Short-time motion of Brownian particles in a shear flow

The short-time motion of Brownian particles in an incompressible Newtonian fluid under shear, in which the fluid inertia becomes important, was investigated by direct numerical simulation of particulate flows. Three-dimensional simulations were performed, wherein external forces were introduced to approximately form Couette flows throughout the entire system with periodic boundary conditions. In order to examine the validity of the method, the mean-square displacement of a single spherical particle in a simple shear flow was calculated, and these results were compared with a hydrodynamic analytical solution that includes the effects of the fluid inertia. Finally, the dynamical behavior of a monodisperse dispersion composed of repulsive spherical particles was examined on short-time scales, and the shear-induced diffusion coefficients were measured for several volume fractions up to 0.50.

Figure 1

Schematic diagrams of a zigzag velocity flow. The arrows represent the velocity vectors of the host fluid in the flow direction. The $x$, $y$, and $z$ axes represent the flow, velocity-gradient, and vorticity directions, respectively.

Figure 2

(Color online) Mean square displacement scaled by $k_{B}T$ of a single spherical particle fluctuating in a Newtonian fluid under shear at $k_{B}T=0.07$ and $\dot{\gamma }=0.005$: (○) flow direction, (◼) vorticity direction, and (△) positional cross correlation. The analytical solution of the MSD in the flow direction $\left(x\right)$ for a Brownian particle in a shear flow: the dashed line represents the LE and the solid line represents the GLE. The dashed-dotted line is the analytical solution of the MSD in the vorticity direction $\left(z\right)$ for the GLE, which is the same form as that for the GLE of a free Brownian particle The analytical solution of the positional cross correlation for a Brownian particle in a shear flow: the dotted line represents the LE and the dashed line represents the GLE. The Brownian time ${\tau }_B\simeq 4.5$, the kinematic time ${\tau }_{\nu }=25$, and the diffusion time ${\tau }_D\simeq 3.4\times {10}^4$. The parameters for both the particle and fluid are $a=5$, $\xi =2$, ${\rho }_p=1$, ${\rho }_f=1$, and $\eta =1$. The time unit is ${\Delta }^2/\nu =1$, and the length unit is $\Delta =1$.

Figure 3

(Color online) The time evolution of the mean-square displacement in the vorticity direction $\left(z\right)$ for several volume fractions $\Phi$ at $k_{B}T=0.07$ and $\dot{\gamma }=0.005$. The Peclet number $\text{Pe}\simeq 170$ and the particle Reynolds number ${\text{Re}}_p=0.125$. The inset shows the volume fraction dependence of $D_z$ scaled by $\dot{\gamma }{a}^2$. The parameters for both the particle and fluid are $a=5$, $\xi =2$, ${\rho }_p=1$, ${\rho }_f=1$, and $\eta =1$.