Electroconvection in a dielectric liquid between two concentric half-cylinders with rigid walls: Linear and nonlinear analysis

We study the linear stability and nonlinear behavior of the electroconvection between two concentric half-cylinders with no-slip conditions on all boundaries. The no-slip condition makes impossible to apply the standard modal approach. Hence, we apply a finite element technique similar to the one we have used in a previous paper about the electroconvection in a rectangular enclosed domain. When compared to the classical problem of electroconvection between two full concentric cylinders, the linear criterion is higher, due to the viscous shear introduced by the lateral walls. As a consequence, the structure of the eigenmodes is very different. There is a repulsion between modes with the same symmetry, forcing pairs of modes to cross each other repeatedly. For inner injection and small value of the ratio between the inner and outer radii the no-slip condition changes the nature of the bifurcation from subcritical to supercritical, while it is always subcritical for outer injection. To understand this behavior, we perform a modal analysis using the eigenfunctions obtained from the linear stability analysis as modal basis. We show that the supercritical branch is originated by the nonorthogonality of the modes when no-slip boundary conditions are imposed. These mechanism explains the previously unexplained appearance of the supercritical branch in the enclosed rectangular configuration.

Figure 1

Computational domain with dimensional boundary conditions for inner injection. For outer injection the boundary conditions on the electric potential are interchanged and the boundary condition for the charge density is applied at the outer electrode.

Figure 2

Computational domain with nondimensional boundary conditions for inner injection. For outer injection the boundary conditions on the electric potential are interchanged and the boundary condition for the charge density is applied at the outer electrode.

Figure 3

Linear threshold computed with Finite Elements (FE) and modal analysis for inner and outer injection for $C=10$. The results computed with FE for rigid walls are also plotted. As $\mathrm{\Gamma }$ approaches 1, the value of $T_c$ goes to the critical value corresponding to the plane case, $T_c^p$, for all the cases. The number of rolls corresponding at each hump is indicated for the inner injection branch in the free walls case.

Figure 4

Eigenvalues of the first four modes vs. $\mathrm{\Gamma }$ for the free walls case for inner injection (a) and outer injection (b). The numbering in the legend corresponds to the number of rolls observed in the half-cylinders configuration. Only a portion of each mode is showed.

Figure 5

Stream function contour plots of the eigenfunctions of the most unstable modes for $\mathrm{\Gamma }=0.05$ (a) and $\mathrm{\Gamma }=0.15$ (b) for the free-walls case. The number of rolls is related with the number $m$ of the corresponding mode in the modal expansion.

Figure 6

First four eigenvalues as a function of $\mathrm{\Gamma }$ for rigid walls. Plot (a) corresponds to inner injection, and plot (b) to outer injection. The number of rolls of the dominant mode is showed up to four.

Figure 7

Stream function contour plots of the eigenfunctions of the most unstable modes for $\mathrm{\Gamma }=0.05$ (a) and $\mathrm{\Gamma }=0.20$ (b) for the rigid walls case and inner injection. The first one is antisymmetric and the second one is symmetric. The counter rolls in the corners appear because of the no-slip boundary conditions for velocity.

Figure 8

Linear stability criteria as a function of $\mathrm{\Gamma }$ for $C=10$ for inner (a) and outer (b) injection in the case of rigid walls. The discrete points correspond to values computed from the numerical simulations, while the solid lines have been computed with the FEM-based stability analysis.

Figure 9

Bifurcation diagrams for inner injection, rigid walls, $\mathrm{\Gamma }=0.5$(a) and $\mathrm{\Gamma }=0.05$(b). The bifurcation is subcritical in the first case and supercritical in the second one.

Figure 10

Electric charge density for rigid walls, inner injection, $\mathrm{\Gamma }=0.05$ and points $B$(a), $C$(b), and $D$(c) in Fig. 9. Point $B$ is on the supercritical branch and points $C$ and $D$ are on the subcritical branch.

Figure 11

Critical values of parameter $T$. $T_{c1}$ is the linear criteria, $T_{c2}$ is the value where the system jumps to the nonlinear branch, $T_f$ is the minimum value to have motion of the fluid. The solid line shows the linear criteria computed with FEM. $T_{c2}$ only exists for small values of $\mathrm{\Gamma }$.

Figure 12

Electric charge density (1) and velocity field (2) for $C=10, \mathrm{\Gamma }=0.05, M=60$: (a) free case with $T=75$, (b) rigid walls with $T=98$, (c) rigid walls with $T=105$.

Figure 13

Bifurcation diagram for outer injection, rigid walls, $\mathrm{\Gamma }=0.05$. The bifurcation is subcritical for all the explored values of $\mathrm{\Gamma }$.

Figure 14

Charge density distribution and streamlines for outer injection, $T=345$ (a) and $T=600$ (b). The axis with the $z$ label in the plots corresponds to the value of the charge density.

Figure 15

Amplitudes of the time evolution of the first eight modes for inner injection with $\mathrm{\Gamma }=0.05, C=10$, and $M=60$. Plot (a) corresponds to free walls and $T=70$, plot (b) to rigid walls and $T=95$, and plot (c) to rigid walls and $T=105$. The vertical axis are plotted in logarithmic scale.

Figure 16

Amplitudes of the time evolution of the eight first modes for outer injection with $\mathrm{\Gamma }=0.05, C=10$ and $M=60$. Plot (a) corresponds to free walls and $T=500$ and plot (b) to rigid walls and $T=600$. The vertical axis are plotted in logarithmic scale.