The direct solution of (4.11) as implemented in the Stokesian Dynamics method is computationally expensive since it involves the costlyO(N2) calculation of the far-field mobility matrix and its costly O(N3) inversion. This limits the simulation toN of the order of a hundred, whereNis the number
of particles in a unit cell with an infinite periodic array of cells. The two-body near-field resistance matrices are stored in a sparse form by including only the nonzero contributions from neighboring particles within a cutoffdistance, thus allowing the construction of the sparseR2B− R∞2Bmatrix in O(N) operations. The speed-limiting part is therefore the construction and inversion of the mobility matrix.
An alternate approach to the SD method called Accelerated Stokesian Dynamics (ASD) was developed by Sierou and Brady [2001] and reduces the computational cost to O(NlogN) opera- tions. In this approach the far-field forces are computed directly using an iterative procedure, thus foregoing the need to even construct the complete mobility matrix (M∞) because only the product of the matrix with a vector is required in the inversion procedure. The hydrodynamic force in (4.11) is split into a far-field and a near-field part as
Fh= Fhff −RFU,nf ·
Up−u∞
+RFE,nf : E, (4.33)
where the subscript ff denotes far-field andnf denotes near-field. The near-field resistance matrix in (4.33) corresponds simply to the relevant component ofR2B− R∞2Bin (4.14). The particle velocities Up−u∞
can be taken from the previous time step or computed as a nested iterative procedure.
The far-field forces (and stresslets) for each particle are computed as an iterative procedure starting with an initial guess, for which the forces in the previous time step may be used. The force and stresslet acting on each particle is distributed over a uniform 3D grid as a set of point forces using a particle-mesh (PME) technique [Hockney and Eastwood 1988]. This allows the use of Hasimoto’s solution [Hasimoto 1959] of the Stokes equation for flow past a periodic array of spheres to calculate the disturbance velocity in the fluid due to the point forces on the mesh nodes.
This is equivalent to calculating the far-field disturbance velocity due to the forces acting on the particles. Next, the disturbance velocity at the center of each particle is found by interpolating from the uniform grid and the Faxén laws are used to calculate the forces and stresslets on the particles.
This procedure is repeated until the forces converge. The new particle velocities are then found from (4.33) by iterative inversion of the sparse RFU,nf matrix. This can be accomplished inO(N) operations by making use of sparse solvers for which only theO(N) multiplication of the resistance matrix with a vector is required.
The advantage of using this technique is that Hasimoto’s solution for a periodic array of point forces involves splitting the velocity field calculation into a wave-space part that can be computed using Fast Fourier Transform (FFT) techniques, and a short-range real-space part that can be com- puted inO(N) operations, based on a splitting parameterα. The wave-space contribution is calcu- lated inO(Nm3logNm) operations whereNmis the total number of grid points in each direction. An optimum value ofαandNmcan be found that will give reasonable accuracy and keep the computa- tional cost down toO(NlogN) operations. Using this method periodic systems withNof the order of a thousand can be simulated on desktop workstations in a reasonable time with good accuracy.
4.4.1 Expansion flow in ASD
The far-field disturbance velocity due to the presence of rigid particles in an expanding fluid can be modeled simply by treating each particle as a point fluid sink with strengthS = 43πa3E, whereEis the rate of expansion in the fluid andais the radius of the particles. The mass conservation equation for the disturbance fluid velocity is now
∇·u=−X
n
Snδ(x−xn) (4.34)
where the summation is over all the particles and xndenotes the position vector for the center of particlen. One can show by solving (4.34) for a single particle that the disturbance velocity at a distancer=afrom the point sink is
v=− 1
4πr3Sx=−1 3Exa3
r3, (4.35)
which is equal to the disturbance velocity due to the presence of a rigid particle of radius ain a fluid expanding at rate E, given in (4.21). Note that in the absence of the particle the expanding fluid would be equivalent to a continuous fluid source with densityEin the volume occupied by the particle. Therefore treating the particles as a fluid source of strengthS essentially cancels out the expanding fluid in the space it is occupying and thus the rigid nature of the particle is ensured.
Each point source is distributed over the PME mesh preserving the total source strength, as is done for the force acting on a particle in ASD. The far-field disturbance velocity in the fluid due
to the point sources on the grid is calculated using Hasimoto’s technique by splitting the velocity field calculation into a wave-space part that can be computed using Fast Fourier Transform (FFT) techniques, and a short-range real-space part. Expressions for the wave-space and real-space contri- butions are derived in the appendix to this chapter. The fluid velocity due to the particle sources is added to the velocity contribution from the forces and stresslets acting on the particles and interpo- lated to find the far-field particle velocities using Faxén laws. Calculation of the pressure moment is not required for determining the disturbance flows. This is equivalent to the asymmetric nature of the mobility matrix in SD. The pressure moment is evaluated after the fact from the converged particle forces and stresslets after the iterative inversion is complete. Calculation of the far-field pressure moment was already implemented in ASD and expressions for the wave-space and real- space contribution are given in the doctoral thesis of Sierou [2002]. The near-field contribution to the pressure moment is computed from the hydrodynamic resistance functions as in the SD method.