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Vibrational convective instability of a binary electrolyte layer between plane horizontal electrodes
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10.1063/1.4804389
/content/aip/journal/pof2/25/5/10.1063/1.4804389
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/5/10.1063/1.4804389

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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) .

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

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

Image of FIG. 7.
FIG. 7.

Critical Rayleigh number vs. the dimensionless vibrational parameter ( ): (a) = 0.001; (b) = 0.5; (c) = 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) .

Image of FIG. 8.
FIG. 8.

Critical wave number vs. the dimensionless vibrational parameter ( ): (a) = 0.001; (b) = 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 = 0.999, the long-wave instability takes place).

Image of FIG. 9.
FIG. 9.

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) = 0.001, (3) = 0.01, (4) = 0.1, (5) = 0.5, and (6) = 0.999.

Image of FIG. 10.
FIG. 10.

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

Image of FIG. 11.
FIG. 11.

(a) Critical Rayleigh number and (b) critical wave number vs. the dimensionless vibrational parameter at negative Ra ( = 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).

Image of FIG. 12.
FIG. 12.

Critical Rayleigh number vs. the dimensionless vibrational parameter at negative Ra ( = 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).

Image of FIG. 13.
FIG. 13.

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

Image of FIG. 14.
FIG. 14.

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

Image of FIG. 15.
FIG. 15.

Parameter , 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) .

Image of FIG. 16.
FIG. 16.

Parameter , 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) .

Image of FIG. 17.
FIG. 17.

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

Tables

Generic image for table
Table I.

The coefficients in the equation for (58) .

Generic image for table
Table II.

The coefficients in the equation for (58) .

Generic image for table
Table III.

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

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/content/aip/journal/pof2/25/5/10.1063/1.4804389
2013-05-20
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Vibrational convective instability of a binary electrolyte layer between plane horizontal electrodes
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/5/10.1063/1.4804389
10.1063/1.4804389
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