Magneto convection of rotating nanofluids in porous medium: metals and semiconductors as nanoparticles

Jyoti Ahuja¹, Urvashi Gupta^2*

¹Energy Research Centre, Panjab University, Chandigarh-160014, India

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

*Corresponding Author E-mail: dr_urvashi_gupta@yahoo.com, urvashi@pu.ac.in

ABSTRACT:

Present paper investigates the onset of thermal convection of a porous nanofluid layer under the combined influence of rotation and magnetic field using Darcy-Brinkman model which is the modification of the Darcy model where in viscous shear effects and change in viscosity are accounted in momentum equation.In the present model, due to the presence of rotation Coriolis force term and due to the presence of magnetic field Lorentz’s force term get introduced in the conservation equations of momentum coupled with the Maxwell’s equations. To investigate the combined effect of these forces on the thermal convection of nanofluid layer normal mode technique and Galerkin type weighted residual method is applied. A comparative analysis of the thermal instability of metallic (Cu, Ag) and semiconducting (TiO₂, SiO₂) nanofluids is examined. It is observed that metals are more stable as compared to semiconductors. Further, it is found that silver nanoparticles stabilize the water based fluid more as compared to copper nanoparticles and in semiconductors TiO₂nanoparticles enhance the stability of the system more as compared to SiO₂ nanoparticles. Stability of the system enhances with the rise in Chandrasekhar number and Taylor number whereas it falls with the rise in porosity. An additional parameter i.e. Darcy number is introduced due to the consideration of Darcy Brinkman model which has a stabilizing effect on the nanofluid layer system.

KEYWORDS: Nanofluid, Brownian motion, Thermophoresis, Darcy number, Chandrasekhar number, Taylor number.

1. INTRODUCTION:

The concept of nanofluids has improvised the heat transfer mechanism by replacing the suspension of micrometer sized particles with nanometer sized particles in conventional fluids. These nanometer sized particles are called as nanoparticles which may be metals, metal oxides, carbides, nitrides and semiconductors. The host liquids may be water, ethylene glycol and propylene glycol etc. The magnificent idea of introducing nanofluids first came into the mind of Choi [1] who claimed the enhanced heat transfer with the addition of nanoparticles. However this enhancement is very small with the low nanoparticle concentration (Kim et al. [2]; Putnam et al., [3]; Witharana et al. [4].) All these reports motivated Buongiorno [5] to develop a system of conservation equations for nanofluids with-in the effect of Brownian motion and thermophoresis. He concluded that effect of these forces result in decrease of viscosity of nanofluid which leads to the heat transfer enhancement. The benchmarking exercise on the problem of thermal convection for nanofluids has been done by Kim et al. [6]. Tzou [7] studied the problem analytically and used eigen function expansion method to solve the conservation equations given by Buongiorno and found that critical Rayleigh number is reduced with the addition of nanoparticles. By using single term Galerkin approximation Nield and Kuznetsov [8] found the expression for thermal Rayleigh number and the condition of overstability in porous medium. After concluding the results of Darcy model Nield and Kuznetsov [9] have taken up Darcy-Brinkman model for a nanofluid layer in porous medium. Further, Bhadauria et al. [10] and Yadav et al. [11] made an extension of the thermal instability problem by introducing Coriolis force term (rotation) in momentum equations for both porous/non-porous mediums and concluded that addition of Coriolis force term in momentum equation made the overall system more stable. Gupta et al. [12,13] studied the effect of magnetic field and Hall currents, respectively, on a nanofluid layer for both free boundaries and have shown that the critical wave number and the associated Rayleigh number undergo a significant increase with the increase in Chandrasekhar number and decrease with the increase in Hall parameter.

Considering the conflicting tendencies of magnetic field and rotation while acting simultaneously, we have examined the combined influence of rotation and magnetic field on a nanofluid layer system. In addition to the body and buoyancy forces; Coriolis force term (with the inclusion of rotation) and Lorentz force term (with the inclusion of magnetic field) are introduced. With the introduction of magnetic field and rotation in the system and with the use of Darcy –Brinkman model, Chandrasekhar number, Taylor number and Darcy number are introduced. Rotation and magnetic field usually have the effect of inhibiting the onset of thermal instability while acting separately.

2. Governing equations in porous medium:

Here the system is comprised by taking a horizontal layer of nanofluid in porous medium which is heated underside. Porous nanofluid layer of porosity ε is bounded by two free boundaries in x-y plane and z-axis is directed vertically upward. The nanofluid layer has attained the temperature at lower and upper boundaries respectively while heating from the bottom. Nanoparticles are distributed in such a way that concentration of nanoparticles isat lower and upper boundaries respectively.

3. Numerical results and discussion:

Let us interpret the impact of magnetic field, permeability and other nanofluid parameters on the stability of the system. Stability of the system w.r.t. nanofluid parameters is examined by analyzing the behavior of thermal Rayleigh number with the variation in respective parameter. Numerical calculations are accomplished by using Eq. (18) for stationary mode of heat transfer and Eq. (25) for oscillatory mode of heat transfer respectively. Graphical interpretation is carried out for metallic nanofluids (Cu, Ag) and semiconductors (TiO₂ and SiO₂) with the help of Mathematica software.

5.1. Metallic nanofluids:

Ultrafine metallic nanoparticles dispersed in base fluids are known as metallic nanofluids. Thermal conductivity of nanofluids gets increased by using nanoparticles of metallic substances due to their high thermal conductivity. In the present paper, nanoparticles of metals (Cu and Ag) are dispersed in the base fluid water for analyzing the stability of metallic nanofluids. The physical properties of nanofluids vary with the variation in shape, size and volumetric fraction of nanoparticles as shown by Yang et al. [15]. The thermophysical properties of the nanofluids, considered in this study, are supposed to be constant except for the density variation, which is determined based on the Boussinesq approximation. The values of nanofluid parameters at (concentration of nanoparticles) are for Cu-water nanofluid and for Ag-water nanofluid.

5.2. Semiconductors:

An electrical conductivity of a semiconductor materials lies between the conductivity of a conductor (like copper), and an insulator (like glass). The physical properties of semiconductor materials are based on the quantum physics which explains the movement of electrons and holes in a crystal lattice. The unique arrangement of the crystal lattice makes silicon and titanium the most commonly used elements in the preparation of semiconducting materials. The values of nanofluid parameters at (concentration of nanoparticles) are for TiO₂-water nanofluid and for SiO₂-water nanofluid.

5.3. Comparative analysis of thermal instability of metallic nanoparticles with semiconductors:

After fixing the values of other parameters (like rotation and magnetic field), a comparative study is carried out for different nanofluids (metallic/semiconducting) by taking the variation in thermal Rayleigh number with wave number. For 0.1 % difference in volume fraction of nanoparticles, the parametric values of different nanofluids are calculated using the physical properties. In Fig. 2, thermal Rayleigh number Ra is plotted with the wave number 

Fig.2. Comparison of metallic nanofluids and semiconductors under the simultaneous influence of rotation and magnetic field

for different nanofluids under consideration. It is observed that Ag-water nanofluid exhibits highest stability as its curve holds higher values of among all other nanofluids whereas curve of SiO2-water nanofluid lies at the bottom representing least stability. It is noticed that density of different nanofluids follow the ordering as Ag>Cu>TiO2>SiO2 which is the same as for the stabilizing influence of nanofluids in the presence of rotation and magnetic field. Thus we may say that nanoparticles having high values of density exhibit more stability.

5.4. Impact of different nanofluid parameters:

Figures 3-10 show the impact of different nanofluid parameters on the thermal instability of the system under the simultaneous influence of rotation and magnetic field. In order to examine the behavior of a particular nanofluid parameter on the thermal instability of the nanofluid, variation in that particular parameter is done by fixing the other parameters as .

Figures 3-4 are plot of with the variation in Taylor number for the nanofluids with nanoparticles as metals and semiconductors respectively. It is noticed from these graphs that rise in the value of rotation parameter, the values of thermal Rayleigh number get incremented for both types of convection; which establishes the stabilizing effect of rotation. Also, in case of stationary convection; the curves showing the effect of Taylor number for Ag–water nanofluid lie above the curves for Cu-water nanofluid which means that Ag-water nanofluid exhibits higher stability as compared to Cu-water nanofluid. On the other hand, if the case of semiconductors is discussed (Fig. 4), it is found that TiO₂-nanoparticles enhance the stability of nanofluid more as compared to SiO₂-nanoparticles with the increase in Taylor number. While in the case of oscillatory motion, influence of Taylor number on the thermal instability of different nanofluids (considered here) is comparatively negligible. In oscillatory mode of convection, values of thermal Rayleigh number for all the types of nanofluids approximately lie on the same curve for the same value of Taylor number. It is also interpreted from the figures that metals inhibit the onset of convection as compared to semiconductors under the combined influence of magnetic field and rotation.


Fig. 3. Impact of Taylor number on the thermal instability of Cu-water and Ag-water nanofluid.	Fig.4. Impact of Taylor number on the thermal instability of TiO₂-water and SiO₂-water nanofluid.


Fig.5. Impact of Chandrasekhar number on the thermal instability of Cu-water and Ag-water nanofluid.	Fig.6. Impact of Chandrasekhar number on the thermal instability of TiO₂-water and SiO₂-water nanofluid.

Figures 5-6 signify the impact of applied magnetic field on the thermal instability of nanofluids with nanoparticles as metals and semiconductors respectively, for stationary as well as oscillatory convections. It can be seen from the figures that stabilizing effect increases as the magnitude of applied magnetic field gets increasesed. Thus the vertical magnetic field enhances the stability of the system for both types of convection. Also, by increasing the magnitude of magnetic field, Ag-water and TiO₂-water nanofluids exhibits higher stability than Cu-water and SiO₂-water nanofluids respectively. Further, these graphical representations show that the heat transfer takes place through oscillatory convection instead of stationary convection.


Fig.7. Impact of porosity on the thermal instability of Cu-water and Ag-water nanofluid	Fig.8. Impact of porosity on the thermal instability of TiO₂-water and SiO₂-water nanofluid.


Fig.9. Impact of Darcy number on the thermal instability of Cu-water and Ag-water nanofluid.	Fig.10. Impact of Darcy number on the thermal instability of TiO₂-water and SiO₂-water nanofluid.

In continuation to observe the impact of other parameters, now the effect of Darcy number and porosity is interpreted in the Figs. 7-10 where it is discovered that Darcy number enhances the stability of the system while the porosity does the reverse by reducing the stability of the system. By increasing the Darcy number and porosity of the medium, Ag-water and TiO₂-water nanofluids exhibits higher stability than Cu-water and SiO₂-water nanofluids respectively.

6. CONCLUSIONS:

In the present paper, magneto-convection of rotating nanofluid layer in porous medium using Darcy-Brinkman model is investigated for both free boundaries. The effects of various nanofluid parameters on the stability of the basic flow are analysed with different types of nanoparticles-metallic (copper (Cu) and silver (Ag)) and semiconducting (Titanium dioxide (TiO₂) and Silica (SiO₂)) with water as the base fluid. The instability sets in through the mode of oscillatory motions instead of stationary convection for the present configuration of nanoparticles. The existence of oscillatory motions in the system is due to the occurrence of two buoyancy forces occurring in the opposite directions i.e. density variation (due to heating from the bottom) and density of nanoparticles (at the bottom). Both the external forces; Lorentz force & Coriolis force enhance the stability of the system for both types of convection: stationary as well as oscillatory motions while porosity of the medium destabilizes the system. Thus magnetic field and rotation postpone the onset of convection while porosity advances the same. With the increase of the distribution of nanoparticles at the lower boundary; thermal Rayleigh number increases i.e. stability of the system enhances and the impact of Darcy number is to contribute towards the stability of the system. Based upon the comparative study of thermal instability carried out by taking metallic and semiconducting nanofluids it is established that metallic nanofluids are more stable than semiconducting nanofluids in the presence of rotation and magnetic field. Also, Ag-water nanofluid exhibit higher stability than Cu-water nanofluid in metals and in semiconductors TiO₂-water nanofluid exhibit higher stability than SiO₂-water nanofluid under the simultaneous influence of magnetic field and rotation.

7 ACKNOWLEDGEMENT:

One of the authors Prof. Urvashi Gupta thanks to Council of Scientific and Industrial Research, New Delhi-110012, India for their financial assistance in the form of Research and Development Project [Ref. No: 25(0247)/15/EMR-II].

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Received on 22.11.2016 Modified on 29.11.2016

DOI: 10.5958/2349-2988.2017.00022.5

Research J. Science and Tech. 2017; 9(1):135-142.