^{1,a)}and A. D. Davydov

^{2}

### Abstract

The stability of a mechanical quasi-equilibrium state of a binary electrolyte layer between planar horizontal electrodes subjected to high-frequency vibration is studied theoretically. It is assumed that reversible anodic metal dissolution and cathodic reactions proceed at the layer boundaries (metal electrode surfaces). A linear analysis of the convective stability is based on a system of equations for averaged fields of hydrodynamic velocity, concentration, and electric potential. An analytical solution to the stability problem with respect to the long-wave perturbations is obtained for positive and negative Rayleigh numbers. It is shown that the Rayleigh number corresponding to the boundary of long-wave instability depends on the direction of vibration, transport properties of the solution, and vibration frequency and amplitude. Approximate analytical solutions of the problem for monotonic instability under horizontal and vertical vibrations are obtained. The stability boundaries of mechanical quasi-equilibrium state of a binary electrolyte layer are determined by solving numerically at various values of transport properties of solution, vibration direction, and electrolysis conditions. It is found that, in contrast to thermal vibrational convection, in the case of concentration vibrational convection, absolute stability cannot be reached. A range of parameters, where the long-wave perturbations are critical, is determined. The monotonic and oscillatory types of instability are analyzed and the results of approximate analytical and numerical solutions are compared and show good agreement.

This work was supported by the Russian Foundation for Basic Research, Project No. 10-03-00517.

I. INTRODUCTION

II. STATEMENT OF PROBLEM: BASIC EQUATIONS

III. LINEAR ANALYSIS OF STABILITY

A. Mechanical quasi-equilibrium

B. Stability of quasi-equilibrium state

C. Long-wave (LW) instability

D. Approximate analytical solution of the problem for monotonic instability under vertical vibration

E. Approximate analytical solution to the problem of monotonic instability under horizontal vibration

IV. NUMERICAL RESULTS

A. Stability in the absence of vibration

B. Stability under high-frequency vibration

1. Stability at positive Rayleigh numbers

2. Stability at negative Rayleigh numbers

3. Long-wave instability

4. Monotonic and oscillatory instability

V. CONCLUSIONS

### Key Topics

- Electrolytes
- 36.0
- Electrodes
- 30.0
- Boundary value problems
- 26.0
- Numerical solutions
- 26.0
- Current density
- 24.0

## Figures

Schematic layer of binary electrolyte solution confined between two horizontal electrodes: (a) arrangement of electrodes corresponding to positive Rayleigh numbers; (b) arrangement of electrodes corresponding to negative Rayleigh numbers.

Schematic layer of binary electrolyte solution confined between two horizontal electrodes: (a) arrangement of electrodes corresponding to positive Rayleigh numbers; (b) arrangement of electrodes corresponding to negative Rayleigh numbers.

Boundaries of long-wave instability of binary electrolyte solution at various values of the dimensionless vibrational parameter at positive (1) and negative (2) Rayleigh numbers.

Boundaries of long-wave instability of binary electrolyte solution at various values of the dimensionless vibrational parameter at positive (1) and negative (2) Rayleigh numbers.

Neutral curves in the absence of vibration: (1–3) A = 0.001, (4–6) A = 0.5, (7–9) A = 0.999; (1, 4, 7) , (2, 5, 8) ; (3, 6, 9) .

Neutral curves in the absence of vibration: (1–3) A = 0.001, (4–6) A = 0.5, (7–9) A = 0.999; (1, 4, 7) , (2, 5, 8) ; (3, 6, 9) .

The ranges of (a) parameter A and (b) critical Rayleigh numbers corresponding to the long-wave (LW) instability in the absence of vibration.

The ranges of (a) parameter A and (b) critical Rayleigh numbers corresponding to the long-wave (LW) instability in the absence of vibration.

(a) Critical Rayleigh number and (b) critical wave number vs. parameter A in the absence of vibration: (1) ; (2) ; (3) .

(a) Critical Rayleigh number and (b) critical wave number vs. parameter A in the absence of vibration: (1) ; (2) ; (3) .

Neutral curves in the case of vibration: ( , GV = 0.001): (a) A = 0.001, (b) A = 0.5, (c) A = 0.999; (1) α = 0, (2) α = π/8, (3) α = π/6, (4) α = π/4, (5) α = π/2, (6) GV = 0 (in the absence of vibration).

Neutral curves in the case of vibration: ( , GV = 0.001): (a) A = 0.001, (b) A = 0.5, (c) A = 0.999; (1) α = 0, (2) α = π/8, (3) α = π/6, (4) α = π/4, (5) α = π/2, (6) GV = 0 (in the absence of vibration).

Critical Rayleigh number vs. the dimensionless vibrational parameter ( ): (a) A = 0.001; (b) A = 0.5; (c) A = 0.999; (1) α = 0; (2) α = π/8; (3) α = π/6; (4) α = π/4; (5) α = π/2; (6) approximate analytical solution (57) ; (7) approximate analytical solution (64) .

Critical wave number vs. the dimensionless vibrational parameter ( ): (a) A = 0.001; (b) A = 0.5; (1) α = 0; (2) α = π/8; (3) α = π/6; (4) α = π/4; (5) α = π/2; (6) approximate analytical solution (57) ; (7) approximate analytical solution (64) (at A = 0.999, the long-wave instability takes place).

Critical wave number vs. the dimensionless vibrational parameter ( ): (a) A = 0.001; (b) A = 0.5; (1) α = 0; (2) α = π/8; (3) α = π/6; (4) α = π/4; (5) α = π/2; (6) approximate analytical solution (57) ; (7) approximate analytical solution (64) (at A = 0.999, the long-wave instability takes place).

Critical Rayleigh number vs. the vibrational parameter in the cases of (a) vertical and (b) horizontal vibration of layer: (1) the heat system; (2–6) the electrochemical system ( ); (2) A = 0.001, (3) A = 0.01, (4) A = 0.1, (5) A = 0.5, and (6) A = 0.999.

Critical Rayleigh number vs. the vibrational parameter in the cases of (a) vertical and (b) horizontal vibration of layer: (1) the heat system; (2–6) the electrochemical system ( ); (2) A = 0.001, (3) A = 0.01, (4) A = 0.1, (5) A = 0.5, and (6) A = 0.999.

(a) Critical Rayleigh number and (b) critical wave number vs. the dimensionless vibrational parameter at negative Ra (A = 0.001, ): (1) α = π/8, (2) α = π/6, (3) α = π/4, (4) α = π/2, and (5) an approximate analytical solution.

(a) Critical Rayleigh number and (b) critical wave number vs. the dimensionless vibrational parameter at negative Ra (A = 0.001, ): (1) α = π/8, (2) α = π/6, (3) α = π/4, (4) α = π/2, and (5) an approximate analytical solution.

(a) Critical Rayleigh number and (b) critical wave number vs. the dimensionless vibrational parameter at negative Ra (A = 0.5, ): (1) α = π/8, (2) α = π/6, (3) α = π/4, (4) α = π/2, (5) an approximate analytical solution (at α = π/8, α = π/6, and α = π/4, the long-wave instability).

(a) Critical Rayleigh number and (b) critical wave number vs. the dimensionless vibrational parameter at negative Ra (A = 0.5, ): (1) α = π/8, (2) α = π/6, (3) α = π/4, (4) α = π/2, (5) an approximate analytical solution (at α = π/8, α = π/6, and α = π/4, the long-wave instability).

Critical Rayleigh number vs. the dimensionless vibrational parameter at negative Ra (A = 0.999, ): (1) α = π/8, (2) α = π/6, (3) α = π/4, (4) α = π/2, (5) an approximate analytical solution (at all values of α, the long-wave instability).

Critical Rayleigh number vs. the dimensionless vibrational parameter at negative Ra (A = 0.999, ): (1) α = π/8, (2) α = π/6, (3) α = π/4, (4) α = π/2, (5) an approximate analytical solution (at all values of α, the long-wave instability).

The neutral curves in the case of layer vibration at negative Rayleigh numbers (A = 0.001, ): (1, 2) α = π/8; (3, 4) α = π/6; (5, 6) α = π/4; (7, 8) α = π/2; (1–4) GV = 0.0002; (5–8) GV = 0.0003.

The neutral curves in the case of layer vibration at negative Rayleigh numbers (A = 0.001, ): (1, 2) α = π/8; (3, 4) α = π/6; (5, 6) α = π/4; (7, 8) α = π/2; (1–4) GV = 0.0002; (5–8) GV = 0.0003.

Negative critical Rayleigh number vs. the dimensionless vibrational parameter under the horizontal layer vibration ( ): (1) the heat system; (2–4) the electrochemical system: (2) A = 0.001, (3) A = 0.5, and (4) A = 0.999.

Negative critical Rayleigh number vs. the dimensionless vibrational parameter under the horizontal layer vibration ( ): (1) the heat system; (2–4) the electrochemical system: (2) A = 0.001, (3) A = 0.5, and (4) A = 0.999.

Parameter A, which correspond to the boundary of long-wave instability region at positive Rayleigh numbers, vs. the dimensionless vibrational parameter at various directions of vibration and various current densities: (1, 2) α = π/2; (3, 4) α = π/4; (5, 6) α = π/6; (7, 8) α = π/8; (9, 10) α = 0; (1, 3, 5, 7, 9) ; (2, 4, 6, 8, 10) .

Parameter A, which correspond to the boundary of long-wave instability region at positive Rayleigh numbers, vs. the dimensionless vibrational parameter at various directions of vibration and various current densities: (1, 2) α = π/2; (3, 4) α = π/4; (5, 6) α = π/6; (7, 8) α = π/8; (9, 10) α = 0; (1, 3, 5, 7, 9) ; (2, 4, 6, 8, 10) .

Parameter A, which correspond to the boundary of long-wave instability region at negative Rayleigh numbers, vs. the dimensionless vibrational parameter at various directions of vibration and various current densities: (1, 2) α = π/2; (3, 4) α = π/4; (5, 6) α = π/6; (7, 8) α = π/8; (1, 3, 5, 7) ; (2, 4, 6, 8) .

Parameter A, which correspond to the boundary of long-wave instability region at negative Rayleigh numbers, vs. the dimensionless vibrational parameter at various directions of vibration and various current densities: (1, 2) α = π/2; (3, 4) α = π/4; (5, 6) α = π/6; (7, 8) α = π/8; (1, 3, 5, 7) ; (2, 4, 6, 8) .

The imaginary part of decrement vs. the vibration direction at the critical Rayleigh numbers and critical wave numbers (GV = 0.001): (1–4) Ra > 0, (5–8) Ra < 0; (1, 3, 5, 7) , (2, 4, 6, 8) ; (1, 2) A = 0.5; (5, 6) A = 0.25; (3, 4, 7, 8) A = 0.001.

The imaginary part of decrement vs. the vibration direction at the critical Rayleigh numbers and critical wave numbers (GV = 0.001): (1–4) Ra > 0, (5–8) Ra < 0; (1, 3, 5, 7) , (2, 4, 6, 8) ; (1, 2) A = 0.5; (5, 6) A = 0.25; (3, 4, 7, 8) A = 0.001.

## Tables

The coefficients b n,m in the equation for (58) .

The coefficients b n,m in the equation for (58) .

The coefficients d n,m in the equation for (58) .

The coefficients d n,m in the equation for (58) .

The dependence of numerically calculated critical Rayleigh number on a degree of basic functions.

The dependence of numerically calculated critical Rayleigh number on a degree of basic functions.

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