Binary Nanofluid Convection for Darcy-Brinkman Model in Hydromagnetics

Jyoti Sharma¹, Urvashi Gupta²*

¹U.I.E.T., Panjab University, Chandigarh-160014, INDIA

²Dr. S. S. Bhatnagar University Institute of Chemical Engineering and Technology, Panjab University, Chandigarh-160014, INDIA

*Corresponding Author Email: dr_urvashi_gupta@yahoo.com

ABSTRACT:

The effect of an externally impressed magnetic field on the onset of convection in a binary nanofluid layer saturating a Darcy-Brinkman model for porous medium is considered in this work. Magnetic field imparts to the fluid rigidity as well as some properties of elasticity which results in the disturbances by new mode of wave propagation. The effects of Brownian motion and thermophoresis due to the presence of nanoparticles and the effects of Dufour and Soret parameters due to the presence of solute have been included in the model under investigation. For analytical study, valid approximations are made in the complex expression for Rayleigh number to get interesting results. The mode of instability is invariably through stationary convection for top heavy distribution of nanoparticles in the fluid. Oscillatory mode of convection is possible for bottom heavy configuration and the frequency of oscillation shows a decrease with an increase in Chandrasekhar number and porosity. It is found that alumina nanoparticles stabilize the water based fluid more than copper nanoparticles which in turn have more stabilizing influence than silver nanoparticles. Fluids do not have much influence of physical properties of nanoparticles on the stability when the particles concentrate more on the bottom of the layer. The stability of the fluid rises appreciably due to the inclusion of magnetic field. Effects of various parameters on the stability of the fluid are studied numerically using Mathematical software and results are shown graphically.

KEYWORDS: Darcy-Brinkman model; Binary nanofluid convection; Chandrasekhar number; Porosity; Thermophoresis; Brownian motion.

1. INTRODUCTION:

Nanotechnology deals with the manufacturing of objects with dimensions of less than hundred nanometers. The idea of suspension of these nanoparticles in a fluid called nanofluid for improving thermal efficiency has been proposed by Choi [1]. Unlike millimeter or micrometer sized particles, nanoparticles in fluid remain stable with very little gravitational settling over long periods of time due to their small size was established by Anoop et al. [2]. Buongiorno [3] derived the conservation equations for nanofluids using the laws of fluid dynamics and the effects of Brownian diffusion and thermophoresis due to the presence of nanoparticles in the fluid. Tzou [4] studied the stability of nanofluids using the method of eigen function expansions to solve conservation equations given by Buongiorno [3]. He established that the nanofluids are less stable than regular fluids. Further, Nield and Kuznetsov [5] and Gupta et al. [6] considered the same set of equations to analyze the convection and magneto convection, respectively, in a nanofluid layer using Galerkin method. The instability of fluids in porous medium has become a demanding problem due to its application in various fields such as storage of agricultural products, geothermal reservoirs and the pollute transport in underground. Initially Darcy model was used for porous medium and by the time it has been extended to evolve as Darcy-Brinkman model. Kuznetsov and Nield [7] studied the instability of nanofluids in porous medium using Darcy- Brinkman model. Linear and nonlinear study on the instability of nanofluids with rotation in porous medium was made using Brinkman model by Chand and Rana [8] and Bhadauria and Agarwal [9] respectively. Convection in a fluid layer in which nanoparticles are suspended in binary fluid is called binary nanofluid convection. Kim et al. [10] studied the effects of Dufour and Soret parameters (which arise due to the presence of solute) on binary nanofluid convection. Double diffusive nanofluid convection in a porous as well as non-porous medium was studied for the first time by Nield and Kuznetsov [11, 12]. The results were approximated by limiting to large nanoparticle Lewis number and large Prandtl number. Further, Gupta et al. [13] and Yadav et al. [14] considered the instability of binary nanofluid layer for bottom and top heavy suspension of nanoparticles, respectively. Recently, Gupta et al. [15] studied the binary nanofluid convection under magnetic field. They have made valid approximations in the complex expressions for analytical study and alumina-water nanofluid is used for numerical investigations.

Convection in binary nanofluids with magnetic field in porous medium has its importance and great application in geophysics (in enhanced oil recovery from underground reservoirs) which motivated us to study the effect of magnetic field on the thermosolutal instability of a nanofluid layer in a porous medium using Darcy-Brinkman model. Due to the inclusion of magnetic field in the system and use of Darcy-Brinkman model, two additional non-dimensional parameters (Chandrasekhar number and Darcy number) are introduced. The problem is solved using normal mode analysis and one term weighted residuals method. For analytical study, we have approximated the expression for Rayleigh number by taking negligible values of Dufour and Soret parameters and limiting to the large values of nanofluid Lewis number and Prandtl number with heat capacity ratio as unity. To validate the results so found, problem is analyzed numerically by releasing the restrictions on parameters using the software Mathematica.

2. GOVERNING EQUATIONS:

A binary nanofluid layer which is heated and soluted from below is taken in the porous medium. The non-dimensional basic conservation equations in the light of Darcy-Brinkman model (Buongiorno [3], Kuznetsov and Nield [7,11], Chandrasekhar [16]) are:

7. CONCLUSIONS:

The binary nanofluid convection in a porous layer under vertical magnetic field is studied using Darcy-Brinkman model. Method of normal modes and single term Galerkin residual approximation is used. The terms corresponding to the effects of Brownian motion and thermophoresis appear in result through equation of conservation of nanoparticles instead of thermal energy equation. The critical wave number lifts with rise in Chandrasekhar number while falls with rise in porosity. The instability in the fluid for top heavy distribution of nanoparticles is through stationary mode only. Top heavy binary nanofluids are less stable than regular binary nanofluids and nanoparticles at the top of the layer make the system so unstable that temperature at the top must be increased in comparison to the bottom or applied magnetic field must be increased to acquire neutral stability. Oscillatory motions come into existence for bottom heavy distribution of nanoparticles and the frequency of oscillation rises with rise in nanoparticle volume fraction at the bottom of the layer and diminishes with rise in Chandrasekhar number, Darcy number and porosity. Bottom heavy fluids are more stable than regular fluids but stability does not get affected much with the rise in nanoparticle concentration at the bottom of the layer. For top heavy distribution of nanoparticles, alumina-water fluid is more stable than copper-water fluid which in turn is more stable than silver- water fluid while bottom heavy distribution does not get influenced by different properties of nanoparticles. The stability of the fluid rises with rise in magnetic field and diminishes with the rise in porosity while solute does not affect the stability of the fluid layer system.

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Received on 15.11.2016 Modified on 20.11.2016

DOI: 10.5958/2349-2988.2017.00014.6

Research J. Science and Tech. 2017; 9(1):93-100.