^{1}Department of Mathematics, Mekapati Rajamohan Reddy Institute of Technology and Science, Udayagiri, Nellore District, A.P, India

^{2}Department of Humanities and Sciences, Annamacharya Institute of Technology and Sciences (Autonomous), Rajampet – 516126, A.P., India

^{3}Department of Mathematics, JNTUA College of Engineering Pulivendula, Pulivendula, A.P, India

^{4}Department of Mathematics, GITAM University, Vishakhaptanam, A.P. - 530045 India

Abstract

The paper aims at investigating the effects of chemical reaction and thermal radiation on the steady two-dimensional laminar flow of viscous incompressible electrically conducting micropolar fluid past a stretching surface embedded in a non-Darcian porous medium. The radiative heat flux is assumed to follow Rosseland approximation. The governing equations of momentum, angular momentum, energy, and species equations are solved numerically using Runge-Kutta fourth order method with the shooting technique. The effects of various parameters on the velocity, microrotation, temperature and concentration field as well as skin friction coefficient, Nusselt number and Sherwood number are shown graphically and tabulated. It is observed that the micropolar fluid helps the reduction of drag forces and also acts as a cooling agent. It was found that the skin-friction coefficient, heat transfer rate, and mass transfer rate are decreased, and the gradient of angular velocity increases as the inverse Darcy number, porous medium inertia coefficient, or magnetic field parameter increase. Increases in the heat generation/absorption coefficient caused increases in the skin-friction coefficient and decrease the heat transfer rate. It was noticed that the increase in radiation parameter or Prandtl number caused a decrease in the skin-friction coefficient and an increase in the heat transfer rate. In addition, it was found that the increase in Schmidt number and chemical reaction caused a decrease in the skin-friction coefficient and an increase in the mass transfer rate.

[1] A. C. Eringen, “Theory of micropolar fluids”, Journal of Mathematics and Mechanics, Vol. 16, pp. 1-18, (1966).

[2] T. Y. Na, and I. Pop, “Boundary-layer flow of micropolar fluid due to a stretching wall”, Archives of Applied Mechanics, Vol. 67, No. 4, pp. 229-236, (1977).

[3] A. Desseaux, and N. A. Kelson, “Flow of a micropolar fluid bounded by a stretching sheet”, Anziam J., Vol. 42, pp. 536-560, (2000).

[4] F. M. Hady, On the solution of heat transfer to micropolar fluid from a non-isothermal stretching sheet with injection, Int. J. Numer. Methods for Heat and Fluid Flow, Vol. 6, No. 6, pp. 99-104, (1966).

[5] O. Aydin and A. Kaya, “Non-Darcin forced convection flow of a viscous dissipating fluid over a flat plate embedded in a porous medium”, Trans Porous Media, Vol. 73, No. 2, pp. 173-186, (2008).

[6] S. S. Das, A. Satapathy, J. K. Das and J. P. Panda, “Mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source”, Int. J. of Heat and Mass Transfer., Vol. 52, No. 25-26, pp. 5962-5969, (2009).

[7] J. Anand Rao and S. Shivaiah, “Chemical reaction effects on an unsteady MHD free convective flow past an infinite vertical porous plate with constant suction and heat source”, Int. J. of Appl. Math and Mech., Vol. 7, No. 8, pp. 98-118,(2011).

[8] S. Y. Ibrahim and O. D. Makinde, “Radiation effect on chemically reacting magneto hydrodynamics (MHD) boundary layer flow of heat and mass transfer through a porous vertical flat plate,” Int. J. Physical Sciences, Vol. 6, No. 6, pp. 1508-1516, (2011).

[9] D. Pal and B. Talukdar, “Combined effects of Joule heating and chemical reaction on unsteady magneto hydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation,” Mathematical and Computer Modelling, Vol. 54, No. 11-12, pp. 3016-3036, (2011).

[10] K. Jhansi Rani and Ch. V. Ramana Murthy, “MHD Flow over a Moving Infinite Vertical Porous Plate with Uniform Heat Flux in the presence of Thermal Radiation”, Advanced in Theoretical and Applied Mathematics, Vol. 6, No. 1, pp. 51-63, (2011).

[11] G. V. Ramana Reddy, N. Bhaskar Reddy and Ch. V. Ramana Murthy, “Heat and mass transfer effects on MHD free convection flow past an oscillating plate embedded in porous medium”, International Journal of Physical Sciences, Vol. 22. No. 2M, pp. 375-380, (2010).

[12] T. S. Reddy, M. C. Raju and S. V. K .V Varma “ unsteady MHD Radiative and Chemically reactive free convection flow near a moving vertical plate in porous medium” Journal of Applied Fluid Mechanics ,Vol. 6, No. 3, pp. 443-451, (2013).

[13] M. C. Raju, S. V. K. Varma, N. A. Reddy, “MHD Thermal diffusion natural convection flow between heated inclined plates in porous medium”, Journal on future engineering and technology, Vol. 6, No. 2, pp. 45-48, (2011).

[14] P.Chandrakala, “Radiation Effects on Flow Past an Impulsively Started Vertical Oscillating Plate with Uniform Heat Flux”, International Journal of Dynamics of Fluids, Vol. 7, No. 1, pp. 1-8, (2011).

[15] R. Choudhury and U. J. Das, “MHD mixed convective heat and mass transfer in a viscoelastic boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction”, Int. J. of statistika and mathematika, Vol. 3, No. 3, pp. 93-101, (2012).

[16] S. Abzal, G. R. Reddy and S. V. K. Varma, “MHD free convection flow and mass transfer unsteady near a moving vertical plate in the presence of thermal radiation”, Annals of faculty engineering hunedoara- International journal of engineering, Tom IX, pp. 29-34, (2011).

[17] V. Ravikumar, M. C. Raju and G. S. S. Raju, “Magnetic field and radiation effects on a double diffusive free convective flow bounded by two infinite impermeable plates in the presence of chemical reaction”, IJSER, Vol. 4, No. 7, pp. 1915-1923, (2013).

[18] R. A. Mohamed, Abdel-Nasser A. Osman, S.M. Abo-Dahab, “Unsteady MHD double diffusive convection boundary-layer flow past a radiate hot vertical surface in porous media in the presence of chemical reaction and heat sink”, Meccanica, Vol. 48, No. 4, pp 931-942, (2013).

[19] M. Y. Malik, T. Salahuddin, Arif Hussain and S Bilal, “MHD flow of tangent hyperbolic fluid over a stretching cylinder: Using Keller box method”, Journal of Magnetism and Magnetic Materials, Vol. 395, pp. 271-276, (2015).

[20] T. Salahuddin, M. Y. Malik, Arif Hussain, S. Bilal and M. Awais, “The effects of transverse magnetic field with variable thermal conductivity on tangent hyperbolic fluid with exponentially varying viscosity”, AIP Advances, Vol. 5, Article ID: 127103, (2015).

[21] T. Salahuddin, Md. Yousaf Malik, Arif Hussain and M. Awais, “MHD flow of Cattanneo-Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness: Using numerical approach”, Journal of Magnetism and Magnetic Materials, Vol. 401, pp. 991-997, (2015).

[22] B. Seshaiah, S. V. K. Varma, M. C. Raju, “The effects of chemical reaction and radiation on unsteady MHD free convective fluid flow embedded in a porous medium with time-dependent suction with temperature gradient heat source”, International Journal of Scientific Knowledge, Vol. 3 No. 2, pp. 13-24, (2013).

[23] R. Rout and H. B. Pattanayak, “Chemical reaction and radiation effects on MHD flow past an exponentially accelerated vertical plate in presence of heat source with variable temperature embedded in a porous medium”, Annals of faculty engineering hunedoara- Int. Jou. Of Engg, Vol. 4, pp. 253-259, (2013).

[24] W. A. Khan, and I. Pop, “The Cheng-Minkowycz problem for the triple–diffusive natural convection boundary layer flow past a vertical plate in a porous medium”, J. Porous Media, Vol.16, No. 7, pp. 637-646, (2013).

[25] G. S. Seth, R. Nandkeolyar and M. S. Ansari, “Effects of thermal radiation and rotation on unsteady hydro magnetic free convection flow past an impulsively moving vertical plate with ramped temperature in a porous medium”, J. Appl. Fluid Mech, Vol.6, No.1, pp. 27-38, (2013).

[26] D. Ch. Kesavaiah, P. V. Satyanarayana, S. Venkataramana, “Effects of the chemical reaction and radiation absorption on an unsteady MHD convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate embedded in a porous medium with heat source and suction”, Int. J. of Appl. Math and Mech., Vol. 7, No. 1, pp. 52-69, (2011).

[27] U. S. Rajput, S. Kumar, “Radiation effects on MHD flow past an impulsively started vertical plate with variable heat and mass transfer”, Int. J. of Appl. Math. and Mech., Vol. 8, No. 1, pp. 66-85, (2012).

[28] R. Muthucumaraswamy, N. Dhanasekar, G. Easwara Prasad, “Effects on first order chemical reaction on flow past an accelerated isothermal vertical plate in a rotating fluid with variable mass diffusion”, Int. J. Math., Vol. 4, No. 41, pp. 28-35, (2013).

[29] B. Devika, P. V. Satya Narayana, S. Venkataramana, “MHD oscillatory flow of a visco elastic fluid in a porous channel with chemical reaction”, Int. J. Engg. Sci. Inv., Vol. 2, No. 2, pp. 26-35, (2013).

[30] K. Chand, K. D. Singh, S. Kumar, “Hall effect on radiating and chemically reacting MHD oscillatory flow in a rotating porous vertical channel in slip flow regime”, Advances in Applied Sciences Research, Vol. 3, No. 4, pp. 2424-2437, (2012).

[31] S. Mukhopadhyay, R. S. R. Gorla, “Effects of partial slip on boundary layer flow past a permeable exponential stretching sheet in presence of thermal radiation”, Heat Mass Transfer, Vol. 48, pp. 1773-1781, (2012). http://dx.doi.org/10.1007/s00231-012-1024-8.

[32] P. K. Kameswaran, S. Shaw, P. Sibanda, P. V. S. N. Murthy, “Homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet”, Int. J. Heat and Mass Transfer, Vol. 57, No. 2, pp. 465-472 (2013).

[33] S. Shaw, P.K. Kameswaran, P. Sibanda, “Homogeneous–heterogeneous reactions in micropolar fluid flow from a permeable stretching or shrinking sheet in a porous medium”, Boundary Value Problems, Vol. 77, No. 1, (2013).

[34] R. Ellahi, S. Aziz, A. Zeeshan, “Non-Newtonian nanofluid flow through a porous medium between two coaxial cylinders with heat transfer and variable viscosity”, Journal of Porous Media, Vol. 16, No. 1, pp. 205-216, (2013).

[35] N. Bachok, A. Ishak, I. Pop, “Boundary layer stagnation-point flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid”, Int. J. f Heat Mass transfer, Vol. 55, No. 25-26, pp. 8122-8128, (2013).

[36] N. S. Akbar, S. Nadeem, R.U. Haq, Z.H. Khan, “Radiation effects on MHD stagnation point flow of nanofluid towards a stretching surface with convective boundary condition”, Chinese Journal of Aeronautics, Vol. 26, No. 6, pp. 1389-1397, (2013), DOI: http://dx.doi.org/10.1016/j.cja.2013.10.008.

[37] M. Sheikholeslami, D. D. Ganji, M. Y. Javed, R. Ellahi, “Effect of thermal radiation on magneto hydrodynamics nanofluid flow and heat transfer by means of two phase model”, J. Magn..Mater. Vol. 374, pp. 36-43, (2015).

[38] M. Turkyilmazoglu, I. Pop, “Heat and mass transfer of unsteady natural convection flow of some nanofluids past a vertical infinite flat plate with radiation effect”, Int. J. Heat Mass Transfer, Vol. 59, pp. 167-171, (2013).

[39] M. Sheikholeslami, D. D. Ganjia, M. M. Rashidib, “Ferro fluid flow and heat transfer in a semi annulus enclosure in the presence of magnetic source considering thermal radiation”, J. Taiwan Inst. Chem. Eng., Vol. 47, pp. 6-17, (2015). http://dxdoi.org/10.1016/j.jtice.2014.09.026.

[40] D. Pal, “Combined effects of non-uniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface”, Commun. Nonlinear Sci. Numer. Simulat., Vol. 16, pp. 1890–1904, (2011).

[41] G. C. Shit, R. Haldar, “Effects of thermal radiation on MHD viscous fluid flow and heat transfer over nonlinear shrinking porous sheet”, Appl. Mathematics Mech. (English Edition), Vol. 32, No. 6, pp. 677-688, (2011).

[42] K. Das, “Impact of thermal radiation on MHD slip flow over a flate plate with

variable fluid properties”, Heat Mass Transfer, Vol. 48, pp. 767-778, (2012).

[43] F. T. Akyildiz, H. Bellout, K. Vajravelu, R.A. Van Gorder, “Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces”, Nonlinear Anal.: Real World Appl., Vol. 12, pp. 2919-2930, (2011).

[44] A. J. Chamkha, R. S. R. Gorla, K. Ghodeswar, “Non-similar solution for natural convective boundary layer flow over a sphere embedded in a porous medium saturated with a nanofluid”, Transp. Porous Media, Vol. 86, No. 1, pp. 13-22, (2011).

[45] N. Bachok, A. Ishak, I. Pop, “Stagnation-point flow over a stretching/shrinking sheet in a nanofluid”, Nanoscale Res. Lett., Vol. 6, pp. 623-632, (2011).

[46] O. D. Makinde, A. Aziz, “Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition”, Int. J. Thermal Sci., Vol. 50, pp. 1326-1332, (2011).

[47] E. M Abo-Eldahaband and M. A El-Aziz, “Flow and heat transfer in a micropolar fluid past a stretching surface embedded in a non-Darcian porous medium with uniform free steam”, Mathematics and Computation, Vol. 162, No. 2, pp. 881-899, (2005).