Abstract
The implication of a uniform horizontal throughflow on the linear stability of buoyancy-driven convection in a vertical fluid layer is studied. The vertical boundaries of the fluid layer are rigid-permeable and holding at uniform but different temperatures. The instability to small-amplitude perturbations is tested by parameterizing the basic stationary flow through the Péclet number and the Prandtl number (representative of liquid mercury, air, water and oil). A modal analysis is performed, and the stability eigenvalue problem is solved numerically using the Chebyshev collocation method. The neutral stability curves are presented, and the critical values of the wave number, wave speed as well as the Grashof number are computed for different prescribed values of the governing parameters. The effect of horizontal throughflow on the stability of fluid flow is found to be dependent on the values of Prandtl number, and their intricacies have been discussed in detail. The results for the case of no throughflow are also highlighted as a particular case from the present study.
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Abbreviations
- a :
-
Wave number in the z-direction
- c :
-
Wave speed
- \(c_{r} \) :
-
Phase velocity
- \(c_{i} \) :
-
Growth rate
- \({\mathbf{e}}_{{\mathbf{z}}} \) :
-
Unit vector in the z-direction
- G :
-
Grashof number
- g :
-
Gravitational acceleration
- g :
-
Gravitational acceleration vector
- h :
-
Half-width of the fluid layer
- M :
-
Order of base polynomial
- N :
-
Number of collocation points
- p :
-
Pressure
- \(\mathrm{Pe}\) :
-
Péclet number.
- \(\mathrm{Pr}\) :
-
Prandtl number
- \({\mathbf{q}}=(u,v,w)\) :
-
velocity vector
- t :
-
Time
- T :
-
Temperature
- \(T_{0} \) :
-
Reference temperature
- \(T_{1} \) :
-
Temperature of the left boundary
- \(T_{2} \) :
-
Temperature of the right boundary
- \(U_{0} \) :
-
Horizontal throughflow velocity
- \(\left( {x,y,z} \right) \) :
-
Cartesian coordinates
- \(\alpha \) :
-
Volumetric thermal expansion coefficient
- \(\beta \) :
-
Temperature gradient
- \(\varepsilon \) :
-
Perturbation parameter
- \(\Theta \) :
-
Amplitude of the perturbed temperature
- \(\kappa \) :
-
Thermal diffusivity
- \(\nu \) :
-
Kinematic viscosity
- \(\rho _{0} \) :
-
Reference density at \(T_{0} \)
- \(\psi \) :
-
Stream function
- \(\Psi \) :
-
Amplitude of the perturbed stream function
- b :
-
basic solution
- c :
-
critical value
- \(\wedge \) :
-
Perturbation fields
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B.M.S. would like to thank the authorities of his University for encouragement and support. The authors are grateful to the referee for most valuable comments that brought this article to its current form.
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Shankar, B.M., Shivakumara, I.S. Instability of natural convection in a vertical fluid layer with net horizontal throughflow. Z. Angew. Math. Phys. 72, 89 (2021). https://doi.org/10.1007/s00033-021-01517-7
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DOI: https://doi.org/10.1007/s00033-021-01517-7