Effect of rotation and suspended particles in Thermosolutal Instability of Maxwell Visco-Elastic fluid in porous medium

Thermal Instability of Rotating Kuvshiniski Viscoelastic Nanofluid in a Porous Medium

Ramesh Chand^*, G. C. Rana,S. K. Kango

¹Department of Mathematics, Government College Nurpur, Himachal Pradesh, India

²Department of Mathematics, NSCBM Govt. College, Hamirpur-177005, Himachal Pradesh, INDIA

³Department of Mathematics, Government College Barsar, Himachal Pradesh, India

*Corresponding Author E-mail: rameshnahan@yahoo.com, drgcrana15@gmail.com

ABSTRACT:

Thermal instability in a horizontal layer of Kuvshiniski viscoelastic nanofluid for more realistic boundary conditions is studied theoretically within the frame work of linear theory. The flux of volume fraction of nanoparticles is taken to be zero on the isothermal boundaries. For porous medium Brinkman model is taken into consideration and the model used for nanofluid incorporates the effect of Brownian diffusion and thermophoresis. The stability criterion for stationary and oscillatory convection have been derived and graphs have been plotted to study the effects of rotation, the Brinkman-Darcy number, the Lewis number, the modified diffusivity ratio and porosity parameter on the stationary convection.

KEYWORDS: Kuvshiniski viscoelastic fluid, rotation, Brownian motion, Galerkin technique, porous medium.

1. INTRODUCTION:

Nanofluids have novel properties that make them potentially useful in wide range of engineering applications where cooling is of primary concern. Nanofluid used as heat transfer, chemical nanofluids, smart fluids, bio nanofluids, medical nanofluids (drug delivery and functional tissue cell interaction) etc. in many industrial applications. Thermal instability of viscoelastic (non-Newtonian) nanofluids in porous media has gained much attention from the researchers because of its engineering and industrial applications. Due to the growing importance of viscoelastic nanofluids in technology and industries, the investigations of such fluids are desirable. The study of thermal instability in a viscoelastic (non-Newtonian) nanofluid in porous medium is relatively of recent origin and it is still in a rudimentary stage. Sheu (2011), Chand and Rana (2012a, 2015a, 2015b), Rana and Chand (2015a), Rana et al. (2016), Yadav et al. (2014, 2016), Umavathi et al. (2015) and Shivakumara et al. (2015) studied the onset of convection in a horizontal layer of porous medium saturated by a viscoelastic nanofluid. Rotation also play important role in the thermal instability of fluid layer and has applications in rotating machineries such as nuclear reactors, petroleum industry bio mechanics etc.

Recently, Nield and Kuznetsov (2014) pointed out that this type of boundary condition on volume fraction of nanoparticles is physically not realistic and it is difficult to control the nanoparticle volume fraction on the boundaries and suggested the normal flux of volume fraction of nanoparticles is zero on the boundaries as an alternative boundary condition which is physically more realistic. Thereafter Chand and Rana (2014, 2015b, 2015c), Chand et al. (2014, 2015) and Rana and Chand (2015a, 2015b) Yadav et al. (2015), Shivakumara and Dhananjaya (2015) investigated the thermal instability of nanofluid problems for more realistic boundary conditions. In this paper we extend the study of Chand and Rana (2012b) by taking Kuvshiniski viscoelastic fluid base fluid for more realistic boundary conditions.

2. MATHEMATICAL FORMULATIONS OF THE PROBLEM:

Considered an infinite horizontal layer of Kuvshiniski visco-elastic nanofluid of thickness‘d’ bounded by plane z = 0 and z = d, heated from below in a porous medium of medium permeability k₁ and porosity ε. Fluid layer is acted upon by a gravity force g(0,0,-g) and is heated from below in such a way that horizontal boundaries z = 0 and z = d respectively maintained at a uniform temperature T₀ and T₁ (T₀> T₁) as shown is Fig.1 The normal component of the nanoparticles flux has to vanish at an impermeable boundaries and the reference scale for temperature and nanoparticles fraction is taken to be T₁and φ₀respectively. It is assumed that nanoparticles are suspended in the nanofluid using either surfactant or surface charge technology. This prevents the particles from agglomeration and deposition on the porous matrix.

Fig.1 Physical configuration of the problem

6. RESULTS AND DISCUSSION:

To study the effect of Lewis number, rotation, Brinkman Darcy number, modified diffusivity ratio, nanoparticles Rayleigh number, and porosity parameter on stationary convection the computations are carried out using equation (36) and for most of nanofluid, the value of Lewis number Le is of range 10 ~10³,thermal Rayleigh number Ra is of range 10²~ 10⁴, modified diffusivity ratio N_A is of range 1 ~ 10, nanoparticles Rayleigh number Rn is of range 1 ~ 10, Taylor number is of the range 10²~ 10⁴, Brinkman-Darcy number is of range 10^-2 ~ 10^-1 and porosity parameter is of the range range 10^-2 ~ 10^-1 [Chand and Rana (2015a, b), Yadav et al. (2016)].

Figs. 2 – 7 demonstrate the neutral curve on the (Ra_s, a) plane for different values of the Lewis number Le, the Taylor number Ta, Brinkman number D̃a, the modified diffusivity ratio N_A,the nanoparticles Rayleigh Rn and porosity parameter ε.

Fig. 2 Variation of the stationary Rayleigh number with wave number for different value of Lewis number

Fig.3 Variation of the stationary Rayleigh number with wave number for different value of Taylor number

Fig. 4 Variation of the stationary Rayleigh number with wave number for different value of Brinkman- Darcy number

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

Fig. 6 Variation of the stationary Rayleigh number with wave number for different value of concentration Rayleigh number

Fig. 7 Variation of the stationary Rayleigh number with wave number for different value of porosity parameter

Fig. 2 shows the variation of stationary Rayleigh number with wave number for different value of Lewis number with the fixed value of other parameters. It is found that stationary Rayliegh number decreases as the value of Lewis number increases, indicating that Lewis number destabilize the stationary convection. It is due to the fact that thermophoresis at a higher value of thermophoretic diffusivity is more supportable to the disturbance in nanofluids, while both thermophoresis and Brownian motion are driving forces in favor of the motion of nanoparticles. It is the good agreement of the result obtained by Chand and Rana (2015a,b).

Fig. 3 shows the variation of Rayleigh number with wave number for different value of the Taylor number, with fixed value of other parameters and it has been found that the Rayleigh number increases with increase in the value of Taylor number thus rotation has stabilizing effect on the system in stationary convection. It is the good agreement of the result obtained by Chand and Rana (2012b).

Fig. 4 shows the variation of Rayleigh number with wave number for different value of the Brinkman Darcy number and it has been found that the Rayleigh number first increases then decreases and finally increases with increase in the value of Brinkman Darcy number, thus Brinkman Darcy number has both the stabilizing and destabilizing effect on the fluid layer. The destabilizing effect of Brinkman Darcy number depends upon the value of the Taylor number. It is the good agreement of the result obtained by Chand and Rana (2012b).

Fig. 5 shows the variation of stationary Rayleigh number with wave number for different value of modified diffusivity ratio with fixed value of other parameters and it is found that stationary Rayliegh number decreases with an increase in the value of modified diffusivity ratio, which indicate that modified diffusivity ratio destabilize the stationary convection. This may lead to an increase in volumetric fraction, which shows that Brownian motion of the nanoparticles will also increase, which may cause destabilizing effect. It is the good agreement of the result obtained by Chand and Rana (2015a,b).

Fig. 6 shows the variation of stationary Rayleigh number with wave number for different values of nanoparticle Rayleigh number with fixed value of other parameters and it is found that stationary Rayliegh number decreases as the value of the nanoparticles Rayleigh number increases, which mean that nanoparticle Rayleigh number has destabilizing effect on the stationary convection. It has destabilizing effect because the heavier nanoparticles moving through the base fluid makes more strong disturbances as compared with the lighter nanoparticles. It is the good agreement of the result obtained by Chand and Rana (2015a,b).

To assess the effect of porous medium on the stability of the system, the variation of the stationary Rayleigh number is shown in Fig. 7 as a function of wave number a for different values of the porosity ε. We found that with an increase in the value of the porosity ε, the stationary Rayleigh number increases, indicating that it delays the onset of convection in nanofluid saturated in porous medium. It is the good agreement of the result obtained by Chand and Rana (2015a,b).

7. CONCLUSIONS:

Thermal convection in a horizontal layer of Kuvshiniski visco-elastic nanofluid in a porous is studied. Brinkman model is used for porous medium. The flux of volume fraction of nanoparticles is taken to be zero on the isothermal boundaries and the eigen value problem is solved using the Galerkin residual method. For stationary convection Kuvshiniski visco-elastic nanofluid behaves like an ordinary Newtonian nanofluid and instability is purely a phenomenon due to buoyancy coupled with the conservation of nanoparticles. It is found that the rotation and porosity parameter have stabilizing effect while Lewis number, the modified diffusivity ratio and the concentration Rayleigh number have destabilizing effect on the stationary convection. Brinkman- Darcy number has both stabilizing and destabilizing effect depending upon the value of the Taylor number.

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Received on 05.11.2016 Modified on 17.11.2016

DOI: 10.5958/2349-2988.2017.00001.8

Research J. Science and Tech. 2017; 9(1):01-08.