Computational fluid dynamics method for determining the rotational diffusion coefficient of cells

Abstract

This work presents a straightforward computational method to estimate rotational diffusion coefficient $\left(D_r\right)$ of cells and particles of various size using continuum fluid mechanics theory. We calculate the torque ( $\Gamma$ ) for cells and particles immersed in fluids to find the mobility coefficient $\mu$ and then obtain $D_r$ by substituting $\Gamma$ in the Einstein relation. Geometries are constructed using triangular mesh, and the model is solved with computational fluid dynamics techniques. This method is less intensive and more efficient than widely used models. We simulate eight different particle geometries and compare the results with previous literature.

FIG. 1.

Meshes of the (A) sphere, (B) ellipsoid, (C) TMV, (D) spherocylinder (E. coli), (E) doublet, (F) triplet, (G) tetrahedron, and (H) biconcave disk (red blood cell, RBC) shapes used in this study. All meshes are re-scaled because the size difference is large among the eight particles.

FIG. 2.

A. Example of the simulation volume and a spherical particle inside. There are two sections, the static part (outer volume) and the moving part (inner part, shown in more detail in B). B. This small cylinder rotates with the particle simultaneously. Significantly simplifying computation because no new mesh updating is needed when the particle rotates. C. A slice of the velocity field in the simulation box with a spherical particle at time t = 10 s.

FIG. 3.

Torque of the particles with angular velocity $\frac{\pi }{2}$. They correspond to the particles in Fig.1: (A) sphere, (B) ellipsoid, (C) TMV, (D) E. coli, (E) doublet, (F) triplet, (G) tetrahedron, and (H) biconcave disk (RBC). The torque when the rotation just starts is unstable because the velocity field is not fully developed. Then it rapidly stabilizes. Frictional torque is computed by the data after the velocity field is fully developed. The plots are dimensionless.

FIG. 4.

Relation of the angular displacement correlation $⟨{\theta }^2⟩$ versus time interval $\Delta t$. The order of the particles is the same as that in Fig. 1 and Fig. 3. A. Sphere. B. Ellipsoid. C. TMV. D. E. coli. E. Doublet. F. Triplet. G. Tetrahedron. H. RBC.