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For anisotropic particles (ellipsoids), we know the drag tensor of individual particles. In the grand mobility matrix, these can be inserted as the block diagonal elements of self-mobility. The off-block-diagonal elements are based on sphere-sphere coupling, and the best we can do for the coupling is to assume the ellipsoids are sphere-like. Some research has been done on collections of spheroids (see [ref]), but it appears quite complicated.

Implementation

• Construct the block-diagonal components of the mobility matrix using the self-mobility tensor
• Approximate the spheroid coupling using an effective radius
• Check to make sure the resulting mobility matrix is not singular (that is, a Cholesky decomposition is still valid)
• If there's an interface, again use the effective radius for the ellipsoid-interface coupling

References

Claeys and Brady - 1993 - Suspensions of prolate spheroids in Stokes flow. P.pdf

The grand mobility matrix should also include pairwise interactions through the interface

In the attached paper,
Eqs. B1-B3 (self interaction)
Eqs. C1-C4 (pair interaction)

Swan and Brady - 2007 - Simulation of hydrodynamically interacting particl.pdf

In the current implementation, the hydrodynamics are pair-wise interactions. The mobility matrix M is constructed pair-wise, and the velocity in over-damped systems is V = M * F. To include many-body hydrodynamics, the mobility matrix must be extended to include higher stress-lets in the fluid field expansion. Then the larger mobility matrix is inverted to obtain the grand resistance matrix, R, which is then truncated to only translational and rotational stress-lets. The truncated R is then inverted to obtain the final mobility M, which now includes many-body terms.

Implementation

• Literature review to find the equations for the higher order stress-let terms
• Construct the larger mobility matrix, M (and allow an option to not include higher order stress-lets)
• Construct the final M by inverted and truncated R, and inverted again to obtain M

References

Brady and Bossis - Stokesian Dynamics.pdf
pages 123-128

Overview

When two particle have a center-to-center distance within 4*radius, there are lubrication hydrodynamic forces. Lubrication forces are expected to be very important in bound nano-dimers. A rough outline for how to implement them:

• Use the already constructed far-field grand mobility matrix, and invert it to obtain a far-field grand resistance matrix
• Sum over all pairs of particles within 4*radius. For each pair, calculate the lubrication resistance matrix and add it to the grand resistance matrix. Additionally, the contribution of the far-field grand resistance matrix for that specific pair must be subtracted since it has now been counted twice.
• Invert the grand resistance matrix to obtain the final grand mobility matrix and use that in the dynamics.

References

Brady and Bossis - Stokesian Dynamics.pdf
Eq.(18) and the surrounding discussion for background theory

Jeffrey and Onishi - 1984 - Calculation of the resistance and mobility functio.pdf
Explicit equations for the lubrication terms in the resistance matrix

1802.08226.pdf
This is supposedly the corrected form of the equations from Jeffrey & Onishi.

2013-W(1).pdf
May have additional useful information

https://github.com/Pecnut/stokesian-dynamics
This appears to be a project that has fully implemented hydrodynamic interactions including lubrication forces. The lubrication forces are discussed explicitly here.

https://github.com/rajeshrin
Another related project

Should have a similar call to trajectory_animation and create static pictures

Is it a good idea to use a dynamic time-step for an over-damped system? The ideal time-step can be chosen at each time-step by calculating the magnitude of the maximum force vector. The time-step would need an upper limit so that the Brownian motion is not abnormally large.

Correlations with hydrodynamic interactions are correct according to the fluctuation-dissipation theorem. Also verify that they are uncorrelated without interactions. And there are single-particle correlations for anisotropic drag particles.

If a single particle is a composition of spheres (like a dimer), then only the net force and net torque effect its dynamics. Stoked would treat this as a single particle. But for certain interactions (sphere electrostatic, collisions, van-der-walls, etc.), the interactions can still be treated more precisely at the sphere-sphere level.

Add a way to tell stoked that a particle is a composite of spheres, and then use that info to obtain the correct forces from sphere-sphere interactions.

Trajectory animation does not work on macOS using the macOS backend (works with PyQT backends)