A new numerical approach for MHD laminar boundary layer flow and heat transfer of nanofluids over a moving surface in the presence of thermal radiation
© Shateyi and Prakash; licensee Springer. 2014
Received: 19 August 2013
Accepted: 1 December 2013
Published: 2 January 2014
Numerical analysis has been carried out on the problem of magnetohydrodynamic boundary layer flow of a nanofluid over a moving surface in the presence of thermal radiation. The governing partial differential equations were transformed into a system of ordinary differential equations using suitable similarity transformations. The resultant ordinary equations were then solved using the spectral relaxation method. Effects of the physical parameters on the velocity, temperature and concentration profiles as well as the local skin-friction coefficient and the heat and mass transfer rates are depicted graphically and/or in tabular form.
MSC: 65PXX, 76-XX.
Many engineering and industrial processes involve heat transfer by means of a flowing fluid in either laminar or turbulent regimes. A decrease in thermal resistance of heat transfer in the fluids would significantly benefit many of these applications/processes. Nanofluids have the potential to reduce thermal resistances, and industrial groups such as electronics, medical, food and manufacturing would benefit from such improved heat transfer. It is well known that conventional heat transfer fluids, such as oil, water and ethylene glycol mixture, are poor heat transfer fluids. Choi  introduced the technique of nanofluids by using a mixture of nanoparticles and the base fluids. The presence of the nanoparticles in the nanofluid increases the thermal conductivity and therefore substantially enhances the heat transfer characteristics of the nanofluid. Nanotechnology has been an ongoing hot topic of discussion in public health as researchers claim that nanoparticles could present possible dangers in health and environment, Mnyusiwella et al. . There are several numerical studies on the modelling of natural convection heat transfer in nanofluids (Kaka and Pramuanjaroekij , Godson et al. , Olanrewaju et al. ). Gbadeyan et al.  numerically studied boundary layer flow induced in a nanofluid due to a linearly stretching sheet in the presence of thermal radiation and induced magnetic field. Khan and Aziz  studied natural convection flow of a nanofluid over a vertical plate with a uniform surface heat flux. Makinde and Aziz  investigated boundary layer flow of a nanofluid past a stretching sheet with a convective boundary conditions. Khan et al.  studied the unsteady free convection boundary layer flow of a nanofluid along a stretching sheet with thermal radiation in the presence of a magnetic field.
Thermal radiation-convection interaction problems are found in the cooling of high temperature components design where heat transfer from surfaces occurs by parallel radiation and convection, the interaction of incident solar radiation with the earth’s surface to produce complex free convection patterns. Hadey et al.  studied the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation. Khan et al.  analyzed the effects of variable viscosity and thermal conductivity on the flow and heat transfer in a laminar liquid film on a horizontal shrinking/stretching sheet. Khan et al.  investigated, by employing the homotopy perturbation transform method (HPTM) and the Padé approximation, the problem of magnetohydrodynamic (MHD) boundary layer flow over a nonlinear porous stretching sheet. Khan et al.  considered a two-dimensional, steady magnetohydrodynamic flow and heat transfer analysis of a non-Newtonian fluid in a channel with a constant wall temperature in the presence of thermal radiation.
As the governing equations modelling MHD flow and heat transfer of nanofluids are highly nonlinear, exact solutions are impossible to obtain. Over the years, numerical methods have been developed, improved, and hybred as a way of getting more accurate solutions. The current study seeks to employ a recently developed numerical technique known as spectral relaxation method  to solve the problem of magnetohydrodynamic boundary layer flow of nanofluids over a moving surface in the presence of thermal radiation. The current method has been successfully employed in [15–20], among others. We apply this method to the problem of MHD flow of a nanofluid past a stretching sheet in the presence of thermal radiation. The governing boundary layer equations are transformed to a system of nonlinear ordinary differential by using suitable local similarity variables.
2 Mathematical formulation
where u and v are the velocity components along the x-axis and y-axis, respectively, is the thermal diffusivity of the fluid, ν is the kinematic viscosity coefficient, k is the thermal conductivity, is the heat flux, is the Brownian diffusion coefficient, is the thermophoresis diffusion coefficient, is the uniform magnetic field strength of the base fluid, σ is the electrical conductivity of the base fluid, τ is the ratio of the nanoparticle heat capacity and the base fluid heat capacity.
The surface moving parameter corresponds to the downstream movement of the plate from the origin, while corresponds to the upstream movement of the plate.
2.1 Similarity transformations
where is the radiation parameter, is the Prandtl number, is the Lewis number, is the local Hartman number, is the Brownian motion parameter and is the thermophoresis parameter.
where is the local Reynolds number.
3 Method of solution
To solve the set of ordinary differential equations (11)-(13) together with the boundary conditions (14) and (15), we employ the Chebyshev pseudo-spectral method known as spectral relaxation method (SRM). This method transforms sets of nonlinear ordinary differential into sets of linear ordinary differential equations. The entire computational procedure is implemented using a program written in MATLAB computer language. The fluid velocity, temperature, the local skin-friction coefficient and the local Nusselt and Sherwood numbers are determined from these numerical computations. The SRM algorithm starts with the assumption of having a system of m nonlinear ordinary differential equations in m unknowns functions, , , where is the independent variable. To solve the resultant iterative scheme, we then use the Chebyshev pseudo-spectral method. The details of the spectral methods can be found in Canuto et al. , Trefethen . Before applying the spectral method, the domain on which the governing equation is defined is transformed to the interval on which the spectral method can then be implemented. We use the transformation to map the interval to .
The above equations form a system of linear decoupled equations which can be solved iteratively for , starting from initial guesses/approximations .
Accuracy of the scheme is established by increasing the number of collocation points N until the solutions are consistent and further increases do not change the value of the solutions.
4 Results and discussion
The system of ordinary differential equations (12)-(13) subject to the boundary conditions (14)-(15) are numerically solved by using spectral relaxation method (SRM). This is a recently developed method, and details of the method are found in . The SRM results presented in this work were obtained using collocation points, and also the convergence was achieved after as few as five iterations. We also take 15 to be the infinity value . We use these default values for the parameters , , , , .
Comparison of SRM solutions for and against those of the as well as those obtained by Olanrewaju et al.  for different values of Pr
Comparison of SRM solutions for and against those of the as well as those obtained by Olanrewaju et al.  for different values of Le
Comparison of SRM solutions for and against those of the as well as those obtained by Olanrewaju et al.  for different values of λ
Computations of and at different values of Nb and Nt
Comparison of SRM solutions for against those of the for different values of λ
Numerical analysis has been carried out on MHD boundary layer flow of nanofluids over a moving surface in the presence of thermal radiation. The governing partial differential equations were transformed into a system of ordinary differential equations using suitable similarity transformations. The resultant equations were then solved using the spectral relaxation method. The accuracy of the SRM is validated against the MATLAB in-built routine for solving boundary value problems as well as previously obtained results. An excellent agreement was observed between our results and those obtained using other methods giving confidence to our present results. We observed that the local temperature rises as the Brownian motion, thermophoresis and radiation effects intensify. Lastly, the Nusselt number decreases while the Sherwood number increases as the Brownian motion and thermophoresis effects are increased.
The authors wish to acknowledge financial support from the University of Venda, NRF and University of Botswana.
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