Modeling and numerical simulations of Brownian rodlike particles with anisotropic translational diffusion

A new kinetic macromodel based on moments of the probability distribution function is proposed to investigate the flow of rodlike Brownian particle suspensions. The rods concentration-orientation coupling is taken into account. A numerical study is presented for rods through the planar channel, with and without introducing a circular obstacle which develops a nonhomogeneous flow. To verify this macromodel, the results are compared with the solution of the associated Fokker-Planck equation taking into consideration an anisotropic translational diffusion tensor. This tensor depends on the local orientation of the rod. Low (smaller than ${10}^3$) Brownian translational Peclet number causes rod migrations across the flow streamlines.

Figure 1

Representation of ${\mathbf{A}}_2$ in an elementary surface for different concentrations and orientations. (a) Two-dimensional random rods with concentration $c_1$. (b) Perfectly aligned rods in the $x$ direction with concentration $c_1$. (c) Two-dimensional random rods with concentration $c_2$. (d) Perfectly aligned rods in the $x$ direction with concentration $c_2$.

Figure 2

FE mesh for a planar channel. BC1, laminar inflow; BC2, pressure outlet; BC3, zero-slip condition; and BC4, symmetry condition. (a) Rectangular planar channel. (b) Planar channel with circular obstacle.

Figure 3

$a_{11}$ and $a_{12}$ versus strain for the streamline along the line $y=0.9H$ in a planar channel, with $|\dot{\gamma }|=1.8/s$. Method 1 is verified with the work of Férec et al. [52]. (a) Evolution of orientation component $a_{11}$ as a function of strain $\left|\gamma \right|$. (b) Evolution of orientation component $a_{12}$ as a function of strain $\left|\gamma \right|$.

Figure 4

(a) $A_{11}/c$ and (b) $A_{12}/c$ components of the orientation tensor, ${\mathbf{A}}_2$, along the channel in a homogeneous system, where concentration equals 1 at the inlet and no rod migrations occur. (a) Distribution of the orientation component $A_{11}/c$ in the $xy$ plane of a planar channel. (b) Distribution of the orientation component $A_{12}/c$ in the $xy$ plane of a planar channel.

Figure 5

Variation of local Peclet numbers with respect to the normalized channel width in a planar channel for the global Peclet numbers ${\mathrm{Pe}}_{\perp }=10,{10}^3$ and ${\mathrm{Pe}}_r=10,{10}^3$. (a) Evolution of the local rotary Peclet number, ${\mathrm{Pe}}_{r_{\mathrm{local}}}$, with respect to the normalized channel width, $y/H$, independent on the translational Peclet number. (b) Evolution of the local translational Peclet number, ${\mathrm{Pe}}_{{\perp }_{\mathrm{local}}}$, with respect to the normalized channel width $y/H$ independent on the rotary Peclet number.

Figure 6

Orientation component $A_{11}/c$ using method 1, across the planar channel, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 7

Orientation component $A_{12}/c$ using method 1, across the planar channel, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 8

Concentration distribution $c$ using method 1, across the planar channel, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 9

Relative change in concentration with respect to the normalized channel width, $y/H$, at the outlet, for ${\mathrm{Pe}}_r={10}^3$ and ${\mathrm{Pe}}_{\perp }=10$, for the four concentration gradients $c_1, c_2, c_3$, and $c_4$.

Figure 10

Orientation component $A_{11}/c$ with respect to the normalized channel width, $y/H$, at the inlet.

Figure 11

Concentration distribution $c$ using method 1, across the planar channel, where homogeneous concentration and aligned rods at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 12

Variation of local Peclet numbers with respect to the normalized channel width at the streamline $y=0.7H$ in a planar channel with the presence of circular obstacle. (a) Evolution of the local rotary Peclet number, ${\mathrm{Pe}}_{r_{\mathrm{local}}}$, with respect to the normalized channel length, $x/H$, independent on the translational Peclet number. (b) Evolution of the local translational Peclet number, ${\mathrm{Pe}}_{{\perp }_{\mathrm{local}}}$, with respect to the normalized channel length $x/H$ independent on the rotary Peclet number.

Figure 13

Variation of local Peclet numbers with respect to the normalized channel length at $x=H$ in a planar channel with the presence of circular obstacle. (a) Evolution of the local rotary Peclet number, ${\mathrm{Pe}}_{r_{\mathrm{local}}}$, with respect to the normalized channel width, $y/H$, independent on the translational Peclet number. (b) Evolution of the local translational Peclet number, ${\mathrm{Pe}}_{{\perp }_{\mathrm{local}}}$, with respect to the normalized channel width, $y/H$, independent on the rotary Peclet number.

Figure 14

Concentration distribution $c$ using method 1, across the planar channel with the presence of circular obstacle, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 15

Orientation component $A_{11}/c$ using method 1, across the planar channel with the presence of circular obstacle, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 16

Orientation component $A_{12}/c$ orientation component using method 1, across the planar channel with the presence of circular obstacle, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 17

Concentration distribution $c$ results using a quadratic closure approximation, across the planar channel, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 18

Orientation component $A_{11}/c$ results using a quadratic closure approximation, across the planar channel, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 19

Concentration distribution $c$ results using the IBOF closure approximation, across the planar channel, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 20

Orientation component $A_{11}/c$ results using the IBOF closure approximation, across the planar channel, where homogeneous concentration and random orientation of the rods are prescribed at the inlet for (a) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }={10}^3$; (b) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }={10}^3$; (c) ${\mathrm{Pe}}_r=10,{\mathrm{Pe}}_{\perp }=10$; and (d) ${\mathrm{Pe}}_r={10}^3,{\mathrm{Pe}}_{\perp }=10$.

Figure 21

FE mesh for components 1 and 2 in method 1. BC1, creeping inflow; BC2, No slip condition; BC3, pressure outlet; BC4, symmetry condition; and BC5, Dirichlet condition for $\mathrm{\Psi }$. (a) Component 1. (b) Component 2.

Figure 22

FE mesh for model 2 BC2: No slip condition, BC3: pressure outlet, BC4: symmetry condition, BC6: creeping inflow $+$ Dirichlet BC for ${\mathbf{A}}_2$.

Figure 23

Percentage of the numerical errors calculated from the normalization of $\mathrm{\Psi }$, for the case of planar channel with homogeneous concentration and planar random orientation at the inlet as a function of Peclet numbers.