Research Journal of Engineering and Technology

Sanjeev Kumar ¹, Veena Sharma ², Kamal Kishor ³, G. C. Rana ⁴

Affiliations
1 Department of Mathematics, Govt. College, Joginder Nagar, Distt. Mandi, (H.P.), IN
2 Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla, IN
3 Department of Mathematics, Govt. Senior Secondary School, Sunder Nagar, IN
4 Department of Mathematics, Govt. College, Naduan, Distt. Hamirpur (H.P.), IN

Abstract

The influence of viscosity, viscoelasticity and medium permeability on the stability of stratified Rivlin-Ericksen viscoelastic fluid is examined for viscoelastic polymeric solutions in the simultaneous presence of a uniform horizontal magnetic field ⃗H (H, 0, 0) and uniform horizontal rotation Ω(Ω, 0, 0). These solutions are known as Rivlin-Ericksen fluids and their rheology is approximated by the Rivlin-Ericksen constitutive relations, proposed by Rivlin and Ericksen [13]. The effects of Coriolis forces on the stability is chosen along the direction of the magnetic field and transverse to that of the gravitational field g (o,o,-g). The system is found to be stable for all wave numbers for stable stratifications and unstable for unstable stratifications for the stratifications in density, viscosity, viscoelasticity, medium permeability and medium porosity. The system can be completely stabilized by large enough magnetic field, which was unstable in the absence of magnetic field; provided the initial configuration is top-heavy density wise. The kinematic viscosity and kinematic viscoelasticity have damping effects on the growth rates with the increase in kinematic viscosity and kinematic viscoelasticity, respectively, for a fixed wave number. The medium permeability has enhancing effects on the growth rates with its increase for a fixed wave number. The above results have also been shown graphically.

Keywords

Rivlin-Ericksen Fluid, Magnetic Field, Rotation, Viscosity, Viscoelasticity, Porous Medium, AMS Classification Number:76A10.

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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, Anukampa Thakur ¹, Priti Bala ¹

Affiliations
1 Department of Mathematics & Statistics, H.P. University, Shimla-171 005, IN

Abstract

A study has been made of the instability of an electrically conducting stratified viscoelastic Rivlin-Ericksen fluid/plasma arranged in a horizontal strata to include a variable horizontal magnetic field. Applying the linear stability theory and normal mode technique to a set of partial differential equations, the solutions of an eigen value problem have been obtained by assuming exponential vertical stratifications in density, neutral gas density, viscosity, viscoelasticity and magnetic field. Using these solutions, the dispersion relation so obtained has been analysed numerically. It has been found that viscosity, viscoelasticity , collisional frequency and magnetic field have stabilizing effects on the growth rates for unstable configuration under certain wave-number band.

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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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