Effect of wave number on the onset of instability in couple-stress fluid and its characterization in the presence of rotation

 

Kamal Singh1 and  Ajaib S. Banyal2*

1Research Scholar, Department of Maths, Singhania University, Pacheri Bari, Jhunjhunu- 333515 (Raj.)

2Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP)-177033 

*Corresponding Author Email: singh_kamal1979@rediffmail.com; ajaibbanyal@rediffmail.com

 

ABSTRACT:

Thermal instability of couple-stress fluid in the presence of uniform vertical rotation is considered. Following the linearized stability theory and normal mode analysis, the paper established the regime for all oscillatory and non-decaying slow motions starting from rest, in a couple-stress fluid of infinite horizontal extension and finite vertical depth in the presence of uniform vertical rotation and the necessary condition for the existence of ‘overstability’ and the sufficient condition for the validity of the ‘exchange principle’ is derived, when the bounding surfaces of infinite horizontal extension, at the top and bottom of the fluid are rigid. Further, the stationary convection at marginal state with free horizontal boundaries is analyzed numerically and graphically, showing that the couple-stress parameter and rotation has stabilizing effect on the system. However, for the constant magnitude of couple-stress parameter and rotation, the wave number has a destabilizing effect for a value less than a critical value, which varies with the magnitude of the couple-stress parameter and rotation; and for higher value than the critical value of the wave number; it has a stabilizing effect on the system.

 

KEYWORDS: Thermal convection; Couple-Stress Fluid; Rotation; PES; Taylor number.

MSC 2000 No.: 76A05, 76E06, 76E15; 76E07.

 

1.  INTRODUCTION:

Right from the conceptualizations of turbulence, instability of fluid flows is being regarded at its root. The thermal instability of a fluid layer with maintained adverse temperature gradient by heating the underside plays an important role in Geophysics, interiors of the Earth, Oceanography and Atmospheric Physics etc. A detailed account of the theoretical and experimental study of the onset of Bénard Convection in Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar[1]. The use of Boussinesq approximation has been made throughout, which states that the density changes are disregarded in all other terms in the equation of motion except the external force term. Sharma et al[2] has considered the effect of suspended particles on the onset of Bénard convection in hydromagnetics. The fluid has been considered here are all Newtonian in all above studies. With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. Stoke[3] proposed and postulated the theory of couple-stress fluid. One of the applications of couple-stress fluid is its use to the study of the mechanism of lubrication of synovial joints, which has become the object of scientific research. According to the theory of Stokes[3], couple-stresses are found to appear in noticeable magnitude in fluids having very large molecules. Since the long chain hylauronic acid molecules are found as additives in synovial fluid, Walicki and Walicka[4] modeled synovial fluid as couple-stress fluid in human joints. Sharma and Thakur[5] have studied the thermal convection in couple-stress fluid in porous medium in hydromagnetics. Sharma and Sharma[6] and Kumar and Kumar[7] have studied the effect of dust particles, magnetic field and rotation on couple-stress fluid heated from below and for the case of stationary convection, found that dust particles have destabilizing effect on the system, where as the rotation is found to have stabilizing effect on the system, however couple-stress and magnetic field are found to have both stabilizing and destabilizing effects under certain conditions.

 

However, in all above studies the case of two free boundaries which is a little bit artificial except the stellar atmospheric case is considered. Banerjee et al[8] gave a new scheme for combining the governing equations of thermohaline convection, which is shown to lead to the bounds for the complex growth rate of the arbitrary oscillatory perturbations, neutral or unstable for all combinations of dynamically rigid or free boundaries and, Banerjee and Banerjee[9] established a criterion on characterization of non-oscillatory motions in hydrodynamics which was further extended by Gupta et al [10]. However no such result existed for non-Newtonian fluid configurations in general and in particular, for Couple-stress viscoelastic fluid configurations. Banyal [11] have characterized the oscillatory motions in couple-stress fluid.

 

6. CONCLUSION:

In this paper, the effect of wave number and uniform vertical rotation on a couple-stress fluid heated from below is investigated and the immediate conclusions of the theorems proved above; and numerical and graphical discussion, are as follows:  

 

(a). The necessary condition for the onset of oscillatory motions and ‘overstability’, for configuration under consideration, is that the inequality (4.17) must be satisfied. Thus the sufficient condition for the non-existence of oscillatory motions and hence the validity of ‘exchange principle’ is that, for the configuration under consideration, which provides a significant improvement to the earlier Which provide a significant improvement to the earlier result by Banyal[14]. The result is also in accordance with corresponding configuration of Newtonian fluid when the couple-stress parameter F=0, by Gupta et al[10].

 

(b). It is observed from figure1 that of the rotation has the stabilizing effect on the onset of instability in the present configuration, from figure 2, the couple stress parameter in the absence of rotation has the stabilizing effect on the onset of instability in the present configuration. However, in the presence of rotation couple-stress parameter may have stabilizing or destabilizing effect as per the conditions given by (5.4), the result which is in accordance with Sharma and Sharma[6].

 

(c). Further, the stationary convection at marginal state for the constant magnitude of couple-stress parameter and uniform vertical rotation, the wave number has a destabilizing effect for a value less than a critical value, which varies with the magnitude of couple-stress parameter and magnetic field, and for higher value than the critical value of the wave number; it has a stabilizing effect on the system when both the horizontal boundaries are free.

                                                               

7. REFERENCES:

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3.              V.K. Stokes, Couple-stress in fluids, Phys. Fluids, 9: 1709-15 (1966).

4.              E. Walicki, A. Walicka, Inertial effect in the squeeze film of couple-stress fluids in biological bearings, Int. J. Appl. Mech. Engg., 4: 363-73 (1999).

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13.           M.B. Banerjee, J.R. Gupta, J. Prakash, On thermohaline convection of Veronis type, J. Math. Anal. Appl., Vol.179: 327-334 (1992).

14.           A.S. Banyal, A Characterization of Oscillatory Motions for Rotatory Convection in Couple-Stress Fluid, Int. J. Current Research and Review (IJCRR), 4(13): 5-14 (2012).    

15.           A.S. Banyal and K. Singh, A Characterization of Rotatory Convection in Couple-Stress Fluid, Int. J. Fluids Engineering, Vol. 3(4): 459-468 (2011).

 

 

 

Received on 18.12.2014            Accepted on 23.12.2014     

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