A Mathematical Model of Magnetohydrodynamic Micropolar Fluid Motion Via Permeable Media with SORET and Dufour Effects

A Mathematical Model of Magnetohydrodynamic Micropolar Fluid Motion Via Permeable Media with SORET and Dufour Effects

Ram Prakash Sharma* S.R. Mishra

Department of Mathematics, JECRC University Jaipur, India

Department Mathematics, Siksha ‘O’ Anusandhan Deemed to be University, Khandagiri, Bhubaneswar 751030, Odisha, India

Corresponding Author Email: 
ramprakash0808@gmail.com
Page: 
250-256
|
DOI: 
https://doi.org/10.18280/mmc_b.870406
Received: 
1 June 2017
|
Accepted: 
5 January 2018
|
Published: 
31 December 2018
| Citation

OPEN ACCESS

Abstract: 

In this article, we have examined 2-dimensional steady magnetohydrodynamic boundary layer motion of viscous micropolar liquid through an extending surface. Simultaneous impacts of Soret and diffusion-thermo are considered. Furthermore, the impact of heat source/sink and first order chemical reaction are also examined. The basic numerical problem i.e. structure of PDE’s is transformed nonlinear into ODE’s through using appropriate transformations. The changed governing equations are explained mathematically through R–K fourth order method. The effect of different parameters on momentum, microrotation, energy, concentration descriptions, shear stress, transfer rate of heat and mass are examined through graphs. Mathematical evaluation is furthermore examined through the existing available outcome as a particular case of our research work.

Keywords: 

porous media, micropolar fluid, MHD, soret and dufour effect, heat source/sink

1. Introduction
2. Formulation of the Problem
3. Results and Discussion
4. Conclusion
Acknowledgement

The authors are grateful to Prof G. C. Sharma, Agra University, Agra, India for his help and valuable suggestions to prepare this research paper and thanks to reviewers also

Nomenclature
  References

[1] Sparrow EM, Cess RD. (1961). The effect of a magnetic field on free convection heat transfer. International Journal of Heat and Mass Transfer 3(4): 267-274. http://doi.org/10.1016/0017-9310(61)90042-4

[2] Gupta AS. (1962). Laminar free convection flow of an electrically conducting fluid from a vertical plate with uniform surface heat flux and variable wall temperature in the presence of a magnetic field. Zeitschrift für angewandte Mathematik und Physik ZAMP 13(4): 324-333. 

[3] Char MI. (1994). Heat transfer in a hydromagnetic flow over a stretching sheet. Heat Mass Transfer 29(8): 495-500. http://doi.org/10.1007/BF01539502

[4] Chiam TC. (1997). Magnetohydrodynamic heat transfer over a non- isothermal stretching sheet. Acta Mechanica 122(1): 169-179. http://doi.org/10.1007/BF01181997

[5] Liu IC. (2004). Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to a transverse magnetic field. International Journal of Heat and Mass Transfer 47(19-20): 4427-4437. http://doi.org/10.1016/j.ijnonlinmec.2004.07.008

[6] Ishak A, Nazar R, Pop I. (2008). Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transfer 44(8): 921-927. http://doi.org/10.2298/TSCI100308198Y

[7] Prasad KV, Pal D, Datti PS. (2009). MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet. Communications in Nonlinear Science and Numerical Simulation 14(5): 2178–2189. http://doi.org/10.1016/j.cnsns.2008.06.021

[8] Harris SD, Ingham BD, Pop I. (2009). Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip. Transport of Porous Media 77(2): 267–285. http://doi.org/10.1007/s11242-008-9309-6

[9] Aziz A. (2010). Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Communications in Nonlinear Science and Numerical Simulation 15(3): 573-580. http://doi.org/10.1016/j.cnsns.2009.04.026

[10] Rohni AM, Ahmad S, Pop I. Merkin JH. (2012). Unsteady mixed convection boundary-layer flow with suction and temperature slip effects near the stagnation point on a vertical permeable surface embedded in a porous medium. Transport of a Porous Media 92(1): 1-14. http://doi.org/10.1007/s11242-011-9883-x

[11] Tao LN. (1960). On combined free and forced convection in channel. ASME Journal of Heat Transfer 82(3): 233-238. http://doi.org/10.1115/1.3679915

[12] Aung W, Worku G. (1986). Developing flow and flow reversal in a vertical channel with symmetric wall temperatures. ASME Journal of Heat Transfer 108(2): 299-304. http://doi.org/10.1115/1.3246919

[13] Trevisan OV, Bejan A. (1990). Combined heat and mass transfer by natural convection in a porous medium. Advances in Heat Transfer 20: 315-352. 

[14] Barletta A. (1998). Laminar mixed convection with viscous dissipation in a vertical channel. International Journal of Heat and Mass Transfer 41(22): 3501-3513. http://doi.org/10.1016/s0017-9310(98)00074-x

[15] Chamkha AJ, Grosan T, Pop I. (2002). Fully developed free convection of a micropolar fluid in a vertical channel. International Communications in Heat and Mass Transfer 29(8): 1119-1127. http://doi.org/10.1016/S0735-1933(02)00440-2

[16] Ishak A, Nazar R, Pop I. (2008). Magnetohydrodynamic (MHD) flow of a micropolar fluid towards a stagnation point on a vertical surface, 2008, Computers & Mathematics with Applications 56(12): 3188-3194. http://doi.org/10.1016/j.camwa.2008.09.013

[17] Lok YY, Amin N, Campean D, Pop I. (2005). Steady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface. International Journal of Numerical Methods for Heat & Fluid Flow 15(7): 654-670. http://doi.org/10.1108/09615530510613861

[18] Ramachandran N, Chen TS, Armaly BF. (1988). Mixed convection in a stagnation flows adjacent to vertical surfaces. ASME Journal of Heat Transfer 110:2(2): 373-37. http://doi.org/10.1115/1.3250494

[19] Srinivas S, Muthuraj R. (2010). MHD flow with slip effects and temperature-dependent heat source in a vertical wavy porous space. Chemical Engineering Communications 197(11): 1387-1403. http://doi.org/10.1080/00986441003626102

[20] Srinivas S, Muthuraj R. (2011). Effects of chemical reaction and space porosity on MHD mixed convective flow in a vertical asymmetric channel with peristalsis. Mathematical and Computer Modelling 54(5-6): 1213-1227.

[21] Nield DA, Bejan A. (1999). Convection in Porous Media. 2nd Edition, Springer, New York. 145-220. http://doi.org/10.1002/0470014164.ch8

[22] Ingham DB, Pop I. (1998). Transport Phenomena in Porous Media. I. Pergamon Oxford.

[23] Ingham DB, Pop I. (2002). Transport Phenomena in Porous Media II, Pergamon, Oxford. http://doi.org/10.1029/EO066i020p00443-03

[24] Bejan A, Khair KR. (1985). Heat and mass transfer by natural convection in a porous medium. International Journal of Heat and Mass Transfer 28: 909-918. http://doi.org/10.1016/0017-9310(85)90272-8

[25] Eckert ERG, Drake RM. (1972). Analysis of Heat and Mass Transfer. McGraw-Hill, New York. 

[26] Kafoussias NG, Williams EM. (1995). Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity. International Journal of Engineering Science 33(9): 1369-1383. http://doi.org/10.1016/0020-7225(94)00132-4

[27] Anghel M, Takhar HS, Pop I. (2000). Dufour and Soret effects on free convection boundary-layer over a vertical surface embedded in a porous medium. Studia Universitatis Babes-Bolyai, Mathematica 4: 11-21. 

[28] Postelnicu A. (2004). Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. International Journal of Heat and Mass Transfer 47(6-7): 1467-1472. http://doi.org/10.1007/s00231-006-0132-8

[29] Postelnicu A. (2007) Influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Heat Mass Transfer 43(6): 595-602. http://doi.org/10.1007/s00231-006-0132-8 

[30] Alam MS, Rahman MM. (2006). Dufour and Soret effects on mixed convection flow past a vertical porous flat plate with variable suction. Nonlinear Analysis: Modelling and Control 11(1): 3-12. http://doi.org/10.1109/HMWC.2014.7000242

[31] Murthy PVSN, Partha MK, Raja Sekhar GP. (2006). Soret and Dufour effects in a non-Darcy porous medium. Journal of Heat Transfer 128(6): 605-610. http://doi.org/10.1115/1.2188512

[32] Rashidi MM, Hayat T, Erfani E, Mohimanian Pour SA, Awatif AH. (2011). Simultaneous  effects of partial slip and thermal-diffusion and diffusion-thermo on steady MHD convective flow due to a rotating disk. Communications in Nonlinear Science and Numerical Simulation 16(11): 4303–4317. http://doi.org/10.1016/j.cnsns.2011.03.015

[33] Mishra SR, Dash GC, Acharya M. (2013). Free convective flow of visco-elastic fluid in a vertical channel with Dufour effect. World Applied Sciences Journal 28: 1275-1280. 

[34] Baag S, Mishra SR, Dash GC, Acharya MR. (2014). Numerical investigation on MHD microploar fluid flow towards a stagnation point on a vertical surface with heat source and chemical reaction. Journal of King Saud University-Engineering Sciences 3(1). http://doi.org/10.1016/j.jksues.2014.06.002

[35]  Bhattacharyya K. (2011). Dual solutions in boundary stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet. International Communications in Heat and Mass Transfer 38(7): 917-922. http://doi.org/10.1016/j.icheatmasstransfer.2011.04.020