Numerical simulations of velocities of the set of rigid spherical particles in Newtonian fluid using the discrete network approximation

Introduction:

Many classical and novel engineering processes involving multiphase flow require to capture the overall or effective behavior of suspensions. It is known that for highly concentrated suspensions of rigid particles, the effective rheological properties (such as effective viscosity, effective permeability, effective viscous dissipation rate) exhibit a singular behavior and its understanding is a fundamental issue. Here we want to present some recent results of our study of this singular behavior.

Formulation:

We consider a mathematical model of a non-colloidal concentrated suspension of neutrally buoyant rigid particles in a Newtonian fluid. The rigid particles are modeled by spheres of equal radii. We consider an irregular (non-periodic) array of spheres. We focus on highly packed suspensions when the concentration of particles is close to maximal, which means that the distance between neighboring particles (the interparticle distance) is much smaller than their sizes. Hence, the main objective of our study is to characterize in a rigorous mathematical framework the dependence of the effective properties (effective viscous dissipation rate) of such suspensions on the irregular geometry of array of particles and applied boundary conditions on the boundary of the fluid region.

Approach:

The approach we have chosen to attack this problem is the discrete network approximation, developed first for the scalar problem (see [3-6]). Then it was generalized to the suspension problem (see [1,2,4]), developing new tools to deal with features which this vectorial problem exhibits. The idea of the discrete network approximation (DNA) approach is in replacing the original continuum problem by the discrete problem on the graph (network), vertices of which correspond to rigid particles of the suspension and edges – to the thin gaps between closely spaced adjacent particles. This network problem amounts to solving a linear system of the dimension equal to the number of particles (N) in the suspension. The unknowns of the system are the translational (3-component vector) and angular (3-component vector) velocities of rigid spheres.

Results:

Here we are presenting some numerical results of the DNA and compare them with ones obtained by the Stokesian dynamics simulations (SDS) introduced and studied in [8].

1. This set of results presents the velocities of the red sphere assuming that the velocities of its neighbors are known (given by SDS): x=[2.422, 0.075, 0.191, 0.052, 0.024, -0.523] (three of the components of the 6-component vector x of solution are the components of the translational velocity of the red particle, the other three – its angular). These results are close to velocities computed based on SDS: x0=[2.447, 0.049, 0.207, 0.079, -0.008, -0.511]. Energies of the system of particles by DNA: E=192.43 and SDS: E0=192.91.

2. Here the velocities xr, xy, xg of three particles are found (red, yellow, green, respectively) and compared with the ones xr0, xy0, xg0, obtained by SDS: xr=[2.507, -0.047, 0.148, 0.061, 0.025, -0.547], xy=[3.753, 0.163, 0.390, 4.515, -0.275, -0.056], xg=[4.516, -0.028, -0.056, 0.005, 0.068, -0.416], and xr0=[2.447, 0.049, 0.207, 0.079, -0.008, -0.511], xy0=[3.783, 0.106, 0.428, -0.057, -0.010, -0.722], xg0=[4.533, -0.280, -0.078, -0.027, 0.127, -0.308]. Energy of the systems are: based on DNA E=473.65 and based on SDS E0=483.63.

Other Results:

The initial data were given on SDS simulations of the periodic arrays of spherical particles based on 32 and 64 spheres. The designed code based on the DNA allows to compute the velocities (translational and angular) and energy of any number of "inner" particles (the red one and red, yellow, green ones in the previous examples) provided the velocities of surrounding particles (the blue ones in previous examples) are given. Other results
Here the velocities of a single inner particle surrounded by "blue" spheres or together with other two inner particles are presented. In addition, the absolute and relative errors and the energies of all systems are computed.

The above numbers show a good agreement between of the results obtained by DNA approach compared to ones obtained by SDS. We note here that the constructed discrete network is an approximation of the stationary Stokes flow, which is one of the reasons for the discrepancy with the Stokesian dynamic simulations.

References:

1.  Berlyand, L., Borcea, L., Panchenko, A.: Network approximation for effective viscosity of concentrated suspensions with complex geometry, SIAM Journal on Mathematical Analysis, 36:5, 2005, pp. 1580-1628.
2. Berlyand, L., and Panchenko, A.: Strong and weak blow up of the viscous dissipation rates for concentrated suspensions, J. Fluid Mech., 578, 2007, pp. 1–34.
3.  Berlyand, L., Gorb, Y. and Novikov A.: Discrete network approximation for highly-packed composites with irregular geometry in three dimensions, in Multiscale Methods in Science and Engineering, B. Engquist, P. Lotstedt, O. Runborg, eds., Lecture Notes in Computational Science and Engineering, 44, Springer, 2005, pp. 21-58.
4.  Berlyand L.V., Gorb Y. and Novikov A., Fictitious fluid approach and anomalous blow-up of the dissipation rate in a 2d model of concentrated suspensions, submitted to Arch. Rat. Mech. Anal., 2006, preprint available on arXiv.org
5.  Berlyand, L., Kolpakov, A.: Network approximation in the limit of small interparticle distance of the effective properties of a high contrast random dispersed composite, Arch. Rat. Math. Anal., 159:3, 2001, pp. 179-227.
6.  Berlyand, L., Novikov, A.: Error of the network approximation for densely packed composites with irregular geometry. SIAM J. Math. Anal., 34:2, 2002, pp. 385-408.
7.  Borcea, L., and Papanicolaou, G.: Network approximation for transport properties of high contrast materials, SIAM J. Appl. Math., 58:2, 1998, pp. 501-539.
8.  Sierou, A. and Brady, J.F.: Accelerated Stokesian dynamic simulations, J. Fluid Mech., 448, 2001, pp. 115–146.