4.3 Stokesian Dynamics
4.3.1 Review of the existing method
The Stokesian Dynamics (SD) technique developed by Durlofsky, Brady and Bossis [1987] success- fully accounts for both the many-body interactions and the near-field lubrication forces by splitting the hydrodynamics into a far-field mobility calculation and a pairwise additive resistance calcula- tion. This method was extended to infinite periodic suspensions [Phillips et al.1988; Brady and Bossis 1988] and has been used successfully to give accurate results for many problems where the system size is of relatively little importance. Following is a brief overview of Stokesian Dynamics.
ForNrigid particles suspended in a Newtonian fluid of viscosityηand densityρ, the motion of the fluid is governed by the Navier-Stokes equations, while the motion of the particles is described by the coupledN-body equation of motion:
m· dUp
dt = Fh+Fp, (4.11)
which simply states that the mass times the acceleration equals the sum of the forces. In (4.11), mis the generalized mass/moment-of-inertia matrix of dimensions 6N ×6N, Up is the particle translational/rotational velocity vector of dimension 6N, andFhandFpare the hydrodynamic and external force-torque vectors acting on the particles, also of dimension 6N each. When the motion on the particle scale is such that the particle Reynolds number is small, the fluid equations of motion becomes linear (Stokes equation) and the hydrodynamic forces and torques acting on the particles in a bulk linear flow can be computed directly from the instantaneous particle configuration:
Fh=−RFU·
Up−u∞
+RFE : E∞. (4.12)
Here,u∞is the translational/rotational velocity of the bulk linear flow evaluated at the particle center
andE∞is the externally imposed rate-of-strain tensor.
The long-range interactions are computed by expanding the force density on the surface of each particle in a series of moments about the center of the particle. The zeroth moment is simply the net force acting on a particle (plus a potential dipole for spherical particles), the first moment can be decomposed into the torque and the stresslet, while higher moments are neglected. This level of truncation gives the minimum set of unknowns per particle required in a bulk linear flow and has been shown to give very accurate results for many hydrodynamic problems. The relation between the far-field forces and stresslets and the far-field contribution to the particle velocities is given succinctly by Faxén laws:
Up−u∞(x)=− 1
6πηaF+ 1+ a2 6∇2
!
uf f, (4.13a)
Ωp−ω∞(x)=− 1
8πηa3T+ 1
2∇×uf f, (4.13b)
−E∞ =− 3
20πηaS+ 1+ a2 10∇2
!
ef f, (4.13c)
where the subscript f f stands for far-field. In the original implementation [Phung 1992] the particle stresslets have 5 independent components corresponding to each particle and therefore are imple- mented as a vector of size 5N in the SD algorithm. The imposed rate of strain E∞ also has 5 independent components and is implemented as a vector of size 5N. With the inclusion of expan- sion flow the rate-of-strain tensor has an additional independent component given by the rate of expansion, so it is implemented as a vector of size 6N with 6 components for each particle. There must also be a corresponding entry for the pressure moment for each prticle, defined as the trace of the stresslet tensor. Therefore the stresslets will also be implemented with a total of 6Nindependent components.
The inverse of the grand resistance matrix, known as the grand mobility matrix (M∞) relates the particle velocities and rate-of-strain relative to the bulk suspension velocity and rate-of-strain respectively, to the force and stresslet acting on each particle. In an infinite suspension the velocity field that would be present at any point in the suspension in the absence of the disturbance flows due to the presence of a particle at that point, is given by the spatial average of the velocity field over the entire suspension (the bulk suspension velocity). The mobility matrix is constructed in pairwise
additive fashion from the Faxén laws for each particle pair and is simply a restatement of the Faxén formulae for all the particles in matrix form. The mobility interactions are summed over an infinite periodic lattice of the particle configuration using the Ewald summation technique as described by Beenakker [1986]. Upon inversion ofM∞ infinite reflections among all the moments and all particles are computed [Durlofsky et al. 1987] and the far-field resistance matrix thus obtained contains the true many-body interactions. The near-field lubrication interactions, which would only be reproduced in (M∞)−1if all multipole moments were included, are added to the resistance matrix in a pairwise additive fashion to complete the grand resistance matrix:
R=(M∞)−1+R2B− R∞2B, (4.14)
whereR2Bis the matrix of exact two-body resistance interactions [Jeffrey and Onishi 1984] andR∞2B is the far-field contribution to the pair interactions from the inversion of the mobility matrix.
An evolution equation for the particle configuration is obtained by integrating (4.11) twice over a time step∆tlarger than the inertial relaxation timeτb = m/6πη0abut small compared with the time over which the configuration changes leading to particle displacements given as
∆x=n
u∞+R−1FU·
RFE : E∞+Fpo
∆t+kT∇ ·R−1FU∆t+X(t)+O(∆t), (4.15)
where X(t) is the additional displacement due to Brownian motion, and (kT∇ ·R−1FU∆t) is a deter- ministic displacement from the configurational-space divergence of theN-particle diffusivity. Thus at each time step the disturbance force on each particle due to the presence of other particles in the imposed flow is calculated and then used to determine the disturbance velocity for each particle.
The displacement due to Brownian motion is added as described later, and the particle stresslets are calculated from (4.3a), (4.3b) and (4.3c). The particle positions are updated to get the new con- figuration and the process is repeated. For more details the reader is referred to the doctoral thesis of Phung [1992] who implemented the SD algorithm in FORTRAN. The same code was modified for the current study. In an expansion/compression flow the size of the simulation cell must also be adjusted at the same rate at which the particle are coming closer so that the particle images also move closer at the same rate and the homogeneity of the infinite suspension is maintained.