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Analytical investigation of Bénard-Marangoni convection heat transfer in a shallow cavity filled with two immiscible fluids

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Abstract

The mechanism of natural and Marangoni convection in a system with two stratified fluid layers without mass transfer at the interface is investigated. The basis of the analytical solution is an assumption of parallel flow over a large portion of the system. The two cases of heat fluxes through horizontal or vertical opposite walls are considered. It is demonstrated that four different patterns of convection can be observed in the present system. The zone of occurrence of these flow patterns are specified in terms of non-dimensional parameters. Velocity and temperature distributions, stream function and Nusselt number are presented over a wide range of the governing parameters. The results obtained are explained in terms of the basic physical mechanisms that govern these flows showing many interesting aspects of the complex interaction between the buoyant and surface tension mechanisms.

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Abbreviations

A :

aspect ratio of cavity,L′/H′

a,b :

bottom heatinga=1,b=0, sidewall heatinga=0,b=1

a 1,a 2 :

constants in equations (13)

C :

temperature gradient along thex direction

G :

parameter, equation (13), related to the inverse of the Bond number

g :

gravitational acceleration

h′ i :

height of layeri

H′ :

height of cavity (h′ 1+h′ 2)

H :

constant, equation (13)

k :

thermal conductivity

K :

parameter, equation (13)

L′ :

length of cavity

Nu:

Nusselt number (1/ΔT)

Ma:

Marangoni number (Sq′H′ 2/α 1 μ 1 k 1 )

Pr:

Prandtl number,ν/α

q′ :

constant heat flux

R c :

Rayleigh number (gβq′H′ 4/ναk)

S :

surface tension gradient with respect to temperature

T :

dimensionless temperature

ΔT′ :

characteristic temperature difference,q′H′/k

ΔT :

wall to wall temperature difference atx=0

u,v :

dimensionless velocity components

x,y :

dimensionless Cartesian coordinates

\(\tilde y\) :

dimensionless coordinate (1−y)

α :

thermal diffusivity

β :

volumetric expansion coefficient

η :

dimensionless position of the interface,h′ 1/H′

\(\tilde \eta\) :

dimensionless length (1−η)

θ :

dimensionless temperature varying withy

μ :

dynamic viscosity

ν :

kinematic viscosity

ρ :

density

σ :

surface tension coefficient

ψ :

stream function

′:

dimensional quantities

-:

relative quantities (layer 2 to layer 1)

i :

in the fluid layeri (i=1, 2)

c :

critical condition

References

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Authors and Affiliations

  1. Department of Mechanical Engineering, École Polytechnique de Montréal, H3C 3A7, Montréal, Québec, Canada

    C. H. Wang & P. Vasseur

  2. Department of Aerospace and Mechanical Engineering, University of Notre Dame, 46556, Notre Dame, Indiana, USA

    Mihir Sen

Authors
  1. C. H. Wang
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  2. Mihir Sen
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  3. P. Vasseur
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Wang, C.H., Sen, M. & Vasseur, P. Analytical investigation of Bénard-Marangoni convection heat transfer in a shallow cavity filled with two immiscible fluids. Appl. Sci. Res. 48, 35–53 (1991). https://doi.org/10.1007/BF01998664

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  • DOI: https://doi.org/10.1007/BF01998664

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