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Linear stability of solutal convection in solidifying mushy layers: permeable mush–melt interface

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Abstract

The linear stability theory is used to investigate analytically the effect of a permeable mush–melt boundary condition on the stability of solutal convection in a mushy layer of homogenous permeability at the near eutectic (solid) limit. The results clearly show that, in contrast to the impermeable mush–melt interface boundary condition, the application of the permeable mush–melt interface boundary condition destabilizes the convection in a mushy layer.

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Abbreviations

Latin Symbols :

 

g * :

Acceleration due to gravity

H * :

Height of the mushy layer

K * :

Permeability function

C :

Mixture composition

c p :

Specific heat of fluid

\(\hat{\varvec{e}}_z\) :

Unit vector in the z-direction.

h fs :

Latent heat of fluid

k 0 :

Characteristic permeability

L :

Length of the mushy layer

p :

Reduced pressure

R :

Rescaled mushy layer Rayleigh number, \(\sqrt{\delta Ra_{\rm m}}\)

Ra m :

Mushy layer Rayleigh number, β* ΔCg * k 0/ (ν * V f *)

s x :

X-component of wavenumber

s y :

Y-component of wavenumber

St :

Reciprocal of Stefan number, h fs / c p ΔT

t :

Time.

T :

Dimensionless temperature, (T * − T L) / (T L − T E)

u :

Horizontal x-component of the filtration velocity

V :

Dimensionless filtration velocity vector, \(u\hat{e}_x +v\hat{e}_y +w\hat{e}_z \).

v :

Horizontal y-component of filtration velocity

V *f :

Velocity of solidifying front

w :

Vertical component of filtration velocity

X :

Space vector, \(x\hat{e}_x +y\hat{e}_y +z\hat{e}_z \)

x :

Horizontal length co-ordinate

y :

Horizontal width co-ordinate

z :

Vertical co-ordinate

Greek Symbols :

 

α :

 = scaled wavenumber, s 2 / π 2

β C :

Solutal expansion coefficient

β T :

Thermal expansion coefficient

δ :

Dimensionless depth of mushy layer

ɛ:

Convection amplitude

\(\phi\) :

Porosity

\(\varphi\) :

Solid fraction = 1- \(\phi \)

κ :

Thermal diffusivity of liquid

μ :

Dynamic viscosity of the fluid

ν :

Kinematic viscosity

\(\Pi (\varphi)\) :

Dimensionless retardability function, k 0/ K *

\(\rho\) :

Fluid density

Γ:

Slope of liquidus line

\(\psi _{mk}\) :

Kronecker delta notation

Superscripts :

 

*:

Dimensional quantities

Subscripts :

 

0:

Simplified parameters

B:

Basic flow quantities

c:

Characteristic values

cr:

Critical values

E:

Eutectic conditions

l:

Liquid conditions

m:

Mush conditions

S:

Solid conditions

st:

Stationary conditions

∞:

Far field conditions

Over :

 

− :

Rescaled quantities

~:

Unscaled quantities

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Natal, King George V Avenue, Durban, 4041, South Africa

    Saneshan Govender

Authors
  1. Saneshan Govender
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Correspondence to Saneshan Govender.

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Govender, S. Linear stability of solutal convection in solidifying mushy layers: permeable mush–melt interface. Transp Porous Med 67, 431–439 (2007). https://doi.org/10.1007/s11242-006-9034-y

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  • DOI: https://doi.org/10.1007/s11242-006-9034-y

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