# UNSTEADY MHD THIN FILM FLOW OF A THIRD GRADE FLUID OVER AN OSCILLATING INCLINED BELT EMBEDDED IN A POROUS MEDIUM

## Main Article Content

## Abstract

In the present work we examine the motion of an incompressible unidirectional magneto-hydrodynamics (MHD) thin film flow of a third grade fluid over an oscillating inclined belt embedded in a porous medium. Moreover, heat transfer analysis has been also discussed in the present work. This physical problem is modeled in terms of nonlinear partial differential equations. These equation together with physical boundary conditions are solved using two analytical techniques namely Optimal Homotopy Asymptotic Method (OHAM) and Homotopy Perturbation Method (HPM). The comparisons of these two methods for different time level are analyzed numerically and graphically. The results exposed that both methods are in closed agreement and they have identical solutions. The effects of various non-dimensional parameters have also been studied graphically.

## Article Details

**Thermal Science**, [S.l.], mar. 2017. ISSN 2334-7163. Available at: <http://thermal-science.tech/journal/index.php/thsci/article/view/2101>. Date accessed: 14 dec. 2017. doi: https://doi.org/10.2298/TSCI141128054G.

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Accepted 2017-03-13

Published 2017-03-13

## References

[2] Aiyesimi, Y. M., et al., Unsteady magneto-hydrodynamic (MHD) thin film flow of a third grade fluid with heat transfer and no slip boundary condition down an inclined plane, Int Journal of phy sci.,8 (2013), 19, pp. 946-955

[3] Aiyesimi, Y. M., et al., MHD Flow of A Third Grade Fluid with Heat Transfer And Slip Boundary Condition Down An Inclined Plane, Mathematical Theory and Modeling,2 (2012), 9, pp. 108-119

[4] Abdallah, I. A., Analytical Solution of Heat and Mass Transfer over a Permeable Stretching Plate Affected by a Chemical Reaction, Internal Heating, Dufour-Souret Effect and Hall Effect, Thermal Science, 13 (2009), 2, pp. 183-197

[5] Gul, T., et al., Thin Film Flow in MHD Third Grade Fluid on a Vertical Belt with Temperature Dependent Viscosity, PLoS ONE 9, (2014), 6, pp.1-12

[6] Gul, T., et al., MHD Thin film flows of a third grade fluid on a vertical belt with slip boundary conditions, J. of Applied Mathematics, (2013), pp.1-14

[7] Bird, R. B., et al., Dynamics of polymeric liquids, 1 Fluid Mechanics second edition, John Wiley & Sons, Inc, (1987).

[8] Hayat, T., et al., Flow of a Second Grade Fluid with Convective Boundary conditions, Thermal Science, 15 (2011), 2, pp. 253-261.

[9] Hayat, T., et al., Unsteady Solutions i n a Third-Grade Fluid Filling the Porous Space, Math, Prob, Eng. (2008), 139560, pp.13.doi:10.1155/2008/139560.

[10] Gamal, M. A., Effect of Magneto-hydrodynamics on thin films of unsteady micro polar fluid through a porous medium, Journal of Modern Physics, 2 (2011), pp. 1290-1304.

[11] Sajid, M., et al., The influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method, Computers and Mathematics with Applications, 56 (2008), pp. 2019–2026.

[12] Idrees, et al., Application of the Optimal Homotopy Asymptotic Method to squeezing flow, Computers and Mathematics with Applications, 59(2010), pp. 3858–3866

[13] Siddiqui, A.M., et al., Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane, Chaos, Solitons & Fractals, 35 (2007) ,1, pp. 140-147

[14] Mabood, F., et al., Application of Optimal Homotopy Asymptotic Method for the Approximate Solution of Riccati Equation, Sains Malaysiana, 42 (2013), 6,pp. 863–867

[15] Ganji, D.D., Rafei, M., Solitary wave solutions for a generalized Hirota–Satsuma coupled KdV equation by homotopy perturbation method, Phys. Latt. A, 356 (2006), 2, pp. 131-137

[16] Lin Jin., Homotopy Perturbation Method for Solving Partial Differential Equations with Variable Coefficients, Int. J. Contemp. Math. Sciences, 3 (2008), 28, pp. 1396-1407

[17] Nawaz, R., et al., Application of Optimal Homotopy Asymptotic Method to Burger Equations, Journal of Applied Mathematics, (2013), pp. 1-8

[18] Hemeda, A.A., Homotopy Perturbation Method for Solving Partial Differential Equations of Fractional Order, Int. Journal of Math. Analysis, 6 (2012),49, pp. 2431 – 2448

[19] Singh, G., Sharma, P. R., Effects of Ohmic Heating and Viscous Dissipation on Steady MHD Flow Near a Stagnation Point on an Isothermal Stretching Sheet, Thermal Science, 13 (2009), 1, pp. 5-12

[20] Rashidi, M.,M., Erfani, E., Analytical method for solving steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohmic heating, Engineering Computations, 29 (2012), 6, pp. 562 – 579.

[21] Rashidi, M.,M., Momoniat, E., Analytic Approximate Solutions for MHD Boundary-Layer Viscoelastic Fluid Flow over Continuously Moving Stretching Surface by Homotopy Analysis Method with Two Auxiliary Parameters. Journal of Applied Mathematics, (2012),780415, pp.19.doi:10.1155/2012/780415.

[22] Rashidi, M.,M., et al, Free convective heat and mass transfer for MHD fluid flow over a permeable vertical stretching sheet in the presence of the radiation and buoyancy effects, Ain Shams Engineering Journal, (2014) 5, 901–912.

[23] Rashidi, M.,M., et al ,Buoyancy effect on MHD flow of Nano fluid over a stretching sheet in the presence of thermal radiation, Journal of Molecular Liquids, 198,(2014),pp.234– 238.