Thermal Instability in a Layer of Nanofluid Subjected to Rotation and Suspended Particles

Ramesh Chand¹, G. C. Rana², Arvind Kumar³ and Vandna Sharma⁴

¹Department of Mathematics, Government College Dhaliara (Kangra), Himachal Pradesh, India

²Department of Mathematics, Sidharth Government College Nadaun (Hamirpur), Himachal Pradesh, India

³Department of Mathematics, Government College Nagrota Bagwan (Kangra), Himachal Pradesh, India

⁴Department of Mathematics, DDM College of Science and Technology Banehra (Una), Himachal Pradesh, India

*Corresponding Author: rameshnahan@yahoo.com

ABSTRACT:

The objective of the present work is to investigate theoretically the combined effect of rotation and suspended particles on the thermal instability in a layer of nanofluid. A linear stability theory, perturbation method and normal mode technique is used to find the solution of fluid layer confined between two free boundaries. For linear theory analysis, critical Rayleigh number has been obtained to study the stability analysis. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection is derived analytically and graphically. The effects of various parameters such as suspended particles, rotation, Lewis number and modified diffusivity ratio on the stationary convection are studied.

KEY WORDS: Prandtl number, Taylor number, Nanofluid, Suspended particles, Lewis number.

1. INTRODUCTION:

Nanofluid is a fluid colloidal mixture of nano (<100 nm) sized particles, in base fluid. Nanoparticles materials may be taken as oxide ceramics (Al₂O₃, CuO), metal carbides (SiC), nitrides (AlN, SiN) or metals (Al, Cu) etc. and base fluids are water, ethylene or tri-ethylene- glycols, oil and other lubricants, bio-fluids, polymer solutions, other common fluids. The term ‘nanofluid’ was coined by Choi (Choi 1995). The onset of convective instability in a layer of fluid heated from below has been extensively studied by researches. A detailed account of thermal instability of Newtonian fluids under varying assumptions of hydrodynamics and hydromagnetics has been given by Chandrasekhar (1961).

Convection of nanofluids based on various assumptions is studied by various authors, Xuan and Li (2003), Tzeng (2005), Buongiorno (2006), Vadasz (2006), Tzou (2008a, 2008b), Alloui et. al. (2010), Kuznetsov and Nield (2010a, b, c), Nield and Kuznetsov (2009, 2010a, b, c, 2011a, b), Sheu (2011a, b), Kim et. al. (2011), Yadav et. al. (2011), Chand and Rana (2012a, b, c).

Most often the fluid is not pure but usually permeated with suspended particles or dust particles. Sufficient motivation for the study of suspended particles is the fact that knowledge concerning fluid-particle mixture is not commensurate with their industrial and scientific importance. Scanlon and Segel (1973), Srivastva (1979) and recently Chand (2011), Rana and Thakur (2012), Rana et al. (2012) studied the effect of suspended particles on Bénard convection and found that the critical Rayleigh number was reduced solely because the heat capacity of pure gas was supplemented by that of particles and it was found that suspended particles destabilize the fluid layer. The rotation also has a significant effect on the onset of thermal instability. It introduces a number of new elements in fluid dynamics, and some of its consequences are unexpected, for example, the role of viscosity is inverted. The origin of this and other consequences of rotation can be attributed to certain theorems relating to vorticity, in the dynamics of rotating fluids. Chandrasekhar (1961) studied the effects of rotation on the onset of thermal instability in detail. In the present paper an attempt has been made to study the combined effect of rotation and suspended particles on thermal convection in a layer of nanofluid layer with free boundaries.

2. Mathematical formulations of the problem

Consider an infinite horizontal layer of nanofluid with suspended particles of thickness ‘d’ bounded by plane z = 0 and z = d, heated from below. Fluid layer is rotating uniform about z-axis with angular velocity W(0, 0, W) and is acted upon by gravity force g (0, 0, -g) as shown in figure 1. The temperature T and volumetric fraction φ of nano particles taken to be T₀ and φ₀at z = 0 and T₁ and φ₁at z = d, (T₀> T₁). The reference temperature is taken to be T_1.

The mathematical equations describing the physical model are based upon the following assumptions

i) Themophysical properties expect for density in the buoyancy force (Boussinesq Hypothesis) are constant;

ii) The fluid phase and nano particles are in thermal equilibrium state;

iii) Nano particles are spherical;

iv) Nanofluid is incompressible, Newtonian and laminar;

v) Each boundary wall is assumed to be impermeable and perfectly thermal conducting;

vi) Radiation heat transfer between the sides of wall is negligible when compared with other modes of the heat transfer.

Figure 1 Physical configuration of the problem

Fig.2 Variation of Rayleigh number with wave number for different value of Taylor number Ta

Fig.3 Variation of Rayleigh number with wave number for different value of suspended particles parameter B

Fig.4 Variation of Rayleigh number with wave number for different value of Lewis number Le

Fig.5 Variation of Rayleigh number with wave number for different value of modified diffusivity ratio NA

Fig. 2 shows the variation of stationary Rayleigh number with wave number for different value of Taylor number and it is found that the Rayliegh number Ra increases as the values of Taylor number parameter Ta inceases, thus rotation stabilizes the fluid layer.

Fig. 3 shows the variation of stationary Rayleigh number with wave number for different value of suspended particles parameter B and it is found that the Rayliegh number Ra decreases as the values of suspended particles parameter B inceases, thus suspended particles has destabilizing effect on the system.

Fig. 4 shows the variation of stationary Rayleigh number with wave number for different value of Lewis number Le and it is found that the Rayliegh number Ra increases as Le number increases, thus Lewis number has stabilizing effect on the system. These results are same as obtained by Chand and Rana (2012a, b, c) and Nield and Kuznetsov (2010a, b, 2011a, b).

Fig. 5 shows the variation of stationary Rayleigh number with wave number for different value of with modified diffusivity ratio N_A and it is found that the Rayliegh number Ra slightly increases as values of modified diffusivity ratio N_A inceases. Thus modified diffusivity ratio N_A has stabilizing effect on the system. These results are same as obtained by Chand and Rana (2012a, b) and Nield and Kuznetsov (2010a, b, 2011a, b, c).

It is observed that parameter B appears in equation (42), thus presence of suspended particles have significant influence oscillatory convection in a layer of nanofluid layer.

It is also noted that the parameter Ta (rotation) does not appear in equation (42), thus rotation has no effect on oscillatory convection.

5. CONCLUSION:

A linear stability analysis for a layer of nanofluid in the presence of rotation and suspended particles for free-free boundaries is investigated. Expressions for Rayleigh number for the stationary and oscillatory convection are obtained. The main conclusions are:

(i) The critical cell size is not a function of any thermo physical properties of nanofluid.

(ii) The suspended particles have destabilizing effect on stationary convection.

(iii) The Lewis number Le, Taylor number Ta and modified diffusivity ratio NA have stabilizing effect on the stationary convection.

(iv) Suspended particles have significant influence oscillatory convection while rotation has no effect on oscillatory convection.

(v) The stability is phenomenon due the buoyancy coupled with the conversation of nanoparticles.

6. ACKNOWLEDGMENT:

The first author is thankful to University Grants Commission (UGC) New Delhi of India for their financial support under Minor Research Project [F. No. 8-3(123)/2011(MRP/NRCB] during this work.

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