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Boundary effects on the Bénard-Marangoni instability under an electric field

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Abstract

This article theoretically studies the Bénard-Marangoni instability problem for a liquid layer with a free upper surface, which is heated from below by a heating coil through a solid plate in ana.c. electric field. The boundary effects of the solid plate, which include its thermal conductivity, electric conductivity and thickness, have great influences on the onset of convective instability in the liquid layer. The stability analysis in this study is based on the linear stability theory. The eigenvalue equations obtained from the analysis are solved by using the fourth order Runge-Kutta-Gill's method with the shooting technique. The results indicate that the solid plate with a higher thermal or electric conductivity and a bigger thickness tends to stabilize the system. It is also found that the critical Rayleigh numberR c, the critical Marangoni numberM c, and the criticala.c. Rayleigh numberE ac become smaller as the intensity of thea.c. electric field increases.

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Abbreviations

a :

wave number of the small disturbance

Bi:

Biot number,hL/K

Bo:

Bond number,ρgL 2/γ

C :

Crispation number,μκ/γL

D :

differentiation with respect toz,∂/∂z

D i,j :

rate of the strain tensor in the fluid

E :

electric field

E a :

a.c. electric Rayleigh number,ε 0 E 20 β 2ΔT 2 L 2/μκ

f e :

electric force per unit volume

g :

gravitational acceleration

k :

unit vector in thez-direction

K :

thermal conductivity

L, L s :

thickness of the liquid and the solid plate, respectively

M :

Marangoni number,τΔTL/νκρ

p :

pressure

Pr:

Prandtl number,ν/κ

R:

Rayleigh number,αgΔTL 3/νκ

t :

time

T :

temperature

V :

velocity, (u, v, w)

x, y, z :

coordinates

Z :

nondimensional surface deflection

α :

thermal expansion coefficient of the fluid density

β :

thermal coefficient of the dielectric constant

ΔT, ΔT s :

difference of temperature across the liquid layer and across the solid plate, respectively

Δφ, Δφ s :

difference of electric potential across the liquid layer and across the solid plate, respectively

γ :

surface tension

ε :

dielectric constant

η :

position of the upper free surface

κ :

thermal diffusivity

μ :

viscosity of fluid

ν :

kinematic viscosity of fluid

ρ :

density of fluid

ρ e :

free charge density

τ :

surface tension gradient with respect to temperature, ∂γ/∂T

σ :

electric conductivity

φ :

electric potential

Ω r , Ω i :

real and imaginary growth rates with time

′:

nondimensional perturbed quantity

c :

critical value

0:

initial value

r :

ratio of the solid plate property to the liquid property

s :

property of the solid plate

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Authors and Affiliations

  1. Department of Applied Mathematics, National Chung Hsing University, Taichung, Taiwan, 40227, Republic of China

    Ming-I Char & Ko-Ta Chiang

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  1. Ming-I Char
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  2. Ko-Ta Chiang
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Char, MI., Chiang, KT. Boundary effects on the Bénard-Marangoni instability under an electric field. Appl. Sci. Res. 52, 331–354 (1994). https://doi.org/10.1007/BF00936836

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