Quincke rotation driven flows

Flows induced by Quincke rotation in particle suspensions are considered. Nonlinear boundary problems for the suspension velocity and particle angular velocity fields in rectangular capillaries are formulated and solved for both no-slip and free boundary conditions. Linear stability analysis shows that the critical electric field strength necessary for the development of macroscopic flow is smaller than the field strength at which spontaneous Quincke rotation of a single particle occurs. This decrease is caused by hydrodynamic synchronization of the particle rotations. In the case of free boundaries interesting intermittent behavior is observed: as the parameters governing the problem pass degenerate eigenmodes revealed by the stability analysis, the nature of the induced flow changes qualitatively.

Figure 1

Sketch of rectangular capillary with a thickness $h$. Rotations of the particles in $xy$ plane create the flow in the $\pm x$ direction [in the case of free boundaries (hatched)]. On hatched walls of the capillary nonslip (Sec. 2) or nonstress (Sec. 2) boundary conditions are applied.

Figure 2

Neutral modes at nonslip boundary conditions. $1-E^2/E_c^2=0.81b^2,w=0.1$. (a) ($n=1$), (b) ($n=2$), (c) ($n=3$), (d) ($n=4$).

Figure 3

Velocity (a) and angular velocity (b) profiles for second neutral mode; $1-E^2/E_c^2=0.81b^2,w=0.1$.

Figure 4

Velocity (a) and angular velocity (b) profiles as solutions of nonlinear boundary problem. From down to up, $1-E^2/E_c^2=(0.5,0.6,0.7)b^2,\varepsilon =b^{2}w(1-w)/4{\pi }^2,a=1,w=0.1$.

Figure 5

The vortex flow of the Quincke suspension is created by nonhomogeneous rotations of the particles in the cross section of the capillary as shown on the right side.

Figure 6

Velocity profile of vortex flow.

Figure 7

Neutral curves of layer with free boundaries ($w=0.1)$: $y=\varepsilon /b^2,x=(E^2/E_c^2-1)/b^2$.

Figure 8

Velocity profiles of two first modes. $w=0.1,\varepsilon /b^2=0.0279,(E^2/E_c^2-1)/b^2=-0.905$.

Figure 9

Neutral modes at point of degeneracy. $E^2/E_c^2-1)/b^2=-0.9049,\varepsilon /b^2=0.0279$ as calculated according the relations (17) and (18) for $w=0.1$.

Figure 10

Velocity profile of the normalized neutral mode at the degeneracy point corresponding to $w=2.1,\varepsilon /b^2=0.025,(E^2/E_c^2-1)/b^2=-0.157$. Pronounced edgelike flow is present.

Figure 11

Solutions of the nonlinear boundary problem Eqs. (14, 15, 16) for the velocity and angular velocity profiles for several values of the viscous penetration length $l_v$. Flatter profiles correspond to a larger viscous penetration length. $E^2/E_c^2=0.8,\varepsilon =0.01,a=1,d^2/l_v^2=(1,1.5,2)$. $d^2/l_v^2$ increases from down to up.