Numerical simulations of capsule deformation using a dual time-stepping lattice Boltzmann method

In this work a quasisteady, dual time-stepping lattice Boltzmann method is proposed for simulation of capsule deformation. At each time step the steady-state lattice Boltzmann equation is solved using the full approximation storage multigrid scheme for nonlinear equations. The capsule membrane is modeled as an infinitely thin shell suspended in an ambient fluid domain with the fluid structure interaction computed using the immersed boundary method. A finite element method is used to compute the elastic forces exerted by the capsule membrane. Results for a wide range of parameters and initial configurations are presented. The proposed method is found to reduce the computational time by a factor of ten.

Figure 1

The initial configuration for a spheroidal capsule in shear flow.

Figure 2

The number of multigrid iterations for solution of lid-driven cavity flow using the multigrid lattice Boltzmann method for various values of the relaxation parameter, $\gamma$.

Figure 3

(a) The spherical and (b) biconcave meshes used in this work. These discretizations have 5120 triangles composed of 2562 Lagrangian nodes.

Figure 4

Computation of the deformation is done by mapping both triangles to a common plane.

Figure 5

(a) $D_{xy}$ for various choices of the iterative tolerance, $\epsilon$. (b) $D_{xy}$ for various time steps, $\mathrm{\Delta }t$.

Figure 6

(a) The deformation parameter and (b) inclination angle for various values of Ca with $V=1$ and $E_b=0$.

Figure 7

(a) Steady-state shape of capsules for various values of Ca. (b) The trajectory of a particle for an initially spherical capsule with Ca = 0.05.

Figure 8

(a) The deformation parameter and (b) inclination angle for initially spherical capsules with $\text{Ca}=0.05$ and $V=1$ for various values of $E_b$.

Figure 9

(a) The deformation parameter and (b) inclination angle for initially spherical capsules $V=5$ and $E_b=0$ for various values of Ca.

Figure 10

(a) The deformation parameter and inclination angle for (a) $\text{Re}=0.05$, (b) $\text{Re}=0.1$, and (c) $\text{Re}=0.25$.

Figure 11

(a) The deformation parameter and (b) inclination angle for various values of Ca for a spheroid with $b/a=0.9$.

Figure 12

(a) The deformation parameter and (b) inclination angle for various values of $Ca$ for a spheroid with $b/a=0.5$.

Figure 13

(a) The trajectory of a particle on an initially spheroidal capsule with $b/a=0.9$ with Ca = 0.05. (b) The trajectory of a particle on an initially spheroidal capsule with $b/a=0.5$ with Ca = 0.05.

Figure 14

The profile of an initially biconcave capsule at (a) $kt=5$, (b) $kt=8$, (c) $kt=11$, (d) $kt=15$, (e) $kt=18$, and (f) $kt=21$. The black dot is the location of a marker point on the membrane.

Figure 15

The profile of an initially biconcave capsule at (a) $kt=4$, (b) $kt=10$, (c) $kt=16$, (d) $kt=22$, (e) $kt=28$, and (f) $kt=30$. The black dot is the location of a marker point on the membrane.

Figure 16

(a) The inclination angle for capsules with various dimensionless shear rates. (b) The inclination angle for capsules with various viscosity ratios.

Figure 17

Classification of simulations with biconcave capsules for a variety of capillary numbers and viscosity ratios.

Figure 18

(a) The speedup as a function of the shear rate, $k$. (b) The speedup as a function of the ratio $\mathrm{\Delta }t/\delta t$. (c) The speedup as a function of the ratio ${\tau }_e/\delta t$.