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Dhiman, Joginder Singh
- On the Effects of Magnetic Field and Temperature-Dependent Viscosity on the Onset Magnetoconvection for General Boundary Conditions
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1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
Source
Research Journal of Science and Technology, Vol 5, No 1 (2013), Pagination: 104-109Abstract
In the present paper, the problem of thermal instability of an electrically conducting fluid layer heated from below and permeated with a uniform vertical magnetic field is studied for all combinations of rigid and dynamically free boundary conditions. The effect of temperature-dependent viscosity on the onset of hydromagnetic thermal convection is investigated both analytically and numerically. The validity of the principle of exchange of stabilities for this general problem has been investigated using the Pellew and Southwell’s method and a sufficient condition for the validity of this principle is also derived. The values of the Rayleigh numbers for each case of boundary combinations are obtained numerically using Galerkin technique. Further, the effect of temperature-dependent viscosity on the onset of stationary convection and consequently on the celebrated ∏2 Q-law of Chandrasekhar for each case of boundary combinations is computed numerically. It is observed that the temperature-dependent viscosity also has the inhibiting effect on the onset of convection as that of magnetic field and the ∏2 Q -law is also valid for this problem.
References
- Banerjee, M. B., Gupta, J. R., Shandil, R. G. and Jamwal, H.S. (1989): Settlement of the long Standing Controversy in Magnetothermoconvection in favour of S. Chandrasekhar, J. Math. Anal. Appl., 144, 356.
- Banerjee, M. B., Shandil, R.G. and Kumar, R. (1995) : On Chandrasekhar’s . –Law, J. Math. Anal. Appl., 191, 460.
- Banerjee, M.B. and Bhowmick, S.K. (1992): Salvaging the Thompson-Chandrasekhar Criterion: A tribute to S. Chandrasekhar, J. Math. Anal. Appl., 167, 57-65.
- Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, Amen. House London, E.C.4, 1961.
- Dhiman, J. S. and Kumar Vijay, (2012): On 2 p Q -Law in Magnetoconvection Problem for General Nature of Boundaries Using Galerkin Method, Research J. Engineering and Technology, 3(2), (2012), 186.
- Finlayson, B.A. (1972): ‘The Method of Weighted Residuals and Variational Principles’; Academic Press, New York.
- Gupta, J.R. and Kaushal, M.B. (1988): Rotatory hydromagnetic double-diffusive convection with viscosity variation, J. Math Phys. Sci., 22 (3), 301.
- Jeffreys, H., Some Cases of Instability in Fluids Motions, Proc. Roy. Soc. London, 1928, A118, 195.
- Kumar, Vijay (2012): A Study of Some Convective Stability Problems with Variable Viscosity, Ph.D. thesis (supervised by J.S. Dhiman), submitted to Himachal Pradesh University, Shimla.
- Low, A. R, On the Criterion for Stability of a Layer of Viscous Fluid Heated From Below, Proc. Roy. Soc. London, 1929, A125, 180.
- Pellew, A. and Southwell, R.V., On Maintained Convective Motion in a Fluid Heated From Below, Proc. Roy. Soc. London, 1940, A 176, 312.
- Straughan, B., Sharp Global Non-Linear Stability for Temperature Dependent Viscosity, Proc. Roy. Soc. London, 2002, A 458, 1773.
- A Modified Analysis of the Onset of Convection in a Micropolar Liquid Layer Heated From Below
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
Source
Research Journal of Science and Technology, Vol 5, No 1 (2013), Pagination: 224-231Abstract
In the present paper, the thermal stability analysis of a micropolar liquid layer heated from below is investigated by utilizing the essential arguments of the modified analysis of Banerjee et al. The principle of exchange of stabilities (PES) is shown to be valid using Pellew and Southwell’s method for the problem, whether the liquid layer is hotter or cooler. A general expression for Rayleigh numbers is derived using Galerkin method valid for all combinations of rigid and dynamically free boundary conditions. The values of critical wave numbers and consequently of critical Rayleigh numbers for each case of boundary combinations are derived and computed numerically, when instability sets in as stationary convection. The effects of microrotation parameters and the coefficient of specific heat variation on critical Rayleigh numbers for each case of boundary conditions are computed numerically. From the obtained results, we conclude that the microrotation viscosity coefficient K and coefficient of specific heat variation for large temperature have stabilizing effects whereas microrotation parameter A has destabilizing effect on the onset of convection.Keywords
Micropolar Liquid, Modified Boussinesq Approximation, Galerkin Method, Principle Of Exchange Of Stabilities, Stationary Convection, Rayleigh Number.References
- Ahmadi, G. (1976), Stability of a micropolar fluid layer heated from below, Int. J. Engng. Sci., 14, 81.
- Banerjee, M.B., Gupta, J.R., Shandil, R.G., Sharma, K.C., Katoch, D.C. (1983), A modified analysis of thermal and thermohaline instability of a liquid layer heated underside, J. Math. Phys. Sci., 17(6), 603.
- Boussinesq, J. (1903), ‘Théorie analytique de la chaleur’, Gauthier Villas, Paris, 2, 172.
- Chandrasekhar, S. (1961), ‘Hydrodynamics and Hydromagnetic Stability’, Oxford University Press, Amen. House London, EC. 4.
- Datta, A.B. and Sastry, V.U.K. (1976), Thermal instability of a horizontal layer of micropolar fluid heated from below, Int. J. Engng. Sci., 14, 631.
- Dhiman, J. S., Sharma, P. K. and Singh G. (2011), Convective stability analysis of a micropolar fluids Layer by variational method, Theo. Appl. Mech. Lett., 1, 04204-1.
- Eringen, A.C. (1966), Theory of micropolar fluids, J. Math. Mech., 16, 1.
- Eringen, A.C. (1998), ‘Microcontinuum Field Theories, II. Fluent Media’, Springer-Verlag, NY, Inc.
- Finlayson, B.A. (1972), ‘The Method of Weighted Residuals and Variational Principles’, Academic Press, NY.
- Lukaszewicz, G. (1999), ‘Micropolar Fluids, Theory and Applications’, Brikhauser. Boston, USA.
- Pellew, A. and Southwell, R. V. (1940), On maintained convective motion in a fluid heated from below, Proc. Roy. Soc. London, E.C.4.
- Rayleigh, L. (1916): On the convective currents in a horizontal layer of fluid when the higher temperature is on the underside, Phil. Mag., 32, 529.
- Schmidt, R.J. and Milverton S.W. (1935): On the instability of the fluid when heated from below, Proc. R. Soc. London, A152, 586.
- On the Bounds for Oscillation in Thermohaline Convection Problems with Temperature-Dependent Viscosity
Abstract Views :198 |
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005 (H.P.), IN
2 Department of Mathematics, D.A.V. College, Malout-152107, Punjab, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005 (H.P.), IN
2 Department of Mathematics, D.A.V. College, Malout-152107, Punjab, IN
Source
Research Journal of Engineering and Technology, Vol 6, No 1 (2015), Pagination: 149-154Abstract
The present paper extends the analysis of Gupta et al. (2001, J. Math. Anal. Appl., 264, 398) of Veronis and Stern type’s thermohaline convection problems for the case of temperature-dependent viscosity. The stability of the oscillatory motions for both types of problems with variable viscosity is discussed in this paper and the upper bounds for the growth rates for neutral or unstable oscillatory perturbations are also prescribed. The obtained results are uniformly valid for all combination of dynamically free and rigid boundaries and are free from a curious condition on the non-negativity of the second derivative of viscosity parameter. Further, various results for an initially top-heavy as well as an initially bottom heavy configurations follow as consequence.Keywords
Thermohaline Convection, Oscillatory Motions, Complex Growth Rate,; Temperature-Dependent Viscosity, Eigenvalue Problem.- On π2 Q-Law in Magnetoconvection Problem for General Nature of Boundaries Using Galerkin Method
Abstract Views :140 |
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. College, Sanjauli, Shimla (H.P.)-171006, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. College, Sanjauli, Shimla (H.P.)-171006, IN
Source
Research Journal of Engineering and Technology, Vol 3, No 2 (2012), Pagination: 186-190Abstract
Chandrasekhar proved his famous -law of stationary convection in magnetoconvection problem for the case of both dynamically free boundaries. For the other two cases of boundary conditions, namely; both rigid boundaries and combinations of rigid and dynamically free boundaries he, on the basis of numerical computations, conjectured that the same law must hold true. In the present paper, we have reinvestigated the onset of thermal instability in an electrically conducting fluid layer heated from below in the presence of a uniform magnetic field. In the present analysis the validity of -law for general nature of bounding surfaces is proved using the Galerkin technique. The obtained results are in good agreement with the numerical results of Chandrasekhar and thus validate his claim.- On the Stability Analysis of a Generalized Double Diffusive Convection Problem
Abstract Views :148 |
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. Post Graduate College, Solan, Distt. Solan (H.P.)-173212, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. Post Graduate College, Solan, Distt. Solan (H.P.)-173212, IN
Source
Research Journal of Engineering and Technology, Vol 3, No 2 (2012), Pagination: 191-195Abstract
The present paper deals with the construction of a generalized setup of eigenvalue problem from the perturbations equations governing Modified Thermohaline convection (derived by Banerjee et al. [1993]) by an appropriate choice of parameters of the fluid. This generalized setup of eigenvalue problem named as; generalized double-diffusive convection problem yields eigenvalue problems for Stern type double-diffusive convection problem and for Dufour-Driven double-diffusive convection problem as a consequence. The stability investigations of this general problem are carried in this paper and some general results concerning the stability or otherwise are derived for the case of both dynamically free boundaries and various consequences of the derived results are also discussed.Keywords
Modified Thermohaline Convection, Stern Type Convection, Dufour-Driven Convection, Eigenvalue Problem.- Quantum Effects on the Magnetogravitational Instability of Viscoelastic Fluid through a Porous Medium
Abstract Views :177 |
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
Source
International Journal of Technology, Vol 6, No 2 (2016), Pagination: 199-205Abstract
In the present analysis, the effect of quantum corrections on the gravitational instability of a viscoelastic fluid through porous medium in the presence of uniform magnetic field has been studied in the transverse and longitudinal mode of wave propagation. For the mathematical formulation of the physical problem Generalized Hydrodynamic and Hass model have been used. A general dispersion relation has been derived by using the normal mode analysis. The general dispersion relation is discussed separately for both the modes of wave propagation under the strongly and weakly coupling limits. It is found that the porosity of the medium and quantum effects modifies the Jeans criterion of instability for both the modes of wave propagation under the strongly and weakly coupling limits. Further, the effects of various parameters on the growth rate of gravitational instability has been numerically studied and depicted graphically.- On the Effect of Non-Uniform Magnetic field on the Jeans Instability of a Viscoelastic Medium
Abstract Views :243 |
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
Source
International Journal of Technology, Vol 4, No 1 (2014), Pagination: 7-12Abstract
In the present paper the problem of Jeans instability of a self gravitating viscoelastic medium in the presence of nonuniform magnetic field for both longitudinal and transverse mode of wave propagation under the kinetic and hydrodynamic limits has been investigated, using the generalized hydrodynamic model. It is found that magnetic field has no effect on the Jeans criterion for the onset of gravitational instability in case of longitudinal mode of wave propagation, whereas it modifies the Jeans criterion in the case of transverse mode of wave propagation and has stabilizing effect on the onset of instability. Further, it is observed that the magnetic field has no effect on the growth rate of Jeans instability of a viscoelastic medium. The effects of shear viscosity, bulk viscosity and Mach number on the Jeans criterion and on the growth rate of Jeans instability have also been studied numerically and the obtained results are depicted graphically, for both strongly coupled plasma (SCP) and weakly coupled plasma (WCP).Keywords
Jeans Instability, Viscoelastic Medium, Non-Uniform Magnetic Field, Wave Propagation, Coupled Plasma.- On the Effect of Non-Uniform Temperature Gradients on the Stability of Modified Thermal Convection Problem
Abstract Views :186 |
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. Degree College, Sunni, Distt. Shimla (H.P.)-171301, IN
3 Govt. Degree College, Kullu (H.P.)-175101, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. Degree College, Sunni, Distt. Shimla (H.P.)-171301, IN
3 Govt. Degree College, Kullu (H.P.)-175101, IN
Source
International Journal of Technology, Vol 4, No 1 (2014), Pagination: 1-6Abstract
The aim of the present paper is to study the effect of non-uniform basic temperature gradients on the onset of modified thermal convection in a layer of fluid heated from below for different combinations of rigid and dynamically free boundary conditions. It is shown that the principle of exchange of stabilities (PES) is valid when the temperature gradient is monotonically decreasing upward, which means that the instability sets in as stationary mode. The expressions for the Rayleigh numbers for each combination of rigid and dynamically free boundary conditions for the stationary case of instability are derived using Galerkin method. The effects of non-uniform temperature gradients and the modification factor which arises due to modified theory of Banerjee et al on the instability are studied from the values of the critical Rayleigh numbers calculated numerically for various temperature profiles and the coefficient of specific heat variation due to temperature variation for the given values of other parameters. It is observed from these values that the Cubic temperature profile is more stabilizing than the inverted parabolic temperature distribution profile. Further, it is also found that the critical Rayleigh numbers for thermally insulating boundaries are lower than those for the corresponding isothermal cases.
Keywords
Thermal Convection, Modified Theory, Temperature Gradient, Stationary Convection, Galerkin Method, Rayleigh Numbers, Boundary Conditions.- Onset of Triply-Diffusive Convection in a Fluid Layer with Suspended Particles and Temperature Dependent Viscosity
Abstract Views :194 |
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Authors
Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
Source
International Journal of Technology, Vol 4, No 1 (2014), Pagination: 1-5Abstract
The present paper mathematically investigates the triply-diffusive convection problem with suspended particles by considering the viscosity to be temperature dependent. The temperature gradient is considered to be destabilizing whereas the solute gradients may be stabilizing or destabilizing. A sufficient condition for the validity of principle of exchange of stabilities (PES) is obtained and a bound for the complex growth rate of an arbitrary oscillatory perturbation, which may be neutral or unstable, is derived for this general problem. Various consequences of the above results are discussed and the analogous results under the individual effect of suspended particles, solute gradients and viscosity variations are also deduced.