Double-diffusive convection in compressible, rotating, couple-stress fluid

C.B. Mehta

Department of Mathematics, Government College Shimla-171006

*Corresponding Author Email:

ABSTRACT:

A layer of compressible, rotating, couple-stress fluid heated and soluted from below is considered. For the case of stationary convection, the compressibility, stable solute gradient and rotation postpone the onset of convection whereas the couple-stress viscosity postpones as well as hastens the onset of convection depending on rotation parameter. The case of overstability is also studied wherein a sufficient condition for the non-existence of overstability is found.

KEYWORDS: Couple-stress fluid, compressibility, rotation, thermosolutal convection.

PACS Nos. : 47.20. Ma, 47. 50. +d

1. INTRODUCTION:

The theoretical and experimental results on thermal convection in a fluid layer, in the absence and presence of rotation, have been given by Chandrasekhar [1]. Veronis [2] has investigated the problem of thermohaline convection in a layer of fluid heated from below and subjected to a stable salinity gradient. Double-diffusive (thermosolutal) convection problems arise in oceanography, limnology and engineering. Brakke [3] explained a double-diffusive instability that occurs when a solution of a slowly diffusing protein is layered over a denser solution of more rapidly diffusing sucrose. Nason et al. [4] found that this instability, which is deleterious to certain biochemical separations, can be suppressed by rotation in the ultracentrifuge.

The theory of couple-stress fluid has been formulated by Stokes [5]. Walicki and Walicka [6] have modelled synovial fluid as a couple-stress fluid in human joints. One of the applications of couple-stress fluid is its use to the study of the mechanisms of lubrications of synovial joints, which has become the object of scientific research. A human joint is a dynamically loaded bearing which has articular cartilage as the bearing and synovial fluid as the lubricant. When a fluid is generated, squeeze-film action is capable of providing considerable protection to the cartilage surface. The shoulder, ankle, knee and hip joints are the loaded-bearing synovial joints of the human body and these joints have a low friction coefficient and negligible wear. Normal synovial fluid is a viscous, non-Newtonian fluid and is generally clear or yellowish. Lin [7] has studied the couple-stress effect on the squeeze film characteristics of hemispherical bearings with reference to synovial joints. Walicki and Walicka [6] have studied the effects of couple-stresses and inertia effects on the characteristics of squeeze-film behaviour in thrust curvilinear bearings with references to synovial joints. On the basis of Stokes’ couple-stress fluid model, Walicki and Walicka [8] have made mathematical modelling of some biological bearings. Sharma et al.[9] have studied a layer of couple-stress fluid permeated with suspended particles, heated from below. For thermal and thermosolutal convection problems, the Boussinesq approximation has been used, which is well justified in the case of incompressible fluids.

When the fluids are compressible, the equations governing the system become quite complicated. Spiegel and Veronis [10] have simplified the set of equations governing the flow of compressible fluids under the assumption that the depth of the fluid layer is much smaller than the scale height as defined by them, if only motions of infinitesimal amplitude are considered. Sharma [11] has studied the thermal instability in compressible fluids in presence of rotation and magnetic field.

Keeping in mind the importance of non-Newtonian fluids, thermosolutal convection and compressibility, the present paper considers a layer of compressible, rotating, couple-stress fluid heated and soluted from below.

6. CONCLUSION:

Stommel et al. [12] and Linden [13] have remarked that the length scales characteristic of double diffusive convecting layers in the ocean may be sufficiently large that the ocean may be sufficiently large that the Earth’s rotation might be important in their formation .Moreover, the rotation of the Earth distorts the boundaries of a hexagonal convection cell in a fluid through a porous medium and distortion plays an important role in the extraction of energy in the geothermal regions.

Couple-stress fluid is an important and useful non-Newtonian fluid. Due to importance of non-Newtonian fluids, compressibility and thermosolutal convection, a layer of compressible, couple-stress fluid heated and soluted from below is studied. During the study it is found that presence of compressibility postpones the onset of convection. The presence of rotation and the stable solute gradient also postpones the onset of convection. In the absence of rotation as well as stable solute gradient the Rayleigh number increases with the increase in couple stress parameters thus postpones the onset of convection.

The couple stress viscosity postpones as well as hastens the onset of convection depending on rotation parameter. If T₁ < (1+x)³{1 + F₁ (1 +x) }², it has a stabilizing effect, whereas it has a destabilizing effect if  T₁ > (1+x)³{1 + F₁ (1 +x) }². In the absence of rotation, couple stress viscosity always postpone the onset of Thermosolutal convection in the presence of compressibility. Overstable comes into play and sufficient condition for the non-existence of overstability is found. For k < n  and k < k^/ , overstability cannot occur and the principle of exchange of stabilities is valid. In the absence of couple stress viscosity (F₁ = 0) the sufficient condition for non-existence of overstability are same as in its presence

7. REFERENCES:

[1] S Chandrasekhar (New York: Dover) (1981)

[2] G Veronis J Marine Res. 23 1 (1965)

[3] M K  Brakke  Arch. Biochem. Biophys. 55, 175 (1955)

[4] P Nason, V Schumaker, B Halsalt and J Schwedes Biopolymers 7 241 (1969)

[5] V K Stokes “Couple-stresses in fluids” Phys. Fluids 9 1709 (1966)

[6] E Walicki and A Walicka  Appl. Mech. Engng. 4 363 (1999)

[7] J R Lin Appl. Mech. Engng. 1, 317 (1996)

[8] E Walicki and A Walicka  Proc. 4^th European and 2^nd Mi MR conference, Harrogate, UK, p. 519, 6-8 July (1998)

[9] R C Sharma, Sunil, Y D Sharma and R S Chandel  Arch. Mech. 54 287 (2002)

[10] E A Spiegel and G Veronis Astrophys. J. 131 442 (1960)

[12] R C Sharma, J. Math. Anal. Appl. 60 227 (1977)

[13] Stommel, H, and Fedorov, K.N Tellus 19,306 (1967)

[14] Lindin, P.F Fluid Dynamics 6, 1 (1974)

Received on 16.08.2016            Accepted on 01.09.2016           

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Int. J. Tech. 2016; 6(2): 248-252.

DOI: 10.5958/2231-3915.2016.00038.9