## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Thermosolutal instability in a horizontal fluid layer affected by rotation

**ABSTRACT**

Thermosolutal convective instability in a horizontal layer affected by rotation is studied. Stationary convection and over-stability cases are considered for different boundary conditions. Analytical solutions were obtained when both boundaries are free and numerical results were obtained for the cases of free and rigid boundaries. The numerical computations of this problem were performed using the method of expansion of Chebyshev polynomials. This method is better suited to the solution of hydrodynamic stability problems than expansions in other sets of orthogonal polynomials. This method not only has high accuracy but also allows stationary and over-stable modes to be treated simultaneously, which is important if perchance the critical eigenvalue flits between the different modes in response to changing parameter values. The results obtained show that the effect of both solute concentration and rotation is to stabilize the system for stationary convection case and for the over-stability case when both boundaries are free. However when both boundaries are rigid some unexpected behavior are obtained in the case of over-stability.

**KEYWORDS**

PAPER SUBMITTED: 2017-12-03

PAPER REVISED: 2018-02-20

PAPER ACCEPTED: 2018-02-21

PUBLISHED ONLINE: 2018-03-04

- Rayleigh, L., On Convection Currents in a Horizontal Layer of Fluid when the Higher Temperature is on the Underside, Philosophical Magazine 32 (1916), 192, pp. 529-546
- Chandrasekhar, S. Hydrodynamic and Hydromagnetic stability, Dover Publications Inc. New York, USA 1961
- Chandrasekhar, S., Elbert, D. The Instability of a Layer of Fluid Heated Below and Subject to Coriolis Forces. II. Proceeding of the Royal Society of London A 231 (1955), pp. 198-210
- Abdullah, A. Thermal Instability of a Non-Linear Magnetic Fluid Under the Influence of both Non-Vertical Magnetic Field and Coriolis Forces, Arabian Journal of Science and Engineering 17 (1992), 4B, pp. 625-633
- Julien, K. et al., Rapidly Rotating Turbulent Rayleigh-Benard Convection, Journal of Fluid Mechanics 322 (1996), pp. 243-273
- Chun, H. et al., Stratified Rotating Boussinesq Equations in Geophysical Fluid Dynamics: Dynamic Bifurcation and Periodic Solutions, Journal of Mathematical Physics 48 (2007), 6, DOI No. 10.1063/1.2710350
- Prosperetti, A., The Effect of Rotation on the Rayleigh-Bénard Stability Threshold, Physics of Fluids 24 (2012), 11, pp. 114101-114116
- Geurts, B., Kunnen, R., Intensified Heat Transfer in Modulated Rotating Rayleigh-Bénard Convection, International Journal of Heat and Fluid Flow 49 (2014), pp. 62-68
- Horn, S., Shishkina, O., Toroidal and Poloidal Energy in Rotating Rayleigh-Bénard Convection , Journal of Fluid Mechanics 762 (2015), pp. 232-255
- Khan, I., Sharidan, S., Rotating MHD Flow of A Generalized Burgers' Fluid Over an Oscillating Plate Embedded in a Porous Medium, Thermal Science 19 (2015), 1, pp. S183-S190
- Sharma, P. et al., Rayleigh Taylor Instability in Dusty Magnetized Fluids with Surface Tension Flowing Through Porous Medium, Thermal Science 20 (2016), 1, pp. 119-130
- Stern, M., The Salt Fountain and Thermohaline Convection, Tellus 12 (1960), pp. 172-175
- Walin, G., Note on the Stability of Water Stratified by both Salt and Heat, Tellus 3 (1964), pp. 389-393
- Veronis, G. On Finite Amplitude Instability in Thermohaline Convection, Journal of Marine Research 23 (1965), 1, pp. 1-17
- Nield, D. The Thermohaline Rayleigh-Jeffreys Problem, Journal of Fluid Mechanics 29 (1967), 3, pp. 545-558
- Nield, D., Bejan, A. Convection in Porous Media, Springer 2006
- Murthy, J., Lee, P., Thermosolutal Convection in a Floating Zone: The Case of an Unstable Solute Gradient, International Journal of Heat and Mass Transfer 31 (1988), 9, pp. 1923 - 1932
- Boudourides, M., Nikoudes, A. The Attractor of Thermosolutal Convection, Physica D 47 (1991), pp. 439-449
- Abdullah, A., Thermosolutal Convection in a Non-Linear Magnetic Fluid. International Journal of Thermal Sciences 39 (2000), 2, pp. 273-284
- Xu, L., Yang, S., Stability Analysis of Thermosolutal Second-Order Fluid in Porous Bénard Layer. Ricerche di Matematica 56 (2007), 1, pp. 149-160
- Jena, S. et al., Thermosolutal Convection in a Rectangular Concentric Annulus: A Comprehensive Study, Transport in Porous Media 98 (2013), 1, pp. 103 - 124
- Sankar, M. et al., Thermosolutal Convection from a Discrete Heat and Solute Source in a Vertical Porous Annulus, International Journal of Heat and Mass Transfer 55 (2012), pp. 4116-4128
- Costa, V., Double-Diffusive Natural Convection in Parallelogrammic Enclosures Filled with Fluid Saturated Porous Media, International Journal of Heat and Mass Transfer 47 (2004), pp. 2699- 2714
- Jagadeesha, R., et al., Numerical Simulation of Double Diffusive Magnetoconvection in an Inclined Parallelogrammic Porous Enclosure with an Internal Heat Source, Materials Today: Proceedings 4 (2017), pp. 10544-10548
- Orzsag, S. Accurate Solution of the Orr-Sommerfeld Equation, Journal of Fluid Mechanics 50 (1971), pp. 689-703
- Hassanien, I.,El-Hawary, H., Chebyshev Solution of Laminar Boundary Layer Flow, International Journal of Computer Mathematics 33 (1989), pp. 127- 132
- Abdullah, A., Lindsay, k., Benard Convection in a Non-Linear Magnetic Fluid, Acta Mechanica 85 (1990), pp. 27-42
- Hassanien, I. et al., Chebyshev Solution of Axisymmetric Stagnation Flow on a Cylinder, Energy Conversion and Management 37 (1996), 1, pp. 67-76
- Straughan, B., Effect of Property Variation and Mode on Convection in a Fluid Overlying a Porous layer, International Journal for Numerical and Analytical Methods in Geomechanics 26 (2002), pp. 75-97
- Banjer, H., Abdullah, A., Thermal Instability in Superposed Porous and Fluid Layers in the Presence of Coriolis Forces, International Journal of Applied Mathematics and Mechanics 7 (2011), pp. 13-27