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### Prakash, Jyoti

- On the Characterization of Nonoscillatory Motions in Maxwell Fluid in a Porous Medium Heated from Below

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1 Department of Mathematics and Statistics, Himachal Pradesh University, Shimla – 171005, IN

#### Authors

**Affiliations**

1 Department of Mathematics and Statistics, Himachal Pradesh University, Shimla – 171005, IN

#### Source

Research Journal of Science and Technology, Vol 5, No 1 (2013), Pagination: 51-54#### Abstract

In the present paper condition for characterizing nonoscillatory motions which may be neutral or unstable in a horizontal layer of Maxwell fluid in a porous medium (modified Darcy-Brinkman-Maxwell Model) heated from below is obtained. It is proved that for a horizontal layer of Maxwell fluid in a porous medium heated from below an arbitrary neutral or unstable mode of the system is definitely nonoscillatory in character and in particular the ‘ principle of the exchange of stabilities’ is valid if (RλP_r)/4∏ ∧ 2≤ 1. The result is uniformly valid for all combinations of free and rigid boundaries.#### Keywords

Maxwell Fluid, Oscillatory Motions, Thermal Convection, Porous Media, Modified – Darcy - Brinkman – Maxwell Model.#### References

- Benard H 1900 Les tourbillions cellulaires dans une nappe liquid. Revenue generale des Sciences pures et appliqués 11 1261-71 and 1309- 28
- Bezan A 2004 Convection Heat Transfer third ed. John Wiley and Sons New Jersey
- Chandrasekhar S 1961 Hydrodynamic and Hydromagnetic stability Clarendon Oxford
- Chen F and Chen C F 1988 Onset of finger Convection in a horizontal porous layer underlying a fluid layer J. Heat Transf. 110(2) 403 – 09
- Drazin P and Reid W 1981 Hydrodynamic Stability Cambridge University Press Cambridge
- Fu C Zhang Z and Tan W 2007 Numerical simulation of thermal convection of a viscoelastic fluid in a porous square box heated from below Physics of Fluids 19 104107(1 – 12)
- Horton C and Rogers F 1945 Convection currents in a porous medium J. Appl. Phys. 16(6) 367-70
- Katto Y and Masuoka T 1967 Criterion for the onset of convective flow in a fluid in a porous medium Int. J. Heat mass Transf. 10(3) 297-309
- Laroze D, Martinez-Mardones J and Bragard J 2007 Thermal convection in a rotating binary viscoelastic liquid mixture Eur. Phys. J. Special Topics 146 291-300
- Li Z and Khayat R E 2005 Finite amplitude Rayleigh-Benard Convection and pattern selection for viscoelastic fluids J. Fluid Mech. 529 221- 51 M. H. Schultz, Spline Analysis, Prentice Hall, Englewood Cliffs, NJ, (1973).
- M. H. Schultz (1973) Spline Analysis Prentice Hall Englewood Cliffs NJ
- Malashetty M S and Swamy M 2007 The onset of convection in a viscoelastic liquid saturated anisotropic porous layer, Trans. Por. Med. 67 203 – 18
- Malashetty M S Swamy M and Kulkarni S 2007 Thermal convection in a rotating porous layer using a thermal nonequilibrium model Phys. Fluids 19 054102 (1-16)
- Mckibbin R and O’Sullivan M J 1980 Onset of convection in a layered porous medium from below J. Fluid Mech. 96(2) 375-93
- Pellew A and Southwell R V 1940 On the maintained convective motion in a fluid heated from below Proc. Roy Soc. A 176 312 – 43
- Rayleigh L 1916 On the convective currents in a horizontal layer of fluid when the higher temperature is on the upper side Phil. Mag. 32 529- 46
- Saravanan S 2009 Centrifugal acceleration induced convection in a magnetic fluid saturated anisotropic rotating porous medium Trans. Por. Med. 77 79 – 86
- Sokolov M and Tanner R I 1972 Convective stability of a general viscoelastic fluid heated from below The Phys. Fluids 15(4) 534 – 39
- Straughan B 2006 Global nonlinear stability in porous convection with a thermal non – equilibrium model Proc. Roy. Soc. A 462 409 – 18
- Tan W and Masuoka T 2007 Stability analysis of a Maxwell Fluid in a Porous medium heated from below Physics Letters A 360 454-60
- Vest C M and Arpaci V S 1969 Overstability of a viscoelastic layer heated from below J. Fluid Mech. 36 (3) 613 – 23
- Yin C Fu C and Tan W 2012 Onset of thermal convection in a Maxwell fluid saturated porous medium. The effects of hydrodynamic boundary and constant flux heating conditions Trans. Porous. Med. 91 777 – 90
- Yoon D Y Kim M C and Choi C K 2004 The onset of oscillatory convection in a horizontal porous layer saturated with viscoelastic liquid Trans. Por. Med. 55 275 – 84
- Zhang Z Fu C and Tan W 2008 Linear and non linear stability analysis of thermal convection for Oldroyd-B fluids in porous media heated from below Phys. Fluids 20 084103(1 – 12)

- On Linear Growth Rates in Thermohaline Convection with Viscosity Variations

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1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

#### Authors

**Affiliations**

1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

#### Source

Research Journal of Science and Technology, Vol 5, No 1 (2013), Pagination: 140-143#### Abstract

In the present paper it is proved that the complex growth rate (where and are real and imaginary parts of p ) of an arbitrary oscillatory motions of growing amplitude, neutral or unstable, for thermohaline convection configuration of Veronis type (Veronis, G., J. Mar. Res., 23(1965)1), with the viscosity variations must lie inside a semicircle in the right half of the prpiplane whose centre is at the origin and radius equals . A similar theorem is also proved for thermohaline convection of Stern type (Stern, M.E., Tellus 12(1960)172). Furthermore the above results are uniformly valid for all combinations of rigid and free bounding surfaces. The results obtain herein, in particular, also yield sufficient conditions for the validity of the ‘principle of the exchange of the stabilities’ for the respective configurations.#### Keywords

Thermohaline Instability, Oscillatory Motions, Veronis Type, Stern Type, Variable Viscosity.#### References

- Banerjee M. B., Katoch D. C., Dube G. S. and Banerjee K., Bounds for growth rate of a perturbation in thermohaline convection, Proc. Roy. Soc. London. A 387(1981)301.
- Banerjee M. B., Gupta J. R. and Shandil R.G., Generalized thermal convection with viscosity variations, J. Math. Phys. Sc. 11(5)(1977)421.
- Brandt A. and Fernando H. J. S., Double Diffusive Convection, Am. Geophys. Union, Washington, (1996).
- Lighthill M. J., Introduction to Boundary Layer Theory in Laminar Boundary Layers, (Ed.: L. Rosenhead), Clarendon Press, Oxford, (1963).
- Prakash J., A Mathematical theorem for thermohaline convection of the Veronis type with viscosity variations, Ind. J. Pure Appl.Math., 26(8)(1995)813.
- Schultz M. H., Spline Analysis, Prentice Hall Ince., Englewood Cliffs. N.J., (1973).
- Stern M. E., The salt fountain and thermohaline convection, Tellus 12 (1960)172.
- Torrance K.E., and. Turcotte D.L., Thermal convection with large viscosity variation, J. Fluid Mech, 47(1971)113.
- Turner J. S., Double diffusive phenomena, Ann. Rev. Fluid mech. 6(1974)37.
- Veronis G., On finite amplitude instability in thermohaline convection, J. Mar. Res., 23(1965)1.

- On Triply Diffusive Convection Analogous to Stern Type with Variable Viscosity

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1 Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, IN

2 J.N. Government Engineering College, Sunder Nagar (H.P.), IN

#### Authors

**Affiliations**

1 Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, IN

2 J.N. Government Engineering College, Sunder Nagar (H.P.), IN

#### Source

Research Journal of Science and Technology, Vol 9, No 1 (2017), Pagination: 111-114#### Abstract

The paper mathematically establishes that triply diffusive convection (analogous to Stern type), with variable viscosity and with one of the components as heat, cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the thermal Rayleigh number |R|, the Lewis number τ_{2}for the second concentration component , μ

_{min}(the minimum value of viscosity μ in the closed interval [0,1]) and the Prandtl number σ satisfy the inequality |R| ≤ 27π

^{4}/4 τ

_{2}(μ

_{min }+ 1/σ) provided D

^{2}μ is positive everywhere. It is further proved that this result is uniformly valid for the quite general nature of the bounding surfaces.

#### Keywords

Triply Diffusive Convection, Variable Viscosity, Concentration Rayleigh Number, Oscillatory Motion, Initially Bottom Heavy Configuration.- On Double-Diffusive Convection in a Binary Viscoelastic Fluid Saturated Anisotropic Porous Layer

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1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla 171005, IN

#### Authors

**Affiliations**

1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla 171005, IN

#### Source

Research Journal of Science and Technology, Vol 9, No 1 (2017), Pagination: 123-126#### Abstract

In the present paper it is mathematically established that the linear growth rate of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude in a double diffusive binary viscoelastic fluid saturated anisotropic porous layer heated from below must lie inside a semicircle in the right half of the (P_{r}, P

_{i})-plane whose centre is at the origin and radius equals λ

_{1}Pr

_{D}Ra

_{T}+√P

_{rD}(λ

_{1}

^{2}Ra

_{T}

^{2}Pr

_{D}+4Ra

_{s})/2, where Ra

_{T}and Ra

_{s}are the Darcy -Rayleigh number and the solute Rayleigh number respectively. Further, it is proved that this result is uniformly valid for quite general nature of the bounding surfaces.

#### Keywords

Double-Diffusive Convection, Viscoelastic Fluid, Porous Medium, Complex Growth Rate, Solute Rayleigh Number.- On Triply Diffusive Convection in Porous Medium:Darcy Brinkman Model

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1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

2 Department of Physics, MLSM College, Sundernagar, H. P., IN

#### Authors

**Affiliations**

1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

2 Department of Physics, MLSM College, Sundernagar, H. P., IN

#### Source

Research Journal of Science and Technology, Vol 9, No 1 (2017), Pagination: 127-130#### Abstract

The present paper deals with the problem of triply diffusive convection analogous to Stern type in porous medium using Darcy-Brinkman model. Bounds are obtained for the complex growth rate of an arbitrary oscillatory perturbation of growing amplitude, neutral or unstable for this configuration which is uniformly valid for any combination of bounding surfaces.#### Keywords

Triply Diffusive Convection, Concentration Rayleigh Number, Porous Medium, Darcy-Brinkman Model.- On the Occurrence of Stationary Convection in Triply Diffusive Convection in Porous Medium : Darcy Model

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1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

#### Authors

**Affiliations**

1 Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, IN

#### Source

Research Journal of Science and Technology, Vol 9, No 1 (2017), Pagination: 131-134#### Abstract

The present paper mathematically establishes that 'the principle of the exchange of stabilities' for triply diffusive convection analogous to double diffusive convection of Stern type in porous medium (Darcy model) is valid in the regime |R|Aσ/2π^{4}≤1, where Ris the Rayleigh number, σ is the thermal Prandtl number and Ais a porous parameter. It is further proved that the above result is uniformly valid for any combination of rigid and free boundaries.