Instability of electroconvection in viscoelastic fluids induced by strong unipolar injection between two coaxial cylinders

In this paper, a detailed two-dimensional numerical study on the nonlinear behaviors of electrohydrodynamic flows of Oldroyd-B viscoelastic dielectric liquid between two coaxial cylinders is conducted. The liquid is subjected to strong unipolar injection from inner annulus. The entire set of coupled equations, including the Navier-Stokes equations, simplified Maxwell's equations, and constitutive equations, is solved by the finite-volume method. Detailed analyses of elastic effects on the flows are presented with corresponding explanation. A bifurcation mode, namely supercritical bifurcation, has been found comparing to the subcritical bifurcation of Newtonian fluids. The linear stability criteria and nonlinear ones corresponding to the onset and stop of the flow motion, respectively, are presented under different elastic conditions. Comparing to Newtonian fluids, the linear stability criteria do not change at low Weissenberg number and decline when Weissenberg number is more than 1, and the nonlinear criteria are larger than that of Newtonian fluids. In addition, viscoelastic fluids show more dynamic behaviors. For example, two charge-void regions rotate in the domain because of azimuthal stress. Due to the first principal normal stress, two parts of charge-void regions first rotate to each other and then are pushed apart. These behaviors are explained in detail and are closely related to the elastic effects.

Figure 1

Sketch of the physical domain and boundary conditions.

Figure 2

Comparison of velocity profiles in lid-driven cavity along the lines $x=0$.5 and $y=0.75$ at time $t=8$ for $Wi=2.0$ and $\beta =0$.5.

Figure 3

Charge-density profile obtained in the hydrostatic regime under different grid sizes, computed numerically and compared to the analytical solution.

Figure 4

Temporal evolutions of the maximum velocity in the logarithmic coordinate, $Wi=0.1$.

Figure 5

Hysteresis loop represented by the maximum velocity $v_{\max }$, $Wi=0.1$.

Figure 6

Distributions of (a) charge density and (b) stream function and velocity field with $T=130$, $Wi=0.1$.

Figure 7

Distributions of first principal normal stress with $T=130$, $Wi=0.1$. Note: we show the contour line of charge density where $q\le 0.05$, which represents the charge-void regions.

Figure 8

Bifurcation diagram for $Wi=0.4$ represented by the maximum velocity $v_{\max }$.

Figure 9

Distributions of charge density and stream function with temporal evolution of $v_{\max }$ for $Wi=0.4$ (a) $T=130$ at $t=810$ and (b) $T=160$ at $t=250$.

Figure 10

(a) Temporal evolution of $v_{\max }$ and (b) distributions of charge density with stream function for $T=115$ when $Wi=0.4$.

Figure 11

(a) Temporal evolution of $v_{\max }$ and (b) distributions of charge density with stream function for $T=112$ when $Wi=0.4$. Note: The charge-void regions move following the direction of the blue arrow at first and turn back following the red arrow.

Figure 12

(a) Temporal evolution of $v_{\max }$ and (b) distributions of charge density with stream function for $T=110$ when $Wi=0.4$. Note: The charge-void regions rotate following the direction of the blue arrow.

Figure 13

(a) Distributions of first principal normal stress for $T=112$ and (b) elastic stress in the azimuthal direction for $T=110$ when $Wi=0.4$. Note: we show the contour line of charge density where $q\le 0.05$, which we consider as charge-void regions.

Figure 14

(a) Phase-space trajectory of the velocity at sample point A and (b) Fourier frequency spectrum of velocity at sample point A for $T=130$ when $Wi=0.4$.

Figure 15

Bifurcation diagram for $Wi=0.7$ represented by the maximum velocity $v_{\max }$.

Figure 16

(a) Bifurcation diagram for $Wi=1$ and (b) temporal evolution of $v_{\max }$ for $T=125$.

Figure 17

(a) Temporal evolution of $v_{\max }$ and (b) distributions of charge density and stream function for $T=115$, $Wi=1$.

Figure 18

Flow chart explicitly showing the change of system due to $Wi$.