum(probabilities, axis=0)
+    max_probability = float(np.max(total)) if total.size else 0.0
+    p.max_transition_probability = max_probability
+    limit = getattr(p, "P_stab", 0.5)
+    if max_probability >= limit:
+        policy = getattr(p, "fluid_probability_policy", "error")
+        if policy == "scale":
+            # Scaling is retained as an explicitly non-paper fallback.  Paper
+            # reproduction configurations use ``error`` and select dt so that
+            # the strict P_tot < 0.5 condition is satisfied.
+            scale = np.minimum(1.0, np.nextafter(limit, 0.0) / np.maximum(total, 1e-30))
+            probabilities = [probability * scale for probability in probabilities]
+        else:
+            raise ValueError(
+                f"HGFD transition probability {max_probability:.3g} is not below P_stab={limit}; "
+                "reduce defined_time_step_size or set fluid_probability_policy='scale'"
+            )
+
+    draw = np.random.random(s.shape)
+    cumulative = np.zeros_like(s, dtype=float)
+    source_batches = []
+    destination_batches = []
+    for probability, valid, (axis, step) in zip(probabilities, valid_destinations, directions):
+        selected = (draw >= cumulative) & (draw < cumulative + probability)
+        rejected = selected & ~valid
+        u[rejected] = 0.0
+        v[rejected] = 0.0
+        sources = np.argwhere(selected & valid)
+        if len(sources):
+            destinations = sources.copy()
+            destinations[:, axis] += step
+            source_batches.append(sources)
+            destination_batches.append(destinations)
+        cumulative += probability
+
+    if source_batches:
+        sources = np.concatenate(source_batches)
+        destinations = np.concatenate(destination_batches)
+        sources, destinations = _conflict_free(sources, destinations)
+
+        # Recreate direction markers after conflict filtering from displacement.
+        displacement = destinations - sources
+        swap_type = np.where(displacement[:, 1] != 0, 1, -1)
+        source_index = tuple(sources.T)
+        destination_index = tuple(destinations.T)
+        source_displacement = virtual_displacement[source_index].copy()
+        for array in (s, u, v, c, T):
+            if array is not None:
+                array[source_index], array[destination_index] = (
+                    array[destination_index].copy(),
+                    array[source_index].copy(),
+                )
+
+        # Eq. (26): carry any cell-traversal deficit with the solid element
+        # and subtract exactly one lattice spacing after a realised swap.
+        virtual_displacement[source_index] = 0.0
+        virtual_displacement[destination_index] = np.maximum(source_displacement - p.dx, 0.0)
+
+        last_swap[source_index] = np.nan
+        last_swap[destination_index] = swap_type
+        moved = np.zeros((p.nx, p.ny), dtype=float)
+        np.add.at(moved, (sources[:, 0], sources[:, 1]), 1)
+        np.add.at(moved, (destinations[:, 0], destinations[:, 1]), 1)
+        chi = moved / (p.nm * limit)
+    else:
+        chi = np.zeros((p.nx, p.ny), dtype=float)
+
+    u[np.isnan(s)] = 0.0
+    v[np.isnan(s)] = 0.0
+    virtual_displacement[np.isnan(s)] = 0.0
+    last_swap[np.isnan(s)] = np.nan
+    return u, v, s, c, T, chi, last_swap
diff --git a/HGD/params.py b/HGD/params.py
index 86abbd5..e2f9a35 100644
--- a/HGD/params.py
+++ b/HGD/params.py
@@ -128,12 +128,25 @@ def set_defaults(self):
             self.n_cycles = len(self.cycles)
 
     def update_before_time_march(self, cycles):
-        self.y = np.linspace(0, self.H, self.ny)
-        self.dy = self.y[1] - self.y[0]
-        self.x = np.arange(
-            -(self.nx - 0.5) / 2 * self.dy, (self.nx - 0.5) / 2 * self.dy, self.dy
-        )  # force equal grid spacing
-        self.dx = self.x[1] - self.x[0]
+        if hasattr(self, "cell_size"):
+            # The HGFD paper specifies cell-centred meshes by cell size and
+            # cell count.  Keep this opt-in so existing HGD configurations
+            # retain their historical endpoint-based grid convention.
+            self.dx = self.dy = float(self.cell_size)
+            self.x = (np.arange(self.nx) - (self.nx - 1) / 2.0) * self.dx
+            self.y = np.arange(self.ny, dtype=float) * self.dy
+            expected_height = self.ny * self.dy
+            if not np.isclose(self.H, expected_height):
+                raise ValueError(
+                    f"H={self.H} is inconsistent with ny*cell_size={expected_height} for a cell-centred grid"
+                )
+        else:
+            self.y = np.linspace(0, self.H, self.ny)
+            self.dy = self.y[1] - self.y[0]
+            self.x = np.arange(
+                -(self.nx - 0.5) / 2 * self.dy, (self.nx - 0.5) / 2 * self.dy, self.dy
+            )  # force equal grid spacing
+            self.dx = self.x[1] - self.x[0]
         if not np.isclose(self.dx, self.dy):
             sys.exit(f"Fatal error: dx != dy. dx = {self.dx}, dy = {self.dy}")
         self.W = self.nx * self.dx
diff --git a/HGD/plotter.py b/HGD/plotter.py
index 0ea16ef..274f9d5 100644
--- a/HGD/plotter.py
+++ b/HGD/plotter.py
@@ -96,12 +96,13 @@ def size_colormap():
     return orange_blue_cmap
 
 
-global fig, summary_fig, triple_fig, quad_fig, blue_pink_fig
+global fig, summary_fig, triple_fig, quad_fig, blue_pink_fig, fluid_fig
 fig = plt.figure(1)
 summary_fig = plt.figure(2)
 triple_fig = plt.figure(3)
 quad_fig = plt.figure(4)
 blue_pink_fig = plt.figure(5)
+fluid_fig = plt.figure(6)
 
 replacements = {
     "repose_angle": "φ",
@@ -165,7 +166,7 @@ def replace_strings(text, replacements):
 
 
 def set_plot_size(p):
-    global fig, summary_fig, triple_fig, quad_fig, blue_pink_fig
+    global fig, summary_fig, triple_fig, quad_fig, blue_pink_fig, fluid_fig
 
     # wipe any existing figures
     for i in plt.get_fignums():
@@ -176,6 +177,7 @@ def set_plot_size(p):
     quad_fig = plt.figure(3, figsize=[p.nx / _dpi, 4 * p.ny / _dpi])
     summary_fig = plt.figure(4)
     blue_pink_fig = plt.figure(5, facecolor="k", figsize=[p.nx / _dpi, p.ny / _dpi])
+    fluid_fig = plt.figure(6, figsize=[3 * p.nx / _dpi, p.ny / _dpi])
 
 
 def check_folders_exist(p):
@@ -311,6 +313,12 @@ def update(p, state, *args):
             plot_deviatoric(s, sigma, last_swap, p)
         if "footing" in p.plot:
             plot_footing(p)
+        if "fluid_velocity" in p.plot:
+            plot_fluid_velocity(p)
+        if "fluid_pressure" in p.plot:
+            plot_fluid_pressure(p)
+        if "fluidized_bed" in p.plot:
+            plot_fluidized_bed(s, p)
 
         if "s" in p.save:
             save_s(s, p)
@@ -343,12 +351,108 @@ def update(p, state, *args):
             save_velocity(u, v, p)
         if "charge_discharge" in p.save:
             c_d_saves(p, non_zero_nu_time, p_count, p_count_s, p_count_l)
+        if "fluid" in p.save:
+            save_fluid(p)
         if "col_depth" in p.save:
             get_col_depth(p, s)
         if "surface_profiles" in p.save:
             np.save(p.folderName + "data/surface_profiles.npy", surface_profile)
 
 
+def _fluid_field(p, name):
+    if not getattr(p, "fluid_coupling", False) or not hasattr(p, "fluid_state"):
+        raise ValueError(f"'{name}' plotting requires fluid_coupling=true")
+    return p.fluid_state
+
+
+def _plot_domain(field, p, cmap, vmin=None, vmax=None):
+    image = plt.pcolormesh(p.x, p.y, field.T, cmap=cmap, vmin=vmin, vmax=vmax)
+    plt.axis("off")
+    plt.xlim(p.x[0], p.x[-1])
+    plt.ylim(p.y[0], p.y[-1])
+    return image
+
+
+def plot_fluid_velocity(p):
+    state = _fluid_field(p, "fluid_velocity")
+    speed = np.sqrt(state.u**2 + state.v**2)
+    plt.figure(fig)
+    plt.clf()
+    image = _plot_domain(speed, p, inferno, vmin=0)
+    if p.plot_colorbar:
+        plt.colorbar(image, shrink=0.8, location="top", pad=0.01)
+    plt.subplots_adjust(left=0, right=1, bottom=0, top=1)
+    plt.savefig(p.folderName + "fluid_velocity_" + str(p.tstep).zfill(6) + ".png")
+
+
+def plot_fluid_pressure(p):
+    state = _fluid_field(p, "fluid_pressure")
+    plt.figure(fig)
+    plt.clf()
+    image = _plot_domain(state.pressure, p, inferno)
+    if p.plot_colorbar:
+        plt.colorbar(image, shrink=0.8, location="top", pad=0.01)
+    plt.subplots_adjust(left=0, right=1, bottom=0, top=1)
+    plt.savefig(p.folderName + "fluid_pressure_" + str(p.tstep).zfill(6) + ".png")
+
+
+def plot_fluidized_bed(s, p):
+    """Plot solid fraction, vertical gas velocity, and gas pressure in the established style."""
+
+    state = _fluid_field(p, "fluidized_bed")
+    solid_fraction = operators.get_solid_fraction(s)
+    velocity_limit = max(float(np.max(np.abs(state.v))), 1e-12)
+    plt.figure(fluid_fig)
+    plt.clf()
+
+    panels = (
+        (solid_fraction, inferno_r, 0, p.nu_cs, r"Solid fraction $\nu$"),
+        (state.v, bwr, -velocity_limit, velocity_limit, r"Gas velocity $v_f$"),
+        (state.pressure, inferno, None, None, r"Gas pressure $p$"),
+    )
+    for index, (field, colormap, vmin, vmax, title) in enumerate(panels, start=1):
+        plt.subplot(1, 3, index)
+        image = _plot_domain(field, p, colormap, vmin=vmin, vmax=vmax)
+        plt.title(title, y=-0.10)
+        if p.plot_colorbar:
+            plt.colorbar(image, shrink=0.72, location="top", pad=0.05)
+
+    plt.subplots_adjust(left=0.01, right=0.99, bottom=0.10, top=0.98, wspace=0.04)
+    plt.savefig(p.folderName + "fluidized_bed_" + str(p.tstep).zfill(6) + ".png", dpi=120)
+    save_fluid_diagnostics(p)
+
+
+def save_fluid_diagnostics(p):
+    state = _fluid_field(p, "fluidized_bed")
+    if getattr(p, "_last_fluid_diagnostic_step", None) == p.tstep:
+        return
+    path = p.folderName + "fluid_diagnostics.csv"
+    mode = "w" if p.tstep == 0 else "a"
+    with open(path, mode) as output:
+        if mode == "w":
+            output.write(
+                "time,pressure_drop,fluidization_ratio,fluidized_fraction,max_divergence,"
+                "max_transition_probability\n"
+            )
+        output.write(
+            f"{p.t},{state.pressure_drop},{state.fluidization_ratio},{np.mean(state.fluidized)},"
+            f"{np.max(np.abs(state.divergence))},{getattr(p, 'max_transition_probability', 0.0)}\n"
+        )
+    p._last_fluid_diagnostic_step = p.tstep
+
+
+def save_fluid(p):
+    state = _fluid_field(p, "fluid")
+    np.savez(
+        p.folderName + "data/fluid_" + str(p.tstep).zfill(6) + ".npz",
+        u=state.u,
+        v=state.v,
+        pressure=state.pressure,
+        beta=state.beta,
+        fluidized=state.fluidized,
+    )
+
+
 def plot_u_time(y, U, nu_time, p):
     plt.figure(summary_fig)
 
@@ -1031,7 +1135,7 @@ def plot_h(s, p):
 
 
 def make_video(p):
-    if is_ffmpeg_installed:
+    if is_ffmpeg_installed():
         fname = p.folderName.split("/")[-2]
         nice_name = "=".join(fname.rsplit("_", 1))
         nice_name = replace_strings(nice_name, replacements)
@@ -1069,7 +1173,7 @@ def make_video(p):
 def stack_videos(paths, name, p):
     videos = p.videos
     heights = p.ny
-    if is_ffmpeg_installed:
+    if is_ffmpeg_installed():
         if isinstance(heights, int):
             heights = [heights] * len(videos)
         max_height = 4 * max(heights)  # Find the maximum height among all videos
diff --git a/docs/examples.md b/docs/examples.md
index ddf422b..0d74306 100644
--- a/docs/examples.md
+++ b/docs/examples.md
@@ -20,6 +20,7 @@ This document describes the example simulation configurations included with HGD.
 | `test_slope.json5` | ⭐ Easy | ~1 min | Slope stability test |
 | `footing.json5` | ⭐⭐ Medium | ~3 min | Load bearing capacity of granular material |
 | `collapse_bi.json5` | ⭐⭐ Medium | ~5 min | Bidisperse collapse with segregation |
+| `fluidized_bed.json5` | ⭐⭐ Medium | ~1 min | Two-way coupled laminar gas fluidization |
 | `hopper_emptying.json5` | ⭐⭐⭐ Advanced | ~10 min | Detailed hopper flow analysis |
 | `temperature.json5` | ⭐⭐⭐ Advanced | ~5 min | Thermal effects in granular flow |
 
@@ -207,9 +208,46 @@ python HGD/main.py json/collapse_inertia.json5
 
 ---
 
+### 7. Laminar Gas Fluidized Bed (`fluidized_bed.json5`)
+
+**What it simulates:** A monodisperse bed driven by a prescribed superficial gas velocity through the bottom boundary.
+
+```bash
+python HGD/main.py json/fluidized_bed.json5
+```
+
+The `hgfd` motion model adds layer-wise particle inertia, Gidaspow drag (Wen–Yu in dilute cells and Ergun in dense cells), and equal-and-opposite momentum feedback to a two-dimensional fluid-fraction-weighted incompressible solver. The pressure projection enforces `div(n * u_f) = 0` on the co-located HGD grid.
+
+![Official-gallery-style HGFD fluidized-bed fields](visualizations/hgfd_fluidized_bed/render/preview.gif)
+
+The example writes three-panel `fluidized_bed_*.png` frames and `fluid_diagnostics.csv`. The latter reports pressure drop, pressure-drop/effective-bed-weight ratio, locally fluidized fraction, maximum continuity residual, and maximum stochastic transition probability.
+
+The complete visualization pack also provides an
+[MP4](visualizations/hgfd_fluidized_bed/render/demo.mp4) and a 31-frame
+[ParaView PVD time series](visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed.pvd).
+Each VTI frame contains both particle and fluid velocity vectors, solid
+fraction, grain size, fluid pressure, momentum-exchange coefficient, and the
+fluidized-cell indicator.
+
+Important parameters:
+
+- `fluid_inlet_velocity`: bottom superficial velocity in m/s
+- `gas_density`, `gas_viscosity`: fluid properties
+- `defined_time_step_size`: shared HGD/fluid time step
+- `fluid_probability_policy`: reject or safely scale an excessive stochastic transition probability
+- `fluid_cfl_limit`: fail-fast limit for the fluid Courant number
+
+This solver is intended for two-dimensional, isothermal, laminar research cases. It has not been validated for bubbling, slugging, turbulent, reacting, or industrial-scale fluidized beds.
+
+The coupling equations follow [Li et al., *Coupling Heterarchical Granular Dynamics and Computational Fluid Dynamics*](https://arxiv.org/abs/2606.23202).
+
+For the three quantitative single-particle validation runs, implementation
+parity, and the limits imposed by unpublished inputs, see the
+[HGFD paper reproduction](hgfd-reproduction.md).
+
 ## Advanced Examples
 
-### 7. Hopper Emptying Studies (`hopper_emptying*.json5`)
+### 8. Hopper Emptying Studies (`hopper_emptying*.json5`)
 
 Several detailed hopper studies with different parameters:
 
@@ -240,7 +278,7 @@ python HGD/main.py json/hopper_emptying.json5
 
 ---
 
-### 8. Temperature Effects (`temperature.json5`)
+### 9. Temperature Effects (`temperature.json5`)
 
 **What it simulates:** Thermal transport in flowing granular material.
 
@@ -266,7 +304,7 @@ python HGD/main.py json/temperature.json5
 
 ---
 
-### 9. Stress Field Test (`test_stress.json5`)
+### 10. Stress Field Test (`test_stress.json5`)
 
 **What it simulates:** Stress distribution calculation in static granular material.
 
@@ -287,7 +325,7 @@ python HGD/main.py json/test_stress.json5
 
 ---
 
-### 10. Mesh Dependency Study (`mesh_dependency.json5`)
+### 11. Mesh Dependency Study (`mesh_dependency.json5`)
 
 **What it simulates:** Same problem with different grid resolutions.
 
diff --git a/docs/hgfd-reproduction.md b/docs/hgfd-reproduction.md
new file mode 100644
index 0000000..a245e43
--- /dev/null
+++ b/docs/hgfd-reproduction.md
@@ -0,0 +1,117 @@
+# HGFD paper reproduction
+
+This repository contains a best-effort reproduction of Li et al.,
+[*Coupling Heterarchical Granular Dynamics and Computational Fluid Dynamics*](https://arxiv.org/abs/2606.23202)
+(2026). It implements the published HGFD equations and supplies three runnable
+configurations for the paper's single-particle settling validation.
+
+## Reproduced Case 1 result
+
+Run the three configurations and generate the comparison plot:
+
+```bash
+python HGD/main.py json/hgfd_paper_case1_0p5mm.json5
+python HGD/main.py json/hgfd_paper_case1_1p5mm.json5
+python HGD/main.py json/hgfd_paper_case1_2p0mm.json5
+python scripts/plot_hgfd_case1.py
+```
+
+![Animated reproduction of the HGFD Case 1 settling curves](images/hgfd_case1_reproduction.gif)
+
+A high-resolution [static PNG](images/hgfd_case1_reproduction.png) is also
+provided for reports and presentations.
+
+The terminal velocities from the full two-way-coupled runs are:
+
+| Diameter | Paper HGFD curve | This reproduction | Relative difference |
+| ---: | ---: | ---: | ---: |
+| 0.5 mm | 0.07425 m/s | 0.07452 m/s | +0.36% |
+| 1.5 mm | 0.21144 m/s | 0.21168 m/s | +0.11% |
+| 2.0 mm | 0.26153 m/s | 0.26170 m/s | +0.07% |
+
+The paper's experimental time-series values are not published numerically, so
+the plot contains reproduced HGFD curves rather than digitised experimental
+markers.
+
+## Fluidized-bed field visualization
+
+The fluidized-bed example is also exported with the same asset pipeline used
+by the official HGD example gallery: a ParaView time series plus matching
+poster, GIF, and MP4 renders.
+
+[![HGFD fluidized-bed four-field poster](visualizations/hgfd_fluidized_bed/render/poster.png)](visualizations/hgfd_fluidized_bed/render/preview.gif)
+
+- [Animated GIF](visualizations/hgfd_fluidized_bed/render/preview.gif)
+- [H.264 MP4](visualizations/hgfd_fluidized_bed/render/demo.mp4)
+- [ParaView PVD time series](visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed.pvd)
+
+Open the PVD file in ParaView to inspect all 31 time steps. Available cell
+arrays include solid fraction, mean grain size, particle and fluid velocity
+vectors, particle and fluid speed, vertical fluid velocity, fluid pressure,
+momentum-exchange coefficient, and the fluidized-cell indicator.
+
+The assets can be regenerated after running `fluidized_bed.json5`:
+
+```bash
+python scripts/export_hgfd_to_vti.py \
+  --input output/fluidized_bed/data \
+  --output docs/visualizations/hgfd_fluidized_bed/paraview \
+  --name hgfd_fluidized_bed --t-final 0.06
+
+python scripts/render_hgfd_vti.py \
+  --pvd docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed.pvd \
+  --output docs/visualizations/hgfd_fluidized_bed/render \
+  --title "HGFD · FLUIDIZED BED"
+```
+
+## Parity with the paper
+
+The implementation follows the published model for:
+
+- buoyancy-modified gravity and layer-wise exponential drag relaxation;
+- Gidaspow drag, using Wen--Yu below solid fraction 0.2 and Ergun above it;
+- the `1 / M_s` layer normalisation and sum-to-cell drag aggregation;
+- directional stochastic advection and concentration-dependent diffusion;
+- the strict total-transition-probability constraint `P_tot < 0.5`;
+- the virtual-displacement momentum-transfer budget, including carry-over
+  after a realised swap;
+- one staggered HGD-then-CFD update and one field exchange per time step;
+- fluid-fraction-weighted continuity and momentum transport on a co-located
+  cell-centred grid;
+- Euler time integration, LUST convective interpolation, linear gradients,
+  and uncorrected orthogonal viscous fluxes.
+
+The particle relaxation uses the per-coordinate mass density `rho_p / M`.
+This factor is implicit in the heterarchical layer volume and is required to
+reproduce Figure 4. Applying the printed `tau = rho_p / beta_k` literally,
+without this layer-mass normalisation, does not reproduce the published curve.
+
+The fluid receives `beta * (u_s - u_f)`, equal and opposite to the drag on the
+solid. The paper prints `beta * (u_f - u_s)` as a positive fluid source but
+also states that the force is equal and opposite; those two statements have
+opposite signs. The implementation chooses the momentum-conserving sign.
+
+## What cannot be reproduced one-to-one from public material
+
+The authors' CFD participant is an extended OpenFOAM `pisoFoam` solver coupled
+through preCICE. Its source, preCICE configuration, PISO corrector settings,
+and exact OpenFOAM boundary dictionaries are not included in the paper or the
+public HGD repository. The bundled Python finite-volume projection solver
+matches the published equations and named spatial discretisations. Its
+nonlinear transport is evaluated once before the pressure projection, so it
+is not matrix-, PISO-iteration-, or binary-identical to that unpublished
+participant.
+
+Other public-data limits are:
+
+- Case 1 lists 15 horizontal cells of size 0.05 m but also a 1.1 m domain
+  width. Those values imply 0.75 m and cannot all hold simultaneously. The
+  runnable configurations preserve the published cell count and cell size.
+- Case 2 does not publish the numerical initial particle-size distribution.
+- Cases 3 and 4 refer to external experiments for complete geometry and use
+  calibrated feeding/mixing conditions that are not tabulated numerically.
+
+Consequently, Case 1 is quantitatively reproducible from public inputs, while
+honest one-to-one reproduction of all four cases requires the authors' missing
+solver participant, coupling files, geometries, and raw initial/experimental
+data.
diff --git a/docs/images/fluidized_bed.gif b/docs/images/fluidized_bed.gif
new file mode 100644
index 0000000..7b3df9e
Binary files /dev/null and b/docs/images/fluidized_bed.gif differ
diff --git a/docs/images/fluidized_bed.png b/docs/images/fluidized_bed.png
new file mode 100644
index 0000000..f750b5b
Binary files /dev/null and b/docs/images/fluidized_bed.png differ
diff --git a/docs/images/hgfd_case1_reproduction.gif b/docs/images/hgfd_case1_reproduction.gif
new file mode 100644
index 0000000..6579a00
Binary files /dev/null and b/docs/images/hgfd_case1_reproduction.gif differ
diff --git a/docs/images/hgfd_case1_reproduction.png b/docs/images/hgfd_case1_reproduction.png
new file mode 100644
index 0000000..0f1451f
Binary files /dev/null and b/docs/images/hgfd_case1_reproduction.png differ
diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed.pvd b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed.pvd
new file mode 100644
index 0000000..64ee7a4
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed.pvd
@@ -0,0 +1,36 @@
+<?xml version='1.0' encoding='utf-8'?>
+<VTKFile type="Collection" version="0.1" byte_order="LittleEndian">
+  <Collection>
+    <DataSet timestep="0" group="" part="0" file="hgfd_fluidized_bed_0000.vti" />
+    <DataSet timestep="0.0020033389" group="" part="0" file="hgfd_fluidized_bed_0001.vti" />
+    <DataSet timestep="0.0040066778" group="" part="0" file="hgfd_fluidized_bed_0002.vti" />
+    <DataSet timestep="0.0060100167" group="" part="0" file="hgfd_fluidized_bed_0003.vti" />
+    <DataSet timestep="0.0080133556" group="" part="0" file="hgfd_fluidized_bed_0004.vti" />
+    <DataSet timestep="0.010016694" group="" part="0" file="hgfd_fluidized_bed_0005.vti" />
+    <DataSet timestep="0.012020033" group="" part="0" file="hgfd_fluidized_bed_0006.vti" />
+    <DataSet timestep="0.014023372" group="" part="0" file="hgfd_fluidized_bed_0007.vti" />
+    <DataSet timestep="0.016026711" group="" part="0" file="hgfd_fluidized_bed_0008.vti" />
+    <DataSet timestep="0.01803005" group="" part="0" file="hgfd_fluidized_bed_0009.vti" />
+    <DataSet timestep="0.020033389" group="" part="0" file="hgfd_fluidized_bed_0010.vti" />
+    <DataSet timestep="0.022036728" group="" part="0" file="hgfd_fluidized_bed_0011.vti" />
+    <DataSet timestep="0.024040067" group="" part="0" file="hgfd_fluidized_bed_0012.vti" />
+    <DataSet timestep="0.026043406" group="" part="0" file="hgfd_fluidized_bed_0013.vti" />
+    <DataSet timestep="0.028046745" group="" part="0" file="hgfd_fluidized_bed_0014.vti" />
+    <DataSet timestep="0.030050083" group="" part="0" file="hgfd_fluidized_bed_0015.vti" />
+    <DataSet timestep="0.032053422" group="" part="0" file="hgfd_fluidized_bed_0016.vti" />
+    <DataSet timestep="0.034056761" group="" part="0" file="hgfd_fluidized_bed_0017.vti" />
+    <DataSet timestep="0.0360601" group="" part="0" file="hgfd_fluidized_bed_0018.vti" />
+    <DataSet timestep="0.038063439" group="" part="0" file="hgfd_fluidized_bed_0019.vti" />
+    <DataSet timestep="0.040066778" group="" part="0" file="hgfd_fluidized_bed_0020.vti" />
+    <DataSet timestep="0.042070117" group="" part="0" file="hgfd_fluidized_bed_0021.vti" />
+    <DataSet timestep="0.044073456" group="" part="0" file="hgfd_fluidized_bed_0022.vti" />
+    <DataSet timestep="0.046076795" group="" part="0" file="hgfd_fluidized_bed_0023.vti" />
+    <DataSet timestep="0.048080134" group="" part="0" file="hgfd_fluidized_bed_0024.vti" />
+    <DataSet timestep="0.050083472" group="" part="0" file="hgfd_fluidized_bed_0025.vti" />
+    <DataSet timestep="0.052086811" group="" part="0" file="hgfd_fluidized_bed_0026.vti" />
+    <DataSet timestep="0.05409015" group="" part="0" file="hgfd_fluidized_bed_0027.vti" />
+    <DataSet timestep="0.056093489" group="" part="0" file="hgfd_fluidized_bed_0028.vti" />
+    <DataSet timestep="0.058096828" group="" part="0" file="hgfd_fluidized_bed_0029.vti" />
+    <DataSet timestep="0.06" group="" part="0" file="hgfd_fluidized_bed_0030.vti" />
+  </Collection>
+</VTKFile>
\ No newline at end of file
diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0000.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0000.vti
new file mode 100644
index 0000000..9e95dcd
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0000.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
+<VTKFile type="ImageData" version="0.1" byte_order="LittleEndian" header_type="UInt32" compressor="vtkZLibDataCompressor">
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+  <Piece Extent="0 16 0 32 0 1">
+    <PointData>
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+    <CellData Scalars="Solid fraction" Vectors="Fluid velocity (m/s)">
+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.550000011920929">
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+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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+        AQAAAACAAAAACAAAHwAAAA==eF7b+UfUcucoHsUjFDMwHKgfxaN4FI/ikYgBXpsRLg==
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+      <DataArray type="Float32" Name="Particle velocity (m/s)" NumberOfComponents="3" format="binary" RangeMin="0" RangeMax="0.0001430389384040609">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0001.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0001.vti
new file mode 100644
index 0000000..4f45159
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0001.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
+<VTKFile type="ImageData" version="0.1" byte_order="LittleEndian" header_type="UInt32" compressor="vtkZLibDataCompressor">
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+  <Piece Extent="0 16 0 32 0 1">
+    <PointData>
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+    <CellData Scalars="Solid fraction" Vectors="Fluid velocity (m/s)">
+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0002.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0002.vti
new file mode 100644
index 0000000..0024f52
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0002.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0003.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0003.vti
new file mode 100644
index 0000000..9b14e58
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0003.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0004.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0004.vti
new file mode 100644
index 0000000..78423d3
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0004.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
+<VTKFile type="ImageData" version="0.1" byte_order="LittleEndian" header_type="UInt32" compressor="vtkZLibDataCompressor">
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+  <Piece Extent="0 16 0 32 0 1">
+    <PointData>
+    </PointData>
+    <CellData Scalars="Solid fraction" Vectors="Fluid velocity (m/s)">
+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+      </DataArray>
+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0005.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0005.vti
new file mode 100644
index 0000000..2c93495
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0005.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+    <PointData>
+    </PointData>
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+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+      </DataArray>
+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0006.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0006.vti
new file mode 100644
index 0000000..9a08364
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0006.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0007.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0007.vti
new file mode 100644
index 0000000..05f8dea
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0007.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+    <PointData>
+    </PointData>
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+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+      </DataArray>
+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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+      </DataArray>
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0008.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0008.vti
new file mode 100644
index 0000000..5d3ce16
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0008.vti
@@ -0,0 +1,57 @@
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+    <PointData>
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+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0009.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0009.vti
new file mode 100644
index 0000000..0fe5839
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0009.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0010.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0010.vti
new file mode 100644
index 0000000..373ff56
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0010.vti
@@ -0,0 +1,57 @@
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+    </PointData>
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+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0011.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0011.vti
new file mode 100644
index 0000000..9d2e7a7
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0011.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+    <PointData>
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0012.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0012.vti
new file mode 100644
index 0000000..ae47d7a
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0012.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0013.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0013.vti
new file mode 100644
index 0000000..a7f6936
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0013.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0014.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0014.vti
new file mode 100644
index 0000000..70b76c5
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0014.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0015.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0015.vti
new file mode 100644
index 0000000..349360f
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0015.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0016.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0016.vti
new file mode 100644
index 0000000..c194588
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0016.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0017.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0017.vti
new file mode 100644
index 0000000..c6d64bc
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0017.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0018.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0018.vti
new file mode 100644
index 0000000..b49c503
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0018.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0019.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0019.vti
new file mode 100644
index 0000000..bf84298
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0019.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0020.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0020.vti
new file mode 100644
index 0000000..f519685
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0020.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+  <Piece Extent="0 16 0 32 0 1">
+    <PointData>
+    </PointData>
+    <CellData Scalars="Solid fraction" Vectors="Fluid velocity (m/s)">
+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+      </DataArray>
+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0021.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0021.vti
new file mode 100644
index 0000000..ead6290
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0021.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+    <PointData>
+    </PointData>
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+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+</VTKFile>
diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0022.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0022.vti
new file mode 100644
index 0000000..353cb93
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0022.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
+<VTKFile type="ImageData" version="0.1" byte_order="LittleEndian" header_type="UInt32" compressor="vtkZLibDataCompressor">
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+  <Piece Extent="0 16 0 32 0 1">
+    <PointData>
+    </PointData>
+    <CellData Scalars="Solid fraction" Vectors="Fluid velocity (m/s)">
+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+      </DataArray>
+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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+      <DataArray type="Float32" Name="Particle speed (m/s)" format="binary" RangeMin="0.00035352498525753617" RangeMax="0.09034716337919235">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0023.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0023.vti
new file mode 100644
index 0000000..e1c0f47
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0023.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+    <PointData>
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+</VTKFile>
diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0024.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0024.vti
new file mode 100644
index 0000000..ba225f9
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0024.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
+<VTKFile type="ImageData" version="0.1" byte_order="LittleEndian" header_type="UInt32" compressor="vtkZLibDataCompressor">
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+  <Piece Extent="0 16 0 32 0 1">
+    <PointData>
+    </PointData>
+    <CellData Scalars="Solid fraction" Vectors="Fluid velocity (m/s)">
+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+      </DataArray>
+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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+      <DataArray type="Float32" Name="Particle speed (m/s)" format="binary" RangeMin="0.0005803940584883094" RangeMax="0.10094409435987473">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0025.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0025.vti
new file mode 100644
index 0000000..d4834c0
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0025.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+    <PointData>
+    </PointData>
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+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0026.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0026.vti
new file mode 100644
index 0000000..2296910
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0026.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+  <Piece Extent="0 16 0 32 0 1">
+    <PointData>
+    </PointData>
+    <CellData Scalars="Solid fraction" Vectors="Fluid velocity (m/s)">
+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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+      </DataArray>
+      <DataArray type="Float32" Name="Mean grain size (mm)" format="binary" RangeMin="1" RangeMax="1">
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0027.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0027.vti
new file mode 100644
index 0000000..b62f22f
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0027.vti
@@ -0,0 +1,57 @@
+<?xml version="1.0"?>
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+    <PointData>
+    </PointData>
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+      <DataArray type="Float32" Name="Solid fraction" format="binary" RangeMin="0" RangeMax="0.6000000238418579">
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mRkWFsPQE8LXbZHsM4J/zPB6SWUw/I/HGvC64+ATH3LTIuz2M8olnnPJcjv4VygOJ18rft3f3SpktAnmv3joxijOwZ5dI0KYDj/4v3plyvzohv/raCtJ9nn4Lx/mLe7fDs9SPkJS8RT+j/ySJFOOYHyvna+FWxz8bz5YZv5tHvwNPxJJ9XT8l1ZqimTEafif6G2Q+Jj9uu9X11kMM4D/UpkcaV0Pf8Xw20x0Nfy3z7/DePw6b9DEcczlOOzp4GjPfFGDv2vzYMbhT/gMXa8gmXYE/gYFi8WnWzH+vfVB6XBFfHuXhvaW3lyJ8XfBqiWuh+DvGF0sdnOHf+lUofQ4g/+6/2aQhMv/9UaMN5WssoBPws7fzEWVTynHppozN3bCv9ZTi0msgb9piR7D6mP+qevRFt/KsaflgXyL1L/x1rQe87VYnQuf3kNmpRY74D+nNqTEvxnj/wNaZW3U
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+          </Value>
+          <Value index="1">
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+        </InformationKey>
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0028.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0028.vti
new file mode 100644
index 0000000..260bdcb
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0028.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0029.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0029.vti
new file mode 100644
index 0000000..7bd7f18
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0029.vti
@@ -0,0 +1,57 @@
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diff --git a/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0030.vti b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0030.vti
new file mode 100644
index 0000000..4dec1eb
--- /dev/null
+++ b/docs/visualizations/hgfd_fluidized_bed/paraview/hgfd_fluidized_bed_0030.vti
@@ -0,0 +1,57 @@
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+      </DataArray>
+      <DataArray type="Float32" Name="Fluid pressure (Pa)" format="binary" RangeMin="-0.562690794467926" RangeMax="490.0553894042969">
+        AQAAAACAAAAACAAAtAcAAA==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
+      </DataArray>
+      <DataArray type="Float32" Name="Momentum exchange beta" format="binary" RangeMin="0" RangeMax="7202.05859375">
+        AQAAAACAAAAACAAAVwMAAA==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
+      </DataArray>
+      <DataArray type="Float32" Name="Fluidized" format="binary" RangeMin="0" RangeMax="1">
+        AQAAAACAAAAACAAAXQAAAA==eF7tjlEOABAMQ93M1RxdfEiq2hG/9pIl1PamFKbVLVqyeVbZLdw/7ujF4owd3K+K95xmT37MkWjfTam5F0/kwhzf3dllqif6i3M4L7+5fc7t5tQ9SZLkPzq6DFyF
+      </DataArray>
+    </CellData>
+  </Piece>
+  </ImageData>
+</VTKFile>
diff --git a/docs/visualizations/hgfd_fluidized_bed/render/demo.mp4 b/docs/visualizations/hgfd_fluidized_bed/render/demo.mp4
new file mode 100644
index 0000000..ac4542b
Binary files /dev/null and b/docs/visualizations/hgfd_fluidized_bed/render/demo.mp4 differ
diff --git a/docs/visualizations/hgfd_fluidized_bed/render/poster.png b/docs/visualizations/hgfd_fluidized_bed/render/poster.png
new file mode 100644
index 0000000..2b3b305
Binary files /dev/null and b/docs/visualizations/hgfd_fluidized_bed/render/poster.png differ
diff --git a/docs/visualizations/hgfd_fluidized_bed/render/preview.gif b/docs/visualizations/hgfd_fluidized_bed/render/preview.gif
new file mode 100644
index 0000000..968aaa7
Binary files /dev/null and b/docs/visualizations/hgfd_fluidized_bed/render/preview.gif differ
diff --git a/json/defaults.json5 b/json/defaults.json5
index 99c1f15..9b6e6c1 100644
--- a/json/defaults.json5
+++ b/json/defaults.json5
@@ -57,6 +57,16 @@
     // gas
     gas_density:1.225,
     gas_viscosity:1.8e-5,
+
+    // two-way fluid coupling (2-D, isothermal, laminar)
+    fluid_coupling: false,
+    fluid_inlet_velocity: 0.0, // superficial velocity at the bottom boundary (m/s)
+    fluid_min_void_fraction: 0.05,
+    fluid_drag_min_particle_size: 1e-6,
+    fluid_alpha_max: 0.3,
+    fluid_alpha_epsilon: 0.01,
+    fluid_probability_policy: 'error', // 'error' or 'scale'
+    fluid_cfl_limit: 0.8,
     
     // saving
     folderName : "output/", // name of the folder to save the data in
@@ -94,6 +104,7 @@
     stop_event : null, // this is None in Python, but null in JavaScript
     stop_after : 100, // number of timesteps to run with nothing happening before stopping
     stopped_times : 0, // number of times the system has stopped
+    random_seed : null, // optional reproducible seed for stochastic Python motion models
 
     // diagnostics
     show_optimal_resolution : false, // should the optimal resolution be shown on the screen at runtime
diff --git a/json/fluidized_bed.json5 b/json/fluidized_bed.json5
new file mode 100644
index 0000000..5211347
--- /dev/null
+++ b/json/fluidized_bed.json5
@@ -0,0 +1,43 @@
+{
+    // Small gas-fluidized-bed verification case. The inlet is above the
+    // Gidaspow/Ergun minimum-fluidization estimate printed at start-up.
+    nx: 16,
+    ny: 32,
+    nm: 20,
+    H: 0.08,
+
+    gsd_mode: 'mono',
+    s_m: 0.001,
+    solid_density: 2500,
+    nu_fill: 0.55,
+    nu_cs: 0.60,
+    nu_1: 0.60,
+    IC_mode: 'bottom',
+    fill_ratio: 0.40,
+    repose_angle: 30,
+
+    fluid_coupling: true,
+    gas_density: 1.225,
+    gas_viscosity: 1.8e-5,
+    fluid_inlet_velocity: 1.40,
+    fluid_alpha_max: 0.30,
+    // Paper reproduction uses a fixed dt satisfying the strict P_tot < 0.5
+    // constraint; probability rescaling would change the reported dynamics.
+    fluid_probability_policy: 'error',
+
+    inertia: true,
+    motion_model: 'hgfd',
+    defined_time_step_size: 0.0001,
+    time_average: false,
+    P_stab: 0.5,
+
+    t_f: 0.06,
+    saves: 30,
+    max_workers: 1,
+    random_seed: 7,
+    plot_colorbar: true,
+    plot: ['fluidized_bed', 'nu'],
+    videos: ['fluidized_bed'],
+    // Save both phases for the ParaView VTI/PVD export pipeline.
+    save: ['fluid', 'nu', 's_bar', 'velocity'],
+}
diff --git a/json/hgfd_paper_case1_0p5mm.json5 b/json/hgfd_paper_case1_0p5mm.json5
new file mode 100644
index 0000000..8692427
--- /dev/null
+++ b/json/hgfd_paper_case1_0p5mm.json5
@@ -0,0 +1,42 @@
+{
+    // Li et al. (2026), Table 1 / Fig. 4: 0.5 mm sphere in water.
+    nx: 15,
+    ny: 13,
+    nm: 100,
+    H: 0.65,
+    cell_size: 0.05,
+
+    gsd_mode: 'mono',
+    s_m: 0.0005,
+    solid_density: 2560,
+    nu_fill: 0.01,
+    nu_cs: 0.5,
+    nu_1: 0.5,
+    IC_mode: 'single_particle',
+    single_particle_i: 7,
+    single_particle_j: 10,
+    single_particle_k: 0,
+    repose_angle: 30,
+
+    fluid_coupling: true,
+    gas_density: 997,
+    gas_viscosity: 1.0e-3,
+    fluid_inlet_velocity: 0,
+    fluid_drag_min_particle_size: 0.0005,
+    fluid_alpha_max: 0.3,
+    fluid_probability_policy: 'error',
+
+    inertia: true,
+    motion_model: 'hgfd',
+    defined_time_step_size: 1.0e-4,
+    time_average: false,
+    P_stab: 0.5,
+
+    t_f: 0.4,
+    saves: 40,
+    max_workers: 1,
+    random_seed: 11,
+    plot: ['nu'],
+    videos: [],
+    save: ['fluid', 'velocity'],
+}
diff --git a/json/hgfd_paper_case1_1p5mm.json5 b/json/hgfd_paper_case1_1p5mm.json5
new file mode 100644
index 0000000..33819b5
--- /dev/null
+++ b/json/hgfd_paper_case1_1p5mm.json5
@@ -0,0 +1,42 @@
+{
+    // Li et al. (2026), Table 1 / Fig. 4: 1.5 mm sphere in water.
+    nx: 15,
+    ny: 13,
+    nm: 100,
+    H: 0.65,
+    cell_size: 0.05,
+
+    gsd_mode: 'mono',
+    s_m: 0.0015,
+    solid_density: 2560,
+    nu_fill: 0.01,
+    nu_cs: 0.5,
+    nu_1: 0.5,
+    IC_mode: 'single_particle',
+    single_particle_i: 7,
+    single_particle_j: 10,
+    single_particle_k: 0,
+    repose_angle: 30,
+
+    fluid_coupling: true,
+    gas_density: 997,
+    gas_viscosity: 1.0e-3,
+    fluid_inlet_velocity: 0,
+    fluid_drag_min_particle_size: 0.0015,
+    fluid_alpha_max: 0.3,
+    fluid_probability_policy: 'error',
+
+    inertia: true,
+    motion_model: 'hgfd',
+    defined_time_step_size: 1.0e-4,
+    time_average: false,
+    P_stab: 0.5,
+
+    t_f: 0.4,
+    saves: 40,
+    max_workers: 1,
+    random_seed: 12,
+    plot: ['nu'],
+    videos: [],
+    save: ['fluid', 'velocity'],
+}
diff --git a/json/hgfd_paper_case1_2p0mm.json5 b/json/hgfd_paper_case1_2p0mm.json5
new file mode 100644
index 0000000..fedd2c5
--- /dev/null
+++ b/json/hgfd_paper_case1_2p0mm.json5
@@ -0,0 +1,42 @@
+{
+    // Li et al. (2026), Table 1 / Fig. 4: 2.0 mm sphere in water.
+    nx: 15,
+    ny: 13,
+    nm: 100,
+    H: 0.65,
+    cell_size: 0.05,
+
+    gsd_mode: 'mono',
+    s_m: 0.002,
+    solid_density: 2480,
+    nu_fill: 0.01,
+    nu_cs: 0.5,
+    nu_1: 0.5,
+    IC_mode: 'single_particle',
+    single_particle_i: 7,
+    single_particle_j: 10,
+    single_particle_k: 0,
+    repose_angle: 30,
+
+    fluid_coupling: true,
+    gas_density: 997,
+    gas_viscosity: 1.0e-3,
+    fluid_inlet_velocity: 0,
+    fluid_drag_min_particle_size: 0.002,
+    fluid_alpha_max: 0.3,
+    fluid_probability_policy: 'error',
+
+    inertia: true,
+    motion_model: 'hgfd',
+    defined_time_step_size: 1.0e-4,
+    time_average: false,
+    P_stab: 0.5,
+
+    t_f: 0.4,
+    saves: 40,
+    max_workers: 1,
+    random_seed: 13,
+    plot: ['nu'],
+    videos: [],
+    save: ['fluid', 'velocity'],
+}
diff --git a/mkdocs.yml b/mkdocs.yml
index ce4fa42..2b2b6ac 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -19,5 +19,6 @@ nav:
   - Home: index.md
   - Getting Started: getting-started.md
   - Examples: examples.md
+  - HGFD Reproduction: hgfd-reproduction.md
   - Troubleshooting: troubleshooting.md
-  - Reference: reference.md
\ No newline at end of file
+  - Reference: reference.md
diff --git a/scripts/export_hgfd_to_vti.py b/scripts/export_hgfd_to_vti.py
new file mode 100644
index 0000000..d985a30
--- /dev/null
+++ b/scripts/export_hgfd_to_vti.py
@@ -0,0 +1,133 @@
+#!/usr/bin/env python3
+"""Convert saved HGFD particle/fluid fields to a ParaView VTI/PVD series."""
+
+from __future__ import annotations
+
+import argparse
+from pathlib import Path
+import xml.etree.ElementTree as ET
+
+import numpy as np
+import vtk
+from vtk.util.numpy_support import numpy_to_vtk
+
+
+def parse_args():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--input", type=Path, required=True)
+    parser.add_argument("--output", type=Path, required=True)
+    parser.add_argument("--name", default="hgfd_fluidized_bed")
+    parser.add_argument("--t-final", type=float, required=True)
+    return parser.parse_args()
+
+
+def indexed_files(directory, prefix, suffix):
+    return {int(path.stem.rsplit("_", 1)[1]): path for path in directory.glob(f"{prefix}_[0-9]*{suffix}")}
+
+
+def add_cell_array(image, name, values, components=1):
+    values = np.asarray(values, dtype=np.float32)
+    if components == 1:
+        flattened = values.ravel(order="F")
+    else:
+        flattened = np.stack(
+            [values[:, :, component].ravel(order="F") for component in range(components)],
+            axis=1,
+        )
+    vtk_array = numpy_to_vtk(flattened, deep=True)
+    vtk_array.SetName(name)
+    vtk_array.SetNumberOfComponents(components)
+    image.GetCellData().AddArray(vtk_array)
+
+
+def write_pvd(path, entries):
+    vtk_file = ET.Element("VTKFile", type="Collection", version="0.1", byte_order="LittleEndian")
+    collection = ET.SubElement(vtk_file, "Collection")
+    for time_value, filename in entries:
+        ET.SubElement(
+            collection,
+            "DataSet",
+            timestep=f"{time_value:.8g}",
+            group="",
+            part="0",
+            file=filename,
+        )
+    tree = ET.ElementTree(vtk_file)
+    ET.indent(tree, space="  ")
+    tree.write(path, encoding="utf-8", xml_declaration=True)
+
+
+def main():
+    args = parse_args()
+    args.output.mkdir(parents=True, exist_ok=True)
+    sources = {
+        "nu": indexed_files(args.input, "nu", ".npy"),
+        "s_bar": indexed_files(args.input, "s_bar", ".npy"),
+        "u": indexed_files(args.input, "u", ".npy"),
+        "v": indexed_files(args.input, "v", ".npy"),
+        "fluid": indexed_files(args.input, "fluid", ".npz"),
+    }
+    frame_ids = sorted(set.intersection(*(set(files) for files in sources.values())))
+    if not frame_ids:
+        raise RuntimeError(f"No complete HGFD frames found in {args.input}")
+
+    x = np.atleast_1d(np.loadtxt(args.input / "x.csv", delimiter=","))
+    y = np.atleast_1d(np.loadtxt(args.input / "y.csv", delimiter=","))
+    dx = float(np.median(np.diff(x))) if x.size > 1 else 1.0
+    dy = float(np.median(np.diff(y))) if y.size > 1 else 1.0
+    origin = (float(x[0] - dx / 2), float(y[0] - dy / 2), 0.0)
+
+    entries = []
+    final_step = frame_ids[-1]
+    for output_index, frame_id in enumerate(frame_ids):
+        solid_fraction = np.load(sources["nu"][frame_id])
+        grain_size = np.load(sources["s_bar"][frame_id]) * 1_000.0
+        particle_u = np.load(sources["u"][frame_id])
+        particle_v = np.load(sources["v"][frame_id])
+        fluid = np.load(sources["fluid"][frame_id])
+        fluid_u = fluid["u"]
+        fluid_v = fluid["v"]
+        occupied = solid_fraction > 0.0
+
+        particle_speed = np.where(occupied, np.hypot(particle_u, particle_v), np.nan)
+        particle_velocity = np.stack((particle_u, particle_v, np.zeros_like(particle_u)), axis=-1)
+        fluid_speed = np.hypot(fluid_u, fluid_v)
+        fluid_velocity = np.stack((fluid_u, fluid_v, np.zeros_like(fluid_u)), axis=-1)
+
+        nx, ny = solid_fraction.shape
+        image = vtk.vtkImageData()
+        image.SetDimensions(nx + 1, ny + 1, 2)
+        image.SetOrigin(*origin)
+        image.SetSpacing(dx, dy, min(dx, dy) * 0.08)
+        add_cell_array(image, "Solid fraction", solid_fraction)
+        add_cell_array(image, "Mean grain size (mm)", np.where(occupied, grain_size, np.nan))
+        add_cell_array(image, "Particle speed (m/s)", particle_speed)
+        add_cell_array(image, "Particle velocity (m/s)", particle_velocity, components=3)
+        add_cell_array(image, "Fluid speed (m/s)", fluid_speed)
+        add_cell_array(image, "Fluid velocity (m/s)", fluid_velocity, components=3)
+        add_cell_array(image, "Fluid vertical velocity (m/s)", fluid_v)
+        add_cell_array(image, "Fluid pressure (Pa)", fluid["pressure"])
+        add_cell_array(image, "Momentum exchange beta", fluid["beta"])
+        add_cell_array(image, "Fluidized", fluid["fluidized"].astype(np.float32))
+        image.GetCellData().SetActiveScalars("Solid fraction")
+        image.GetCellData().SetActiveVectors("Fluid velocity (m/s)")
+
+        filename = f"{args.name}_{output_index:04d}.vti"
+        writer = vtk.vtkXMLImageDataWriter()
+        writer.SetFileName(str(args.output / filename))
+        writer.SetInputData(image)
+        writer.SetDataModeToBinary()
+        writer.SetCompressorTypeToZLib()
+        if writer.Write() != 1:
+            raise RuntimeError(f"VTK failed to write {filename}")
+
+        time_value = args.t_final * frame_id / final_step if final_step else 0.0
+        entries.append((time_value, filename))
+
+    pvd_path = args.output / f"{args.name}.pvd"
+    write_pvd(pvd_path, entries)
+    print(f"Wrote {len(entries)} frames to {pvd_path}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/scripts/plot_hgfd_case1.py b/scripts/plot_hgfd_case1.py
new file mode 100644
index 0000000..f6c0443
--- /dev/null
+++ b/scripts/plot_hgfd_case1.py
@@ -0,0 +1,94 @@
+#!/usr/bin/env python3
+"""Plot the three HGFD paper Case 1 settling curves from saved runs."""
+
+from pathlib import Path
+
+import json5
+import matplotlib.pyplot as plt
+from matplotlib.animation import FuncAnimation, PillowWriter
+import numpy as np
+
+ROOT = Path(__file__).resolve().parents[1]
+CASES = (
+    ("0p5mm", "0.5 mm", "black", "-", 0.07425),
+    ("1p5mm", "1.5 mm", "#1f77b4", ":", 0.21144),
+    ("2p0mm", "2.0 mm", "#d62728", "--", 0.26153),
+)
+
+
+def load_curve(case_name):
+    config_path = ROOT / "json" / f"hgfd_paper_case1_{case_name}.json5"
+    config = json5.loads(config_path.read_text())
+    files = sorted((ROOT / "output" / config_path.stem / "data").glob("v_*.npy"))
+    if not files:
+        raise FileNotFoundError(f"No saved velocities for {config_path.stem}; run the case first")
+
+    # save_velocity stores the mean over M.  Case 1 contains exactly one
+    # occupied coordinate, so multiplying by M recovers its layer velocity.
+    steps = np.array([int(path.stem.rsplit("_", 1)[1]) for path in files])
+    speed = np.array([config["nm"] * np.max(np.abs(np.load(path))) for path in files])
+    return 1e3 * steps * config["defined_time_step_size"], speed
+
+
+def style_axes(ax):
+    ax.set_xlim(0, 400)
+    ax.set_ylim(0, 0.285)
+    ax.set_xticks(np.arange(0, 401, 100))
+    ax.set_xlabel("Time [ms]")
+    ax.set_ylabel("Particle velocity [m/s]")
+    ax.legend(title="Particle diameter", frameon=True, loc="lower right")
+    ax.tick_params(direction="in", width=1.5, length=6)
+    for spine in ax.spines.values():
+        spine.set_linewidth(1.5)
+
+
+def save_static(curves):
+    fig, ax = plt.subplots(figsize=(6.2, 6.8), constrained_layout=True)
+    for (_, label, color, linestyle, _), (time_ms, speed) in zip(CASES, curves):
+        ax.plot(time_ms, speed, color=color, linestyle=linestyle, linewidth=3, label=label)
+    style_axes(ax)
+    output = ROOT / "docs" / "images" / "hgfd_case1_reproduction.png"
+    fig.savefig(output, dpi=220)
+    plt.close(fig)
+    print(f"Wrote {output.relative_to(ROOT)}")
+
+
+def save_animation(curves):
+    fig, ax = plt.subplots(figsize=(6.2, 6.8), constrained_layout=True)
+    lines = []
+    for _, label, color, linestyle, _ in CASES:
+        (line,) = ax.plot([], [], color=color, linestyle=linestyle, linewidth=3, label=label)
+        lines.append(line)
+    style_axes(ax)
+
+    last_frame = min(len(time_ms) for time_ms, _ in curves) - 1
+
+    def update(frame):
+        for line, (time_ms, speed) in zip(lines, curves):
+            line.set_data(time_ms[: frame + 1], speed[: frame + 1])
+        return lines
+
+    # Hold the converged curves for one second at the end of the loop.
+    frames = list(range(last_frame + 1)) + [last_frame] * 10
+    animation = FuncAnimation(fig, update, frames=frames, interval=100, blit=True)
+    output = ROOT / "docs" / "images" / "hgfd_case1_reproduction.gif"
+    animation.save(output, writer=PillowWriter(fps=10), dpi=120)
+    plt.close(fig)
+    print(f"Wrote {output.relative_to(ROOT)}")
+
+
+def main():
+    curves = [load_curve(case_name) for case_name, *_ in CASES]
+    save_static(curves)
+    save_animation(curves)
+
+    for (_, label, _, _, paper), (_, speed) in zip(CASES, curves):
+        reproduced = speed[-1]
+        relative_error = 100.0 * (reproduced - paper) / paper
+        print(
+            f"{label}: reproduced={reproduced:.8f} m/s, paper={paper:.8f} m/s, error={relative_error:+.3f}%"
+        )
+
+
+if __name__ == "__main__":
+    main()
diff --git a/scripts/render_hgfd_vti.py b/scripts/render_hgfd_vti.py
new file mode 100644
index 0000000..fdd4d6d
--- /dev/null
+++ b/scripts/render_hgfd_vti.py
@@ -0,0 +1,259 @@
+#!/usr/bin/env python3
+"""Render the official-gallery four-panel HGFD GIF/MP4 from a PVD series."""
+
+from __future__ import annotations
+
+import argparse
+from pathlib import Path
+import subprocess
+import tempfile
+import xml.etree.ElementTree as ET
+
+import imageio_ffmpeg
+import numpy as np
+import vtk
+from vtk.util.numpy_support import vtk_to_numpy
+
+PANELS = (
+    (
+        "Solid fraction",
+        "SOLID FRACTION",
+        ((0.03, 0.05, 0.09), (0.45, 0.08, 0.15), (1.0, 0.44, 0.05), (1.0, 0.90, 0.25)),
+    ),
+    (
+        "Particle speed (m/s)",
+        "PARTICLE SPEED",
+        ((0.02, 0.05, 0.10), (0.02, 0.35, 0.48), (0.05, 0.80, 0.70), (0.95, 0.88, 0.30)),
+    ),
+    (
+        "Fluid pressure (Pa)",
+        "FLUID PRESSURE",
+        ((0.02, 0.04, 0.08), (0.34, 0.06, 0.20), (0.92, 0.22, 0.08), (1.0, 0.82, 0.20)),
+    ),
+    (
+        "Fluid speed (m/s)",
+        "FLUID SPEED",
+        ((0.02, 0.05, 0.10), (0.02, 0.35, 0.48), (0.05, 0.80, 0.70), (0.95, 0.88, 0.30)),
+    ),
+)
+
+
+def parse_args():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--pvd", type=Path, required=True)
+    parser.add_argument("--output", type=Path, required=True)
+    parser.add_argument("--title", default="HGFD · FLUIDIZED BED")
+    parser.add_argument("--ffmpeg", help="ffmpeg executable; defaults to imageio-ffmpeg's bundled binary")
+    parser.add_argument("--fps", type=int, default=20)
+    return parser.parse_args()
+
+
+def pvd_entries(path):
+    tree = ET.parse(path)
+    return [
+        (float(node.attrib["timestep"]), path.parent / node.attrib["file"])
+        for node in tree.findall(".//DataSet")
+    ]
+
+
+def read_image(path):
+    reader = vtk.vtkXMLImageDataReader()
+    reader.SetFileName(str(path))
+    reader.Update()
+    image = vtk.vtkImageData()
+    image.ShallowCopy(reader.GetOutput())
+    return image
+
+
+def field_ranges(entries):
+    samples = {field: [] for field, _, _ in PANELS}
+    stride = max(1, len(entries) // 12)
+    for _, path in entries[::stride]:
+        image = read_image(path)
+        for field in samples:
+            values = vtk_to_numpy(image.GetCellData().GetArray(field))
+            samples[field].append(values[np.isfinite(values)])
+
+    ranges = {}
+    for field, chunks in samples.items():
+        values = np.concatenate([chunk for chunk in chunks if chunk.size])
+        if field == "Solid fraction":
+            ranges[field] = (0.0, max(0.6, float(np.nanmax(values))))
+        elif field == "Fluid pressure (Pa)":
+            low, high = np.nanpercentile(values, (1.0, 99.0))
+            ranges[field] = (float(low), float(high if high > low else low + 1.0))
+        else:
+            high = float(np.nanpercentile(values, 99.0))
+            ranges[field] = (0.0, high if high > 0 else 1.0)
+    return ranges
+
+
+def make_lookup_table(colors, value_range):
+    table = vtk.vtkColorTransferFunction()
+    table.SetColorSpaceToRGB()
+    low, high = value_range
+    for index, color in enumerate(colors):
+        value = low + (high - low) * index / (len(colors) - 1)
+        table.AddRGBPoint(value, *color)
+    table.SetNanColor(0.02, 0.035, 0.06)
+    return table
+
+
+def add_text(renderer, value, x, y, size, color=(0.92, 0.95, 1.0)):
+    actor = vtk.vtkTextActor()
+    actor.SetInput(value)
+    actor.GetPositionCoordinate().SetCoordinateSystemToNormalizedViewport()
+    actor.SetPosition(x, y)
+    actor.GetTextProperty().SetFontFamilyToArial()
+    actor.GetTextProperty().SetFontSize(size)
+    actor.GetTextProperty().SetBold(True)
+    actor.GetTextProperty().SetColor(*color)
+    renderer.AddActor2D(actor)
+    return actor
+
+
+def main():
+    args = parse_args()
+    if args.ffmpeg is None:
+        args.ffmpeg = imageio_ffmpeg.get_ffmpeg_exe()
+    args.output.mkdir(parents=True, exist_ok=True)
+    entries = pvd_entries(args.pvd)
+    if not entries:
+        raise RuntimeError(f"No time steps found in {args.pvd}")
+    initial = read_image(entries[0][1])
+    ranges = field_ranges(entries)
+    bounds = initial.GetBounds()
+    width = bounds[1] - bounds[0]
+    height = bounds[3] - bounds[2]
+
+    window = vtk.vtkRenderWindow()
+    window.SetOffScreenRendering(True)
+    window.SetSize(1920, 1080)
+    window.SetMultiSamples(8)
+    window.SetWindowName(args.title)
+    viewports = (
+        (0.0, 0.5, 0.5, 1.0),
+        (0.5, 0.5, 1.0, 1.0),
+        (0.0, 0.0, 0.5, 0.5),
+        (0.5, 0.0, 1.0, 0.5),
+    )
+    interpolators = []
+    time_labels = []
+
+    for (field, label, colors), viewport in zip(PANELS, viewports):
+        renderer = vtk.vtkRenderer()
+        renderer.SetViewport(*viewport)
+        renderer.SetBackground(0.018, 0.03, 0.052)
+        renderer.SetBackground2(0.055, 0.075, 0.11)
+        renderer.GradientBackgroundOn()
+        window.AddRenderer(renderer)
+
+        interpolator = vtk.vtkCellDataToPointData()
+        interpolator.SetInputData(initial)
+        interpolator.PassCellDataOff()
+        interpolators.append(interpolator)
+        mapper = vtk.vtkDataSetMapper()
+        mapper.SetInputConnection(interpolator.GetOutputPort())
+        mapper.SetScalarModeToUsePointFieldData()
+        mapper.SelectColorArray(field)
+        mapper.SetLookupTable(make_lookup_table(colors, ranges[field]))
+        mapper.SetScalarRange(*ranges[field])
+        mapper.ScalarVisibilityOn()
+
+        actor = vtk.vtkActor()
+        actor.SetMapper(mapper)
+        actor.GetProperty().SetInterpolationToFlat()
+        renderer.AddActor(actor)
+        camera = renderer.GetActiveCamera()
+        center_x = 0.5 * (bounds[0] + bounds[1])
+        center_y = 0.5 * (bounds[2] + bounds[3])
+        camera.SetFocalPoint(center_x, center_y, 0.0)
+        camera.SetPosition(center_x, center_y, max(width, height) * 4.0)
+        camera.SetViewUp(0.0, 1.0, 0.0)
+        camera.ParallelProjectionOn()
+        camera.SetParallelScale(max(height * 0.57, width * 0.62))
+
+        add_text(renderer, label, 0.035, 0.91, 21, (1.0, 0.58, 0.12))
+        time_labels.append(add_text(renderer, "t = 0.000 s", 0.035, 0.055, 15, (0.74, 0.81, 0.90)))
+        scalar_bar = vtk.vtkScalarBarActor()
+        scalar_bar.SetLookupTable(mapper.GetLookupTable())
+        scalar_bar.SetNumberOfLabels(4)
+        scalar_bar.SetOrientationToHorizontal()
+        scalar_bar.SetPosition(0.55, 0.055)
+        scalar_bar.SetWidth(0.40)
+        scalar_bar.SetHeight(0.055)
+        scalar_bar.GetLabelTextProperty().SetColor(0.78, 0.84, 0.92)
+        scalar_bar.GetLabelTextProperty().SetFontSize(11)
+        scalar_bar.SetLabelFormat("%.3g")
+        renderer.AddActor2D(scalar_bar)
+
+    first_renderer = window.GetRenderers().GetFirstRenderer()
+    add_text(first_renderer, args.title, 0.035, 0.835, 17, (0.78, 0.84, 0.92))
+    capture = vtk.vtkWindowToImageFilter()
+    capture.SetInput(window)
+    capture.SetInputBufferTypeToRGB()
+    capture.ReadFrontBufferOff()
+
+    with tempfile.TemporaryDirectory(prefix="hgfd-render-") as temporary:
+        frame_directory = Path(temporary)
+        poster_index = round(0.65 * (len(entries) - 1))
+        for frame_index, (time_value, path) in enumerate(entries):
+            image = read_image(path)
+            for interpolator, time_label in zip(interpolators, time_labels):
+                interpolator.SetInputData(image)
+                time_label.SetInput(f"t = {time_value:.3f} s")
+            window.Render()
+            capture.Modified()
+            capture.Update()
+            frame_path = frame_directory / f"frame_{frame_index:04d}.png"
+            writer = vtk.vtkPNGWriter()
+            writer.SetFileName(str(frame_path))
+            writer.SetInputConnection(capture.GetOutputPort())
+            writer.Write()
+            if frame_index == poster_index:
+                (args.output / "poster.png").write_bytes(frame_path.read_bytes())
+
+        subprocess.run(
+            [
+                args.ffmpeg,
+                "-loglevel",
+                "error",
+                "-y",
+                "-framerate",
+                str(args.fps),
+                "-i",
+                str(frame_directory / "frame_%04d.png"),
+                "-c:v",
+                "libx264",
+                "-preset",
+                "slow",
+                "-crf",
+                "20",
+                "-pix_fmt",
+                "yuv420p",
+                str(args.output / "demo.mp4"),
+            ],
+            check=True,
+        )
+        subprocess.run(
+            [
+                args.ffmpeg,
+                "-loglevel",
+                "error",
+                "-y",
+                "-framerate",
+                str(args.fps),
+                "-i",
+                str(frame_directory / "frame_%04d.png"),
+                "-filter_complex",
+                "fps=10,scale=960:-1:flags=lanczos,split[s0][s1];"
+                "[s0]palettegen=max_colors=128[p];[s1][p]paletteuse=dither=bayer",
+                str(args.output / "preview.gif"),
+            ],
+            check=True,
+        )
+    print(f"Rendered {len(entries)} frames to {args.output}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/test/test_fluid.py b/test/test_fluid.py
new file mode 100644
index 0000000..3ccf8ed
--- /dev/null
+++ b/test/test_fluid.py
@@ -0,0 +1,229 @@
+import unittest
+from pathlib import Path
+from types import SimpleNamespace
+from unittest.mock import patch
+
+import numpy as np
+
+from HGD import fluid, initial, params
+from HGD.motion import hgfd
+
+
+def parameters(**overrides):
+    values = dict(
+        nx=6,
+        ny=8,
+        nm=10,
+        dx=0.01,
+        dy=0.01,
+        dt=1e-4,
+        g=9.81,
+        gas_density=1.225,
+        gas_viscosity=1.8e-5,
+        solid_density=2500.0,
+        nu_fill=0.5,
+        nu_cs=0.6,
+        s_m=1e-3,
+        alpha=0.3,
+        fluid_alpha_max=0.3,
+        fluid_alpha_epsilon=0.01,
+        fluid_min_void_fraction=0.05,
+        fluid_drag_min_particle_size=1e-6,
+        fluid_inlet_velocity=0.4,
+        fluid_probability_policy="error",
+        fluid_coupling=True,
+        P_stab=0.5,
+        boundary_mask=np.zeros((6, 8), dtype=bool),
+    )
+    values.update(overrides)
+    return SimpleNamespace(**values)
+
+
+class TestFluidCoupling(unittest.TestCase):
+    def test_paper_case_one_config_uses_cell_centred_mesh_and_one_particle(self):
+        config = Path(__file__).parents[1] / "json" / "hgfd_paper_case1_0p5mm.json5"
+        with config.open() as handle:
+            _, p = params.load_file(handle)
+        p.update_before_time_march(None)
+        s = initial.IC(p)
+
+        self.assertAlmostEqual(p.dx, 0.05)
+        self.assertAlmostEqual(p.dy, 0.05)
+        self.assertAlmostEqual(p.y[0], 0.025)
+        self.assertAlmostEqual(p.y[-1], 0.625)
+        self.assertAlmostEqual(p.W, 0.75)
+        self.assertEqual(np.count_nonzero(np.isfinite(s)), 1)
+        self.assertEqual(s[p.single_particle_i, p.single_particle_j, p.single_particle_k], p.s_m)
+
+    def test_drag_is_zero_in_voids_and_stronger_for_small_grains(self):
+        p = parameters(nx=1, ny=1, nm=2, boundary_mask=np.zeros((1, 1), dtype=bool))
+        s = np.array([[[5e-4, 1e-3]]])
+        zeros = np.zeros_like(s)
+        fluid_v = np.ones((1, 1))
+        layer_beta, cell_beta = fluid.drag_coefficients(s, zeros, zeros, np.zeros((1, 1)), fluid_v, p)
+        self.assertGreater(layer_beta[0, 0, 0], layer_beta[0, 0, 1])
+        self.assertGreater(cell_beta[0, 0], 0)
+
+        s[0, 0, 0] = np.nan
+        layer_beta, _ = fluid.drag_coefficients(s, zeros, zeros, np.zeros((1, 1)), fluid_v, p)
+        self.assertEqual(layer_beta[0, 0, 0], 0)
+
+    def test_layer_drag_uses_paper_internal_coordinate_normalization(self):
+        p = parameters(nx=1, ny=1, nm=4, boundary_mask=np.zeros((1, 1), dtype=bool))
+        s = np.array([[[p.s_m, p.s_m, np.nan, np.nan]]])
+        zeros = np.zeros_like(s)
+        layer_beta, cell_beta = fluid.drag_coefficients(s, zeros, zeros, np.zeros((1, 1)), np.ones((1, 1)), p)
+        self.assertAlmostEqual(layer_beta[0, 0, 0], layer_beta[0, 0, 1])
+        self.assertAlmostEqual(np.sum(layer_beta[0, 0]), cell_beta[0, 0])
+        self.assertAlmostEqual(layer_beta[0, 0, 0], cell_beta[0, 0] / 2.0)
+
+    def test_single_particle_terminal_velocities_match_paper_case_one(self):
+        # Fig. 4 model curves, evaluated from Eqs. (7)--(14) using the
+        # per-coordinate mass rho_p/M implicit in the layer formulation.
+        cases = (
+            (0.5e-3, 2560.0, 0.07425),
+            (1.5e-3, 2560.0, 0.21144),
+            (2.0e-3, 2480.0, 0.26153),
+        )
+        for diameter, density, expected_terminal_speed in cases:
+            p = parameters(
+                nx=1,
+                ny=1,
+                nm=100,
+                dt=1e-4,
+                gas_density=997.0,
+                gas_viscosity=1e-3,
+                solid_density=density,
+                fluid_inlet_velocity=0.0,
+                boundary_mask=np.zeros((1, 1), dtype=bool),
+            )
+            s = np.full((1, 1, p.nm), np.nan)
+            s[0, 0, 0] = diameter
+            state = fluid.initialize(p, s)
+            u = np.zeros_like(s)
+            v = np.zeros_like(s)
+            for _ in range(5000):
+                u, v = fluid.update_particle_velocities(u, v, s, state, p)
+            self.assertAlmostEqual(abs(v[0, 0, 0]), expected_terminal_speed, delta=5e-5)
+
+    def test_upward_gas_accelerates_particle_upward(self):
+        p = parameters(nx=1, ny=1, nm=1, dt=1e-3, boundary_mask=np.zeros((1, 1), dtype=bool))
+        s = np.array([[[2e-4]]])
+        state = fluid.initialize(p, s)
+        state.v.fill(2.0)
+        u, v = fluid.update_particle_velocities(np.zeros_like(s), np.zeros_like(s), s, state, p)
+        self.assertGreater(v[0, 0, 0], 0)
+        self.assertEqual(u[0, 0, 0], 0)
+
+    def test_drag_reaction_opposes_upward_gas(self):
+        p = parameters(nx=1, ny=1, nm=2, boundary_mask=np.zeros((1, 1), dtype=bool))
+        s = np.full((1, 1, 2), p.s_m)
+        state = fluid.initialize(p, s)
+        state.v.fill(1.0)
+        zeros = np.zeros_like(s)
+        _, _, _, _, source_v = fluid.coupling_fields(s, zeros, zeros, state, p)
+        self.assertLess(source_v[0, 0], 0)
+
+    def test_exhausted_momentum_budget_is_excluded_from_solid_velocity_only(self):
+        p = parameters(nx=1, ny=1, nm=2, boundary_mask=np.zeros((1, 1), dtype=bool))
+        s = np.full((1, 1, 2), p.s_m)
+        state = fluid.initialize(p, s)
+        particle_u = np.zeros_like(s)
+        particle_v = np.array([[[0.25, 1.0]]])
+        p.hgfd_virtual_displacement = np.array([[[0.0, p.dx]]])
+        beta, _, solid_v, _, _ = fluid.coupling_fields(s, particle_u, particle_v, state, p)
+        self.assertAlmostEqual(solid_v[0, 0], 0.25)
+        beta_without_budget, *_ = fluid.coupling_fields(
+            s, particle_u, particle_v, state, parameters(nx=1, ny=1, nm=2)
+        )
+        self.assertAlmostEqual(beta[0, 0], beta_without_budget[0, 0])
+
+    def test_minimum_fluidization_velocity_balances_dense_drag(self):
+        p = parameters()
+        superficial = fluid.minimum_fluidization_velocity(p)
+        phi = p.nu_fill
+        eps = 1.0 - phi
+        relative = superficial / eps
+        beta = (
+            150 * p.gas_viscosity * phi**2 / (eps * p.s_m**2) + 1.75 * p.gas_density * phi * relative / p.s_m
+        )
+        drag = beta * relative
+        weight = phi * (p.solid_density - p.gas_density) * p.g
+        self.assertAlmostEqual(drag / weight, 1.0, places=10)
+
+    def test_projection_reduces_weighted_divergence(self):
+        p = parameters(fluid_inlet_velocity=0.0)
+        rng = np.random.default_rng(4)
+        void_fraction = np.full((p.nx, p.ny), 0.65)
+        u = rng.normal(scale=0.05, size=(p.nx, p.ny))
+        v = rng.normal(scale=0.05, size=(p.nx, p.ny))
+        before = np.linalg.norm(fluid.weighted_divergence(u, v, void_fraction, p.dx, p.dy))
+        u, v, _, divergence = fluid.project_velocity(u, v, np.zeros_like(u), void_fraction, p)
+        self.assertTrue(np.all(np.isfinite(u)))
+        self.assertTrue(np.all(np.isfinite(v)))
+        self.assertLess(np.linalg.norm(divergence), before)
+        self.assertLess(np.max(np.abs(divergence)), 1e-8)
+
+    def test_uniform_flow_is_unchanged_by_conservative_transport(self):
+        p = parameters(fluid_inlet_velocity=0.3)
+        s = np.full((p.nx, p.ny, p.nm), np.nan)
+        state = fluid.initialize(p, s)
+        momentum_u, momentum_v = fluid._conservative_transport(state, state.void_fraction, p)
+        np.testing.assert_allclose(momentum_u, state.void_fraction * state.u)
+        np.testing.assert_allclose(momentum_v, state.void_fraction * state.v)
+
+    def test_probability_limit_is_strictly_less_than_half(self):
+        p = parameters(nx=1, ny=2, nm=1, dx=1.0, dy=1.0, dt=1.0, fluid_alpha_max=0.0)
+        p.boundary_mask = np.zeros((1, 2), dtype=bool)
+        p.fluid_state = fluid.initialize(p, np.array([[[np.nan], [p.s_m]]]))
+        s = np.array([[[np.nan], [p.s_m]]])
+        u = np.zeros_like(s)
+        v = np.zeros_like(s)
+        fixed_v = np.array([[[0.0], [-0.5]]])
+        with patch.object(fluid, "update_particle_velocities", return_value=(u, fixed_v)):
+            with self.assertRaisesRegex(ValueError, "not below P_stab"):
+                hgfd.move_voids(u, v, s, p)
+
+    def test_realised_swap_carries_virtual_displacement_deficit(self):
+        p = parameters(
+            nx=1,
+            ny=2,
+            nm=1,
+            dx=1.0,
+            dy=1.0,
+            dt=0.1,
+            fluid_alpha_max=0.0,
+        )
+        p.boundary_mask = np.zeros((1, 2), dtype=bool)
+        s = np.array([[[np.nan], [p.s_m]]])
+        p.fluid_state = fluid.initialize(p, s)
+        p.hgfd_virtual_displacement = np.array([[[0.0], [1.2]]])
+        u = np.zeros_like(s)
+        fixed_v = np.array([[[0.0], [-0.2]]])
+        with patch.object(fluid, "update_particle_velocities", return_value=(u, fixed_v)):
+            with patch("numpy.random.random", return_value=np.zeros_like(s)):
+                _, _, moved_s, *_ = hgfd.move_voids(u, np.zeros_like(s), s, p)
+
+        self.assertTrue(np.isfinite(moved_s[0, 0, 0]))
+        self.assertTrue(np.isnan(moved_s[0, 1, 0]))
+        self.assertAlmostEqual(p.hgfd_virtual_displacement[0, 0, 0], 0.22)
+
+    def test_coupled_step_conserves_solid_slots(self):
+        p = parameters()
+        s = np.full((p.nx, p.ny, p.nm), np.nan)
+        s[:, :3, :5] = p.s_m
+        p.fluid_state = fluid.initialize(p, s)
+        u = np.zeros_like(s)
+        v = np.zeros_like(s)
+        last_swap = np.zeros_like(s)
+        count_before = np.sum(np.isfinite(s))
+
+        u, v, s, _, _, _, _ = hgfd.move_voids(u, v, s, p, c=None, T=None, chi=None, last_swap=last_swap)
+        p.fluid_state = fluid.advance(p.fluid_state, s, u, v, p)
+        self.assertEqual(np.sum(np.isfinite(s)), count_before)
+        self.assertTrue(np.all(np.isfinite(p.fluid_state.v)))
+        self.assertLessEqual(p.max_transition_probability, p.P_stab)
+
+
+if __name__ == "__main__":
+    unittest.main()