Stokeslet Fluid Node#

Module: mime.nodes.environment.stokeslet.fluid_node Class: StokesletFluidNode Stability: experimental Algorithm ID: MIME-NODE-012 Version: 1.0.0 Verification Mode: Mode 2 (Independent)

Summary#

Quasi-static regularised Stokeslet boundary-element method (BEM) fluid solver for confined microrobot fluid-structure interaction. Computes drag force and torque on a rigid body either via a precomputed 6×6 resistance matrix (standalone mode) or via LU-backsubstitution against a background flow field (Schwarz coupling mode). Has no Mach-number restriction, so it remains valid at the high rotation rates where LBM-only solvers fail.

The node is one half of MIME’s hybrid low-Reynolds solver: it resolves the body drag exactly (no Mach constraint, no diffuse-band bias), while IBLBM and FVM-IBM resolve the volumetric background flow with the body removed.

Governing Equations#

Quasi-static incompressible Stokes flow,

\[ \nabla p = \mu \nabla^2 \mathbf{u}, \qquad \nabla \cdot \mathbf{u} = 0, \]

solved in boundary-integral form using the regularised Stokeslet Green’s function [@Cortez2005]:

\[ u_i(\mathbf{x}) = \frac{1}{8\pi\mu} \int_{\partial B} S^{\varepsilon}_{ij}(\mathbf{x}-\mathbf{y})\, f_j(\mathbf{y})\, dS_{\mathbf{y}}, \]

where the regularised kernel \(S^{\varepsilon}_{ij}(\mathbf{r})\) blends the singular Stokeslet with a smoothing parameter \(\varepsilon\) scaled to the surface-element spacing.

Cylinder confinement uses the analytical Liron–Shahar image correction \(G_{\text{wall}}\) [@LironShahar1978]:

\[ A_{\text{conf}} = A_{\text{body}} + G_{\text{wall}}(R_{\text{cyl}}). \]

Discretization#

  • Surface mesh of \(N_{\text{body}}\) collocation points with quadrature weights \(w_k\) (regularised Stokeslet quadrature).

  • System matrix \(A\) is dense \(3N_{\text{body}} \times 3N_{\text{body}}\). Standalone mode reduces it to a \(6\times 6\) resistance matrix by solving six rigid-body forcing problems at init.

  • Schwarz mode keeps the full \(A\) factorisation; per-step cost is one LU solve.

  • When wall_table is provided, the analytical \(G_{\text{wall}}\) is baked into \(A\) at init — there is no double-counting with a coupled LBM wall (LBM resolves the volumetric background; \(G_{\text{wall}}\) is an image correction on the body perturbation).

Implementation Mapping#

Equation Term

Implementation

Notes

Regularised Stokeslet kernel

mime.nodes.environment.stokeslet.kernel

JAX-vectorised

BEM system matrix \(A\)

mime.nodes.environment.stokeslet.bem.assemble_system_matrix

Liron–Shahar image \(G_{\text{wall}}\)

mime.nodes.environment.stokeslet.cylinder_wall_table.assemble_image_correction_matrix_from_table

Precomputed table

6×6 resistance matrix

mime.nodes.environment.stokeslet.resistance.compute_resistance_matrix

Standalone mode

Schwarz RHS / solve

mime.nodes.environment.stokeslet.fluid_node.StokesletFluidNode._update_schwarz

LU backsubstitution

Force/torque integration

mime.nodes.environment.stokeslet.bem.compute_force_torque

\(\int \mathbf{f}\,dS\), \(\int \mathbf{r}\times\mathbf{f}\,dS\)

Assumptions and Simplifications#

  1. Quasi-static Stokes flow on the body (\(\partial_t \mathbf{u}\) ignored on the BEM scale).

  2. Newtonian, incompressible fluid; single \(\mu\).

  3. Rigid body — no surface deformation.

  4. Confined mode assumes a long, straight, circular cylinder of radius \(R_{\text{cyl}}\) and warns if the body centroid is more than \(0.05\,R_{\text{cyl}}\) off-axis.

  5. No double-counting check between the wall table and a coupled LBM background — those are documented to compute different things.

  6. The BEM solve has no spatial stencil, so halo_width() returns the empty {} default — the node is pointwise and shardable, unlike IBLBM and FVM-IBM. See Node API migration.

Validated Physical Regimes#

Parameter

Verified Range

Notes

Body Reynolds

\(\le 0.1\)

Stokes regime; rotation Re can be higher

\(\kappa = a/R_{\text{cyl}}\)

\(\le 0.3\)

Sphere benchmark to <4% error

Mesh density

mean-spacing-derived \(\varepsilon\)

Robust on geodesic meshes

Known Limitations and Failure Modes#

  1. Standalone mode cannot represent pulsatile background flow or multi-robot wake coupling — those need Schwarz mode with LBM/FVM.

  2. Off-axis bodies invalidate the cylindrical wall table; the node raises if the body extends outside the cylinder and warns at moderate offsets.

  3. Dense \(3N \times 3N\) memory cost limits standalone \(N\) to a few thousand surface points on a single GPU.

  4. The Force-Coupling-Method body_force_density flux is a stub that returns zeros until the Level-3 LBM coupling is wired.

Stability Conditions#

The BEM system is solved by direct LU factorisation, so there is no explicit-timestep stability condition. Conditioning degrades for \(\varepsilon\) much smaller than the surface spacing; the node defaults \(\varepsilon = \) mean-spacing \(/2\).

State Variables#

Field

Shape

Units

Description

drag_force

(3,)

N

Hydrodynamic force on the body

drag_torque

(3,)

N·m

Hydrodynamic torque on the body

body_traction

(N, 3)

Pa

Surface traction (Schwarz mode only)

Parameters#

Parameter

Type

Default

Units

Description

mu

float

—

Pa·s

Dynamic viscosity

body_mesh

SurfaceMesh

—

—

Body surface mesh + weights

wall_mesh

SurfaceMesh | None

None

—

Optional explicit wall mesh (standalone)

interface_mesh

SurfaceMesh | None

None

—

Presence enables Schwarz mode

wall_table

WallTable | None

None

—

Liron–Shahar image table

R_cyl

float | None

None

m

Cylinder radius (required with wall_table)

epsilon

float | None

spacing/2

m

Regularisation length

Boundary Inputs#

Field

Shape

Default

Coupling Type

Description

body_velocity

(3,)

zeros

replacive

Body linear velocity [m/s]

body_angular_velocity

(3,)

zeros

replacive

Body angular velocity [rad/s]

body_orientation

(4,)

(1,0,0,0)

replacive

Body orientation quaternion

background_flow

(N, 3)

zeros

replacive

LBM velocity at body surface (Schwarz mode)

Boundary Fluxes (outputs)#

Field

Shape

Units

Description

drag_force

(3,)

N

Drag force on the body

drag_torque

(3,)

N·m

Drag torque on the body

body_traction

(N, 3)

Pa

Surface traction (Schwarz mode only)

body_force_density

(N, 3)

N/m³

Stub for Level-3 FCM coupling

MIME-Specific Sections#

Anatomical Operating Context#

Compartment

Flow Regime

Re Range

Viscosity Range

Blood vessel (confined UMR)

quasi-Stokes

0 – 0.1

1 – 4 mPa·s

Silicone vessel benchmarks

quasi-Stokes

0 – 0.1

1 mPa·s (water)

Clinical Relevance#

For confined helical UMRs, viscous drag dominates inertia and the near-wall lubrication forces set the achievable swim speed. BEM gives unbiased drag at the resolution needed for closed-loop control — surface-method accuracy here is what lets the IDE’s PID trim work without retuning per geometry.

References#

  • [@Cortez2005] Cortez et al. (2005). The method of regularized Stokeslets in three dimensions. Phys. Fluids 17(3).

  • [@LironShahar1978] Liron & Shahar (1978). Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech.

  • [@Purcell1977] Purcell (1977). Life at Low Reynolds Number.

Verification Evidence#

  • MIME-VER-stokeslet-001: sphere in unconfined flow vs. Stokes drag (<4%)

  • MIME-VER-stokeslet-002: confined sphere vs. Liron–Shahar reference solution

  • MIME-VER-stokeslet-003: rotational drag of helical UMR vs. de Jongh (2025)

  • Unit tests: tests/nodes/environment/stokeslet/

Changelog#

Version

Date

Change

1.0.0

2026-04-10

Initial implementation — standalone + Schwarz modes