Thermosolutal Convection in Compressible Couple-Stress Fluid with Fine Dust
Chander Bhan Mehta1, Susheel Kumar2 and Sanjeev Gangta3
1Department of Mathematics, Centre of Excellence, Sanjauli , Shimla, India
2Department of Mathematics, St. Bede’s College, Shimla, India
3Department of Mathematics, Govt Boys Sr. Sec. School, Kotkhai, Shimla,India
*Corresponding Author E-mail: chanderbmehta@gmail.com
ABSTRACT:
A layer of compressible, couple-stress fluid permeated with suspended particles (fine dust), heated and soluted from below is considered. For the case of stationary convection; the compressibility, couple-stress and stable solute gradient postpone the set of convection where as the suspended particles hasten the onset of convection. The graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The stable solute gradient is found to introduce the overstability modes in the system, which ware non-existing, in its absence. The case of overstability is also considered wherein the sufficient conditions for non-existence of overstability are obtained.
KEYWORDS: Couple-stress fluid, double-diffusive convection, suspended particles.
1. INTRODUCTION:
The theory of Bénard convection in a layer of Newtonian fluid heated from below has been discussed in detail by Chandrasekhar (1981). Chandra (1938) performed careful experiments in an air layer and found a contradiction between the theory and his experiments. He found that convections occurred at much lower gradients than predicted if the layer depth was less than 7mm and called this motion, “columnar instability”. However, for layers deeper than 10mm, a Bénard-type cellular convection was observed. Thus, there is a contradiction between the theory and the experiment. Scanlon and Segal (1973) have considered the effect of suspended particles on the onset of Bénard convection and found that the critical Rayleigh number was reduced solely because the heat capacity of the pure fluid was supplemented by that of the particles. Palaniswamy and Purushotham (1981) have considered the stability of shear flow of stratified fluids with fine dust and found the effect of fine dust to increase the region of instability. Veronis (1965) has investigated the problem of thermohaline convection in a layer of fluid heated from below and subjected to a stable salinity gradient. The physics is quite similar to Veronis’s (1965) thermohaline convection in the stellar case in that helium acts like salt in raising the density and in diffusing more slowly than heat. Such similarity also exists in several geophysical situations. This problem of the onset of thermal convection in the presence of a solute gradient is of great importance because of its application to oceanography, atmospheric physics, astrophysics and geophysics. The heat and solute being two diffusing components, thrmosolutal convection is the general term used in dealing with such phenomena. The Boussinesq approximation has been used, in all the above studies on thermal and thermohaline convection, which is well justified in the case of incompressible fluids.
When the fluids are compressible, the equations governing the system become quite complicated. To simplify the set of equations governing the flow of compressible fluids, Spiegel and Veronis (1960) made the assumptions:
(i) The depth of fluid layer is much smaller than the scale height as defined by them, if only motions of infinitesimal amplitude are considered and (ii) the fluctuations in temperature, density and pressure, introduced due to motion, do not exceed their static variations, in non-linear investigations. Under the above assumptions, the flow equations are found to be the same as those for incompressible fluids except that the static temperature gradient is replaced by its excess over the adiabatic. The thermal instability in compressible fluids in the presence of magnetic field and rotation has been considered by Sharma (1977).
Stokes (1966) has formulated the theory of couple-stress fluid. Walicki and Walicka (1999) modeled synovial fluid as a couple-stress fluid in human joints. Normal synovial fluid is clear or yellowish and is a viscous, non-Newtonian fluid.
The present paper deals with the effect of suspended particles (or fine dust) on the thermosolutal convection in compressible, couple-stress fluid. The motivations for the present study are the decades-old contradiction between the theory and experiment (columnar instability) and the fact that the knowledge concerning fluid-particle mixtures is not commensurate with their scientific and industrial importance. With the growing importance of non-Newtonian fluids in modern technology and industry,
investigations on such fluids are desirable. Further motivation for this study is its usefulness in chemical technology, industry and biomechanics (e.g. physiotherapy).Compressibility of the fluid is an important aspect.
2. Description of the problem and perturbation equations
Consider an infinite horizontal, compressible couple-stress fluid layer of thickness d permeated with suspended particles and bounded by the planes z = 0 and z = d. This layer is heated and soluted from below such that a steady adverse temperature gradient b (= |dT/dz|) and adverse solute gradient b/ (= |dC/dz|) are maintained.
References
[1]. Chandra K. (1938): “Instability of fluids heated from below”, Proc. Roy. Soc.( Lon), vol. A164,pp.231-242.
[2]. Chandrasekhar S. (1981): “Hydrodynamic and Hydromagnetic Stability”. –NewYork ; Dover Publication.
[3].Palaniswamy V.I and Purushotham C.M. (1981): “Stability of shear flow of stratified fluids with fine dust”, Phys. Fluids, vol.27, pp.1224-1228.
[4]. Scanlon J.W. and Segal L.A. (1973): “Some effects of suspended particles on the onset of Bénard convection”, Phys.Fluids, vol.16, pp 1573 -1578.
[5]. Sharma R.C. (1977): “Thermal instability in compressible fluids in the presences of rotation and magnetic field”, J.Math. Anal. Appl., vol. 60, pp. 227-235.
[6]. Sharma R.C. and Gangta S. (2002): “On compressible couple-stress fluid heated and soluted from below in porous medium”, Communicated for publication.
[7].Spiegel E.A. and Veronis G. (1960): “On the Boussinesq approximation for a compressible fluid”, Astrophys.J. vol,131,pp.442-444.
[8]. Stokes V.A.(1966): “Couple-stresses in fluids”, Phys. Fluids, vol. 9, pp. 1709-1715.
[9]. Veronis G. (1965): “On finite amplitude instability in thermohaline convection”, J. Marine Res., vol. 23, pp.1-17.
[10].Walicki E. and Walicka A. (1999): “Inertia effects in the squeeze film of a couple-stress fluid in biological bearing”. –Appl. Mech. Eng., vol.4, pp. 363-373.
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Received on 20.01.2014 Accepted on 31.01.2014 © EnggResearch.net All Right Reserved Int. J. Tech. 4(1): Jan.-June. 2014; Page 140-144 |