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Influence of viscosity variation on the stationary Bénard-Marangoni instability with a boundary slab of finite conductivity

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Summary

The onset of stationary Bénard-Marangoni instability in a variable-viscosity fluid layer with a boundary slab of finite conductivity is studied. The relations of the viscosity and surface tension of the fluid with the temperature are exponential and linear, respectively. The asymptotic solutions of the long wavelength, for small values of the conductivity and thickness of the solid, are achieved and are very well compared with the numerical results. As the viscosity ratio increases, the validity of the asymptotical solutions is extended for cases of larger values of the thermal conductivity and thickness ratios.

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Abbreviations

a :

wave number

B :

viscosity parameter

Bi:

Biot number,qd l /k l

d l :

thickness of fluid layer

d r :

depth ratio,d s /d l

d s :

thickness of solid layer

D f :

thermal diffusivity of fluid

D s :

thermal diffusivity of solid layer

g :

gravitational acceleration

k:

unit vector in thez-direction

k l :

thermal conductivity of fluid

k r :

thermal conductivity ratio,k s /k l

k s :

thermal conductivity of solid layer

M:

Marangoni number, τΔT l d l /µ0 D f

P :

pressure

Pr:

Prandtl number,v 0/D f

q :

heat transfer coefficient

R:

Rayleigh number, αgΔT l d 3 l /v 0 D f

t :

time

T l :

temperature of fluid

T s :

temperature of solid plate

T 0 :

reference temperature

ω:

nondimensional velocity in thez-direction

W :

eigenfunction ofw component

x, y, z :

Cartesian coordinates

α:

thermal expansion coefficient of the fluid density

γ:

surface tension

Γ:

parameter,M/R

ΔT l :

difference of temperature across the fluid layer

θ:

perturbed temperature

Θ:

eigenfunction of temperature

μ:

dynamic viscosity of fluid

ν:

kinematic viscosity of fluid

ϱ:

density of fluid

τ:

negative rate of change of the surface tension, −∂γ/∂T l

ω:

complex growth rate of disturbances

c :

critical state

l :

property of the liquid

r :

ratio of the solid plate property to the liquid property

s :

property of the solid plate

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  1. M. -I. Char & C. -C. Chen

    Present address: Department of Applied Mathematics, National Chung Hsing University, 402, Taichung, Taiwan, ROC

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    1. M. -I. Char
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    Char, M.I., Chen, C.C. Influence of viscosity variation on the stationary Bénard-Marangoni instability with a boundary slab of finite conductivity. Acta Mechanica 135, 181–198 (1999). https://doi.org/10.1007/BF01305751

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