Traveling Wave Solutions of 3D Fractionalized MHD Newtonian Fluid in Porous Medium with Heat Transfer

Document Type: Research Paper


Department of Mathematics, NED University of Engineering & Technology, Karachi-75270, Pakistan


In the present paper, we get exact solutions of Magnetohydrodynamic (MHD) of the fractionalized three-dimensional flow of Newtonian fluid with porous and heat transfer through the traveling wave parameter. The governing equations are produced dependent on established Navier-stokes equations which can be diminished to ordinary differential equation by wave parameter ξ=ax+by+nz+Utα/Γ(α+1). The new exact solutions are established for three various cases. In special cases the solution for Newtonian fluid with and without MHD and porous effects can also be found from the general solution by putting M+Φ→0 and solutions for simple Newtonian fluid can also be obtained by putting α→1 in general solutions. Finally, the effect of the parameter of interest on the stream motion, as well as difference among the Newtonian fluids is examined by 2D and 3D graphical interpretations. 


Main Subjects

[1] Khater, M.M., Zahran, E.H., Shehata, M.S., Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system, J. Egyptian Math. Soc., 25, 2017, 8-12.

[2] Khan, K., Akbar, M.A., Traveling wave solutions of nonlinear evolution equations via the enhanced -expansion method, J. Egyptian Math. Soc., 22, 2014, 220-226.

[3] Seadawy, A.R., Lu, D., Yue, C., Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability, J. Taibah Univ. Sci., 11, 2017, 623-633.

[4] Khan, N. A., Naz, F., Khan, N. A., Ullah, S., MHD nonaligned stagnation point flow of second grade fluid towards a porous rotating disk, Nonlin. Eng., 8, 2019, 231-249.

[5] Khan, K., Akbar, M.A., Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method, Ain Shams Eng. J., 4, 2013, 903-909.

[6] Bruzn, M.S., Gandaries, M.L., Traveling wave solutions of the  equation with generalized evolution, Math. Methods Appl. Sci., 41, 2018, 5851-5857.

[7] Huang, B., Xie, S., Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics, Adv. Differ. Equations, 29, 2018, doi:10.1186/s13662-017-1441-6.

[8] Redi, R.T., Anulo, A.A., Some new traveling wave solutions of modified Camassa Holm equation by the improved  expansion method, Math. Comput. Sci., 3, 2018, 23.  

[9] Ellahi, R., Mohyud-Din, S.T., khan, U. , Exact traveling wave solutions of fractional order boussinesq-like equations by applying exp-function method, Results Phys., 8, 2018, 114-120.

[10] Kabir, M.R., Datta, B.K., New exact traveling wave solutions to burgers equation, J. Sci. Res. Reports, 10, 2018, 1-9.

[11] Mohyud-Din, S.T., Bibi, S., Exact Solutions for nonlinear fractional differential equations using -expansion method, Alexandria Eng. J., 57, 2018, 1003-1008.

[12] Momani, S., Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365, 2007, 345-350.

[13] Ghaneai, H., Hosseini, M.M., Mohyud-Din, S.T., Modified variational iteration method for solving a neutral functional-differential equation with proportional delays, Int. J. Numer. Meth. Heat Fluid Flow, 22, 2012, 1086-1095.

[14] Odibat, Z.M., Abdullah, P., Ghazi, B., Application of variational iteration method differential equations of fractional order to Nonlinear, Int. J. Nonlinear Sci. Numer. Simul., 7, 2006, 27-34.

[15] Merdan, M., Gokdogan, A., Yildirim, A., Mohyud-Din, S.T., Numerical simulation of fractional Fornberg-Whitham equation by differential transformation method, Abstr. Appl. Anal., Article ID 965367, 2012, 8.

[16] Arikoglu, A., Ozkol, I., Solution of fractional differential equations by using differential transform method, Chaos, Solitons and Fractals, 34, 2007, 1473-1481.

[17] Ahmad, J., Mohyud-Din, S.T., Solving fractional vibrational problem using restarted fractional Adomian’s decomposition method, Life Sci. J., 10, 2016, 210-216.

[18] Odibat, Z.M., Momani, S., Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput., 181, 2006, 767-774.

[19] Zhang, S., Zong, Q.A., Liu, D., Gao, Q., A generalized exp-function method for fractional riccati differential equations, Commun. Fract. Calc., 1, 2010, 48-51.

[20] Bekir, A., Güner, Ö., Cevikel, A.C., Fractional complex transform and exp-function methods for fractional differential equations, Abstr. Appl. Anal., Article ID 4264628, 2013.

[21] Bekir, A., Uygun, F., Exact travelling wave solutions of nonlinear evolution equations by using the  expansion method, Arab J. Math. Sci., 18, 2012, 73-85.

[22] Biswas, A., Bhrawy, A.H., Abdelkawy, M.A., Alshaery, A.A., Hilal, E.M., Arabia, S., Symbolic computation of some nonlinear fractional differential equations, Rom. J. Phys., 59, 2014, 433-442.

[23] Ahmad, J., Mohyud-din, S.T., An efficient algorithm for some highly nonlinear fractional PDEs in mathematical physics, PLoS One, 9, 2014, 1-17.

[24] Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228, 2009, 7792-7804.

[25] Narayana, P. S., Babu, D. H., Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, J. Taiwan Inst. Chem. Eng., 59, 2016, 18-25.

[26] Shehzad, S.A., Hayat, T., Alhuthali, M.S., Asghar, S., MHD three-dimensional flow of Jeffrey fluid with Newtonian heating, J. Cent. South Univ., 21, 2014, 1428-1433.

[27] Khan, N. A., Mahmood, A., Jamil, M., Khan, N.U., Traveling wave solutions for MHD aligned flow of a second grade fluid, Int. J. Chm. React. Engg, 8, 2010, A163.

[28] Jamil, M., Ahmed, A., New traveling wave solutions of MHD Micropolar fluid in porous Medium, submitted for publication, J. Intell. Fuzzy Syst., 2019.

[29] Azhar, W. A., Fetecau C., Vieru D., MHD free convection flow of a viscous fluid in a rotating system with damped thermal transport, Hall current and slip effects, Euro. Phys. J. Plus, 133, 2018, 353.

[30] Dolat K., Khan A., Khan I., Ali F., Karim F. U., Tlili I., Effects of relative magnetic field, chemical reaction, heat generation and Newtonian heating on convection flow of Casson fluid over a moving vertical plate embedded in a porous medium, Scientific Reports, 9, 2019, 400.

[31] Abdelsalam, S.I.,Vafai, K., Combined effects of magnetic field and rheological properties on the peristaltic flow of a compressible fluid in a microfluidic channel, Eur. J. Mech. B/Fluids, 65, 2017, 398-411.

[32] Abdelsalam, S.I., Bhatti, M.M., The impact of impinging TiO 2 nanoparticles in Prandtl nanofluid along with endoscopic and variable magnetic field effects on peristaltic blood flow, Multidiscip. Model. Mater. Struct., 14(3), 2018, 530-548.

[33] Abdelsalam, S.I., Bhatti, M.M., The study of non-Newtonian nanofluid with hall and ion slip effects on peristaltically induced motion in a non-uniform channel, RSC Adv., 19, 2018, 7904-7915.

[34] Elmaboud, Y.A., Abdelsalam, S.I., Mekheimer, K.S.,Vafai, K., Electromagnetic flow for two-layer immiscible fluids, Eng. Sci. Technol.: An Int. J., 22, 2019, 237-248.

[35] Pavlov, K.B., Magnetohydrodynamic flow of an incompressible viscous caused by deformation of plane surface, Magnitnaya Gidrodinamika, 10(4), 1974, 146-147.

[36] Afzal, S., Asghar, S., Ahmad, A., Three-dimensional MHD flow over a shrinking sheet: Analytical solution and stability analysis, Chin. Phys. B, 26, 2017, 1-5.

[37] Ishak, A., Nazar, R., Pop, I., Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder, Energy Convers. Manag, 49, 2008, 3265-3269.

[38] Shoaib, M., Perveen, R., Rana, M.A., Slip effect on three dimensional mhd flow of viscous fluid along an infinite plane with periodic suction, Int. Bhurban Conf. Appl. Sci. Technol., Islamabad, Pakistan, 2017, 575-580.

[39] Khan, N.A., Khan, S., Ara, A., Flow of micropolar fluid over an off centered rotating disk with modified Darcy's law, Propuls. Power Res., 6, 2017, 285-295.

[40] Ahmad, M., Ahmad, I., Sajid, M., Magnetohydrodynamic time-dependent three dimensional flow of Maxwell  fluid over a stretching surface through porous space with variable thermal condition, J. Brazilian Soc. Mech. Sci. Eng., 38(6), 2016, 767–1778.

[41] Aurangzaib, Sharif Uddin, M., Bhattacharyya, K., Shafie, S., Micropolar fluid flow and heat transfer over an exponentially permeable shrinking sheet, Propuls. Power Res., 5, 2016, 310-317.

[42] Jamil, M., Khan, N.A., Mahmood, A., Din, Q., Some exact solutions for the flow of a Newtonian fluid with heat transfer via prescribed vorticity, J. Prime Res. Math., 6, 2010, 38-55.

[43] Abbas, Z., Javed, T., Sajid, M., Ali, N., Unsteady MHD flow and heat transfer on a stretching sheet in a rotating fluid, J. Taiwan Inst. Chem. Eng., 41, 2010, 644-650.

[44] Khan, N.A., Ara, A., Jamil, M., Yildirim, A. , Traveling wave solutions for MHD Aligned flow of a second grade fluid, A symmetry independent approach, J. King Saud Uni. Sc., 24, 2011, 63-67.

[45] Khan, N.A., Khan, H., Traveling wave solutions for (3+1) dimensional equations arising in fluid mechanics, Nonlinear Eng., 3, 2014, 209-214.

[46] Zhao, Y., Chen, L., Zhang, X.R., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf., 75, 2016, 206-217.