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Inertial and coriolis effects on oscillatory flow in a horizontal dendrite layer

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Abstract

Flow instability due to oscillatory modes of disturbances in a horizontal dendrite layer during alloy solidification is investigated under an external constraint of rotation. The flow in the dendrite layer, which is modeled as flow in a porous layer and with the inertial effects included, is assumed to rotate about the vertical axis at a constant angular velocity. The investigation is an extension of the work in Riahi (On stationary and oscillatory modes of flow instablity in a rotating porous layer during alloy solidification. J. Porous Media, 6, 177–187, 2003), which was for the case in the absence of the inertial effects. Results of the stability analyses indicate, in particular, that the Coriolis effect can enhance the physical domain for the oscillatory flow, while the inertial effect tends to reduce such domain. Sufficiently strong inertial effect can eliminate presence of the oscillatory mode only for the rotation rate beyond some value. The effect of interaction between the local volume fraction of solid and the flow associated with the Coriolis term was found to be stabilizing.

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Abbreviations

a :

Horizontal wave number

a :

Horizontal wave number vector

a 1 :

x-Component of a

a 2 :

y-Component of a

a c :

Critical a

C :

Scaled concentration ratio

\({\tilde{C}}\) :

Dimensional compositon

C 0 :

Far field composition

C e :

Eutectic composition

C l :

Specific heat per unit volume

C r :

A concentration ratio

C s :

Composition of dendrites

d :

Dendrite layer thickness

g :

Acceleration due to gravity

G :

1 + S/C

G t :

(G−1)/(CG 2)

i :

Pure imaginary number

K :

A permeability reciprocal

K 1 :

A permeability parameter

k :

Thermal diffusivity

k s :

Solute diffusivity

L :

An inertial parameter

L a :

Latent heat of solidification

\({\tilde{L}}\) :

An inertial parameter

M :

Liquidus slope

P :

Scaled modified pressure

\({\tilde{P}}\) :

Modified pressure

P 0 :

A constant

P B :

Modified basic pressure

Q :

Rotation rate

R :

Scaled Rayleigh number

\({\tilde{R}}\) :

Rayleigh number

R c :

Critical R

S :

Scaled Stefan number

S t :

Stefan number

t :

Scaled time variable

T :

Coriolis parameter

\({\tilde{t}}\) :

Time variable

T e :

Eutectic temperature

T L :

Liquidus temperature

T ∞ :

Far field temperature

u :

Scaled Darcy’s velocity vector

U :

Velocity vector

\({{\bf \tilde{u}}}\) :

Darcy’s velocity vector

\({\tilde{u}}\) :

x-Component of u

W :

Poloidal function for u

V :

Solidification speed

\({\tilde{v}}\) :

y-Component of u

\({\tilde{w}}\) :

z-Component of u

x :

A scaled horizontal variable

x :

Unit vector along x-axis

\({\tilde{x}}\) :

A horizontal variable

y :

A scaled horizontal variable

y :

Unit vector along y-axis

\({\tilde{y}}\) :

Another horizontal variable

z :

Scaled vertical variable

z :

Unit vector along z-axis

\({\tilde{z}}\) :

Vertical variable

α* :

Themal expansion coefficient

β* :

Solute expansion coefficient

β:

β*−Mα*

ΔC :

C 0–C e

ΔT :

T L (C 0)−T e

Δ:

Horizontal Laplacian operator

δ:

Dimensionless depth of the porous layer

\({\nabla}\) :

Gradient operator

\({{\tilde{\theta}}}\) :

Temperature

θB :

Basic temperature

θ:

Perturbation Temperature

θ∞ :

T ∞/ΔT

ν:

Kinematic viscosity

Π:

Permeability

Π(0):

A reference permeability

\({\phi}\) :

Perturbation to solid fraction

\({\phi_{\rm B}}\) :

Basic solid fraction

\({{\tilde{\phi}}}\) :

Local volume fraction of solid

ρ0 :

A reference density

σ:

Complex growth rate of disturbance

σ i :

Disturbance frequency

σr :

Real growth rate of disturbance

ψ :

Toroidal function for u

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Authors and Affiliations

  1. Department of Mathematics, University of Texas-Pan American, 1201W. University Dr., Edinburg, Texas, 78541-2999, USA

    D. N. Riahi

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  1. D. N. Riahi
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Correspondence to D. N. Riahi.

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Riahi, D.N. Inertial and coriolis effects on oscillatory flow in a horizontal dendrite layer. Transp Porous Med 69, 301–312 (2007). https://doi.org/10.1007/s11242-006-9091-2

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  • DOI: https://doi.org/10.1007/s11242-006-9091-2

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