Dual Solutions in MHD Boundary Layer Nanofluid Flow and Heat Transfer with Heat Source/Sink considering Viscous Dissipation

 

Ruchika Dhanai,  Puneet Rana, Lokendra Kumar

Department of Mathematics, Jaypee Institute of Information Technology, A-10, Sector-62, Noida-201307, Uttar Pradesh, India.

*Corresponding Author: puneetranaiitr@gmail.com

 

ABSTRACT:

In this present analysis, the numerical investigation of steady, magneto-hydrodynamic boundary-layer nanofluid past a permeable shrinking sheet has been discussed considering Brownian motion and thermophoresis. The effect of viscous dissipation, heat source/sink and suction/injection are taken into account and controlled by the non-dimensional parameters. After using appropriate similarity transformation, the final system of ordinary differential equation is solved numerically by shooting technique. The dual solutions exist for   whereas unique solution is obtained at critical values   and solution does not exist for   for other fixed parameters. The current study shows that the effect of nonlinear parameter, Hartmann number and heat source/sink on skin friction and rate of heat transfer. The results are validated for the limiting cases.

 

KEY WORDS: MHD, Shrinking sheet, Nanofluids, Dual solution, Viscous dissipation.

 

1. INTRODUCTION

Motivated by the importance of stretching/shrinking sheet in various industries and engineering processes like glass blowing, annealing and thinning of copper wires, the boundary layer flow over a stretching sheet was studied by Crane [1]. This study has recently been extended to nanofluid by Khan and Pop [2] using two-component slip mechanism based model. Then after, Rana and Bhargava [3] have employed finite element method for the numerical computation of flow and heat transfer characteristic over a nonlinearly stretching sheet. Moreover, analytical solution of the boundary layer flow over an exponential stretching sheet has been investigated Nadeem and Lee [4] using homotopy analysis method. 

 

The study of magneto-hydrodynamic has numerous applications in engineering, agriculture and petroleum industries. The problem of natural convection under the effect of a magnetic field has also applications in geophysics and astrophysics [5]. Fang and Zhang [6] have given exact solution for MHD flow equation of fluid over a shrinking sheet. They reported two solution branches for  but for M=1 single solution branch is obtained only in case of suction and when  there is also single branch of solution for both suction and injection. In 2011, Hamad [7] investigated the analytical solution of electrical conducting nanofluid flow over a linearly stretching sheet under the influence of magnetic field. He found that momentum boundary layer thickness decreases but thermal boundary thickness increases with magnetic field. Rana et al. presented unsteady MHD transport phenomena over a stretching sheet in a rotating nanofluid [8]. Numerical investigation of the MHD flow and heat transfer of nanofluid between two horizontal plates in rotating system using Cu, Ag, Al2O3 and TiO2 nanoparticles in water has been computed by Sheikholeslami et al. [9] and it is noticed that heat transfer is the highest for TiO2 nanoparticles. Currently, much attention has been devoted to work in the presence of magnetic field [10-14]. Most of the studies are also based on temperature dependent heat source/sink which may affect the heat transfer characteristics. Chamkha and Aly [15], and Rana and Bhargava [16] have considered the effect of heat generation and absorption (source/sink) in their study.

 

The main concern of current study is to investigate the dual solution for the combined effects of magnetic field, mass suction transfer, and viscous dissipation for steady boundary layer nanofluid flow over a power- law stretching/shrinking sheet in the presence of heat source/sink by using Kuznetsov and Nield revised nanofluid model [17]. The variation of skin friction, rate of heat transfer, temperature and nanoparticle concentration are obtained using shooting method [18] with Runge-Kutta fourth order method and presented graphically in this paper.

 

 

2. NANOFLUID TRANSPORT MODEL

 

Fig 2. The effect of magnetic field M on velocity profile .

Fig 3. The effect of heat source/sink parameter Q on rate of heat transfer  for different values of mass transfer parameter s.

Fig 4. The effect of Eckert number Ec on rate of heat transfer  for different values of mass transfer parameter s.

Fig 5. Temperature profile  for different value of mass transfer parameter.

Fig 6. Temperature profile  for different values of Prandtl number Pr and the effect of Lewis number on nanoparticle volume fraction .

Figs. 1 presents the effect of power-law parameter m with and without magnetic field M on the skin friction . The two branches of solution are reported in the certain domain of mass transfer parameter s and terminated at critical value  which varies with magnetic field and power law parameter. It can also be seen that beyond  ( ), no solution exists and at only unique solution exists. The first solution of skin friction increases and second solution decreases with magnetic field whereas the both solutions decrease with power-law parameter. In the presence of magnetic field, a nanofluid experience Lorentz force which slows down the motion. This can also be justified from Fig. 2, where magnitude of velocity decreases as magnetic field increases.

Further, the effect of heat source/sink Q and viscous dissipation parameter Ec (Eckert number) on the rate of heat transfer  for different values of mass transfer parameter s is investigated in Figs. 3-4 and dual solutions are captured for the rate of heat transfer, which is decreasing with increasing values of Q and Ec for first solution. The critical value remains unaffected with the both parameters, such that no solution exists for . The temperature profile is discussed with the variations of mass suction parameter s and Prandtl number Pr in Figs. 5-6. Temperature increases as mass transfer parameter increases while it decreases with increasing value of Pr. Fig. 6 depicts that both first and second solutions of nanoparticle concentration is lower at the surface for large value of Lewis number.

 

5. CONCLUSION:

In this paper, we have investigated the two branches of solution for MHD boundary layer flow and heat transfer of nanofluid over a power-law stretching/shrinking surface, taking into consideration the effects of heat source/sink and viscous dissipation. Following results are obtained:

·        The critical value  is found for the existence of both first and second solutions.

·        At the surface, Skin friction decrease with increasing value of power-law parameter whereas Skin friction increases as magnetic field increases and magnitude of velocity is lower for higher value of magnetic field.

·        The rate of heat transfer decreases with heat source/sink parameter and Eckert number.

·        Temperature of nanofluid increases with mass transfer parameter and decreases with Pr.

 

 

6. REFERENCES:

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2.       Khan W.A., Pop I., “Boundary-layer flow of a nanofluid past a stretching sheet”, International Journal of Heat and Mass Transfer, Vol. 53, pp. 2477-2483, 2010.

3.       Rana P., Bhargava R., “Flow and heat transfer over a nonlinearly stretching sheet: A numerical study”, Communication in Nonlinear Science and Numerical Simulation, Vol. 17, pp. 212-226, 2012.

4.       Nadeem S., Lee C., “Boundary layer flow of nanofluid over an exponentially stretching surface”, Nanoscale Research letters, Vol. 7, 2012.

5.       Ganesan P., Palani G., “Finite difference analysis of unsteady natural convection MHD flow past an inclined plate with variable surface heat and mass flux”, International Journal of Heat and Mass Transfer, Vol. 47, pp. 4449-4457, 2004.

6.       Fang T., Zhang J., “Closed-form exact solution of MHD viscous flow over a shrinking sheet”, Communication Nonlinear Science Numerical Simulation, Vol. 14, pp. 2853-2857, 2009.

7.       Hamad M.A.A., “Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field”, International Communications in Heat and Mass transfer, Vol. 38, pp. 487-492, 2011.

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9.       Sheikholeslami M., Hatami M., Ganji D.D., “Nanofluid flow and heat transfer in a rotating system in the presence of a magnetic field”, Journal of Molecular Liquids, Vol. 190, pp. 112-120, 2014.

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11.     Sheikholeslami M., Bandpy M.G., Ellahi R, Zeeshan A., “Simulation of MHD CuO–water nanofluid flow and convective heat transfer considering Lorentz forces”, Journal of Magnetism and Magnetic Materials, Vol. 369, pp. 69-80, 2014.

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15.     Chamkha A. J., Aly A. M.  “MHD free convection flow of a nanofluid past a vertical plate in the presence of heat generation or absorption effects”, Chemical Engineering Communications, Vol. 198, pp. 425-441, 2011.

16.     Rana P., Bhargava R., “Numerical study of heat transfer enhancement in mixed convection flow along a vertical plate with heat source/sink utilizing nanofluids”, Communication Nonlinear Science Numerical Simulation, Vol. 16, pp. 4318-4334, 2011.

17.     Kuznetsov A.V., Nield D.A., “Natural convection boundary-layer of a nanofluid past a vertical plate: A revised model”, International Journal of Thermal Sciences, Vol. 77, pp.126-129, 2014.

18.     Na T.Y., “Computational Method in Engineering Boundary Value Problems”, Academic Press, New York, 1979.

 

 

Received on 11.12.2014                    Accepted on 29.01.2015

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Research J. Engineering and Tech. 6(1): Jan.-Mar. 2015 page 142-148

DOI: 10.5958/2321-581X.2015.00021.5