Research Journal of Engineering and Technology

Veena Sharma ¹, Abhilasha ², Sumit Gupta ³

Affiliations
1 Department of Mathematics $ Statistics, H.P.University, Shimla-5, IN
2 Govt. Degree College, Amb, Una (H.P.), IN
3 APG University, Solan (H.P.), IN

Abstract

The instability of two immiscible superposed, non-viscous, electrically conducting and counter-streaming fluids with upper fluid layer lighter than the lower fluid layer in the presence of uniform magnetic field has been investigated numerically. Using the linear theory and normal mode analysis, the exact solutions of eigen value problem have been obtained for stress free bounding surfaces and the dispersion relation so obtained has been analysed to examine the effects of magnetic field and velocity of streaming fluids on the growth rates of the unstable mode of perturbation on the physical system. It has been found that the magnetic field has a slight stabilizing effect on the system; whereas the velocity of the counter-streaming fluids has a large enough stabilizing effect towards neutral stability.

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Veena Sharma ¹, Priti Bala ¹, Sumit Gupta ¹, Radhe Shyam ¹

Affiliations
1 Department of Mathematics and Statistics, H.P. University, Shimla-5, IN

Abstract

An investigation is made on the onset of thermal instability in a triply diffusive three dimensional fluid layer in porous medium , since there are important fluid mechanical systems in which the density depends on three or more stratifying agencies having different diffusivities, in which the onset of convective instability in isothermal ternary and non-isothermal ternary fluids is termed as triple diffusion. Special attention is given to systems with k₁ >>k₂ , k₃ and p₁ >>K₂/K₃ where K_i is the molecular diffusivity of the ith component and Pis the molecular diffusivity of the ith component and p₁ is the thermal Prandtl number. The perturbation equations are analyzed in terms of normal modes after linearizing the relevant set of hydrodynamic equations to derive a characteristic equation. For the stationary convection, the stable solute gradients are found to have stabilizing effects on the system, due to which the onset of instability is postponed and the medium permeability parameter has a destabilizing effect on the system..The presence of stable solute gradients and medium permeability parameter also introduce oscillatory modes which were nonexistent in their absence. The case of overstability (with increasing amplitude of perturbations) is also considered and the sufficient conditions for the non existence of stable solutions are derivedr. These analytical results are confirmed numerically and the effects of various parameters on the stability analysis for both the cases of stationary convection and overstability are depicted graphically using the software Mathematica version 2007.

Keywords

Thermal Instability, P₁, Thermal Prandtl Number, Solute Rayleigh Numbers, Medium Permeability.

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