Journal of Applied and Computational Mechanics
Comparative Study of Plane Poiseuille Flow of Non-isothermal Couple Stress Fluid of Reynold Viscosity Model using Optimal Homotopy Asymptotic Method and New Iterative Method
Document Type : Research Paper
Authors
Department of Mathematics, Abdul Wali Khan University, Garden Campus, Mardan, KP, Pakistan
Abstract
In this paper, we have explored the steady Poiseuille flow of couple stress fluid between two parallel plates under the influence of non-isothermal effects of Reynold viscosity model, using Optimal Homotopy Asymptotic Method (OHAM) and New Iterative Method (NIM). We obtained expressions for velocity profile, temperature distribution, average velocity, volume flux and shear stress. The solutions obtained using these methods are in the form of infinite series; therefore, they can be easily computed. Comparative results of solutions obtained by both methods are given using different tables and graphs.
Keywords
Main Subjects
[1] Sadaf, H, Akbar, M.U. and S. Nadeem., Induced magnetic field analysis for the peristaltic transport of non-Newtonian nanofluid in an annulus, Mathematics and Computers in Simulation, 148, 2018, 16-36.
[2] Raju, C. S. K., S. Saleem, M. M. Al-Qarni, and S. Mamatha Upadhya., Unsteady nonlinear convection on Eyring–Powell radiated flow with suspended graphene and dust particles, Microsystem Technologies, 25(4), 2019, 1321-1331.
[3] Sadiq, M.A., Khan, A.U., Saleem, S. and Nadeem, S., Numerical simulation of oscillatory oblique stagnation point flow of a magneto micropolar-nanofluid, RSC advances, 9(9), 2019, 4751-4764.
[4] Bazighifan, O., Ahmad, H. & Yao, S.-W., New Oscillation Criteria for Advanced Differential Equations of Fourth Order, Mathematics, 8,2020, 728. doi:10.3390/math8050728.
[5] Malik, M.Y., Salahuddin, T., Hussain, A. and Bilal, S., MHD flow of tangent hyperbolic fluid over a stretching cylinder: using Keller box method, Journal of Magnetism and Magnetic Materials, 395, 2015, 271-276.
[6] Tan, W.C. and Xu, M.Y., The impulsive motion of flat plate in a generalized second grade fluid, Mechanics Research Communications, 29(1), 2002, 3-9.
[7] Ahmad, H., Khan, T. A. & Yao, S. W., An efficient approach for the numerical solution of fifth-order KdV equations, Open Mathematics, 18, 2020, 738–748. doi:10.1515/math-2020-0036.
[8] Farooq, M., Islam, S., Rahim, M.T. and Haroon, T., Heat transfer flow of steady couple stress fluids between two parallel plates with variable viscosity, Heat Transfer Research, 42(8), 2011, 737-780.
[9] Ahmad, H., Khan, T. A., Durur, H., Ismail, G. M. & Yokus, A., Analytic approximate solutions of diffusion equations arising in oil pollution, Journal of Ocean Engineering and Science, 2020, doi:https://doi.org/10.1016/j.joes.2020.05.002.
[10] Yokus, A., Durur, H. & Ahmad, H., Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, Facta Universitatis, Series: Mathematics and Informatics, 35, 2020, 523–531.
[11] Chen, C.I., Chen, C.K. and Yang, Y.T., Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate conditions, Heat and Mass Transfer, 40(3-4), 2004, 203-209.
[12] Ahmad, H., Variational Iteration Method with an Auxiliary Parameter for Solving Telegraph Equations, Journal of Nonlinear Analysis and Application, 2018, 2018, 223–232.
[13] Yokus, A., Durur, H., Ahmad, H. & Yao, S.-W., Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation, Mathematics, 8, 2020, 908.
[14] Fetecau, C. and Fetecau, C., Decay of a potential vortex in an Oldroyd-B fluid, International Journal of Engineering Science, 43(3-4), 2005, 340-351.
[15] Stokes, V.K., Couple stresses in fluids, Physics of Fluids, 9(9), 1966, 1709-1715.
[16] Devakar, M. and Iyengar, T.K.V., Run up flow of a couple stress fluid between parallel plates, Nonlinear Analysis: Modelling and Control, 15(1), 2010, 29-37.
[17] Srinivasacharya, D. and Kaladhar, K., Analytical solution of MHD free convective flow of couple stress fluid in an annulus with Hall and Ion-slip effects, Nonlinear Analysis: Modelling and Control, 16(4), 2011, 477-487.
[18] Rao, S.L. and Iyengar, T.K.V., Analytical and computational studies in couple stress fluid flows, UGC Research project C-8-4/82 SR III, 1985.
[19] Srinivasacharya, D., Stokes flow of an incompressible Couple stress fluid past an approximate sphere, Ph. D. Thesis, 1995.
[20] Tsai, C.Y., Novack, M. and Roffe, G., Rheological and heat transfer characteristics of flowing coal-water mixtures (No. DOE/MC/23255-2763), General Applied Science Labs., Inc., Ronkonkoma, NY (USA), 1988.
[21] Chinyoka, T. and Makinde, O.D., Computational dynamics of unsteady flow of a variable viscosity reactive fluid in a porous pipe, Mechanics Research Communications, 37(3), 2010, 347-353.
[22] Abouelregal AE, Ahmad H, Yao SW. Functionally Graded Piezoelectric Medium Exposed to a Movable Heat Flow Based on a Heat Equation with a Memory-Dependent Derivative, Materials, 13(18), 2020, 3953.
[23] Herisanu, N., Marinca, V., Dordea, T. and Madescu, G., A new analytical approach to nonlinear vibration of an electrical machine, Proceedings of the Romanian Academy Series A-Mathematics Physics Technical Sciences Information Science, 9(3), 2008, 229-236.
[24] Marinca, V. and Herisanu, N., Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, International Communications in Heat and Mass Transfer, 35(6), 2008, 710-715.
[25] Ahmad I, Ahmad H, Thounthong P, Chu YM, Cesarano C. Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method, Symmetry, 12(7), 2020, 1195.
[26] Ahmad, H., Seadawy, A. R., Khan, T. A. & Thounthong, P., Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations, Journal of Taibah University for Science, 14, 2020, 346–358.
[27] Marinca, V. and Herisanu, N., Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method, Journal of Sound and Vibration, 329(9), 2010, 1450-1459.
[28] Mabood, F., Khan, W.A. and Ismail, A.I.M., Analytical solution for radiation effects on heat transfer in Blasius flow, International Journal of Modern Engineering Sciences, 2(2), 2013, 63-72.
[29] Srivastava MH, Ahmad H, Ahmad I, Thounthong P, Khan NM., Numerical simulation of three-dimensional fractional-order convection-diffusion PDEs by a local meshless method, Thermal Science, 2020, DOI: 10.2298/TSCI200225210S.
[30] Abdou, M.A. and Elgarayhi, A., On the homotopy asymptotic method of quantum zakharov-kuznetsov equation in ion acoustic waves, Walailak Journal of Science and Technology, 13(5), 2016, 365-373.
[31] Ahmad, H. & Khan, T. A., Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations, Journal of Low Frequency Noise Vibration and Active Control, 38, 2019, 1113–1124.
[32] Abouelregal A, Ahmad H., A Modified Thermoelastic Fractional Heat Conduction Model with A Single-Lag and Two Different Fractional-Orders, Journal of Applied and Computational Mechanics, 2020, DOI: 10.22055/JACM.2020.33790.2287.
[33] Ullah, H., Islam, S., Idrees, M. and Nawaz, R., Application of optimal homotopy asymptotic method to doubly wave solutions of the coupled Drinfel’d-Sokolov-Wilson Equations, Mathematical Problems in Engineering, 2013, 2013, 362816.
[34] Ahmad, H., Seadawy, A. R. & Khan, T. A., Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm, Mathematics and Computers in Simulation, 2020, DOI: 10.1016/j.matcom.2020.04.005.
[35] Sheikholeslami, M., Hatami, M. and Ganji, D.D., Micropolar fluid flow and heat transfer in a permeable channel using analytical method, Journal of Molecular Liquids, 194, 2014, 30-36.
[36] Anjum, A., Mir, N.A., Farooq, M., Javed, M., Ahmad, S., Malik, M.Y. and Alshomrani, A.S., Physical aspects of heat generation/absorption in the second-grade fluid flow due to Riga plate: Application of Cattaneo-Christov approach, Results in Physics, 9, 2018, 955-960.
[37] Ahmad, H., Khan, T. A. & Cesarano, C., Numerical Solutions of Coupled Burgers′ Equations, Axioms, 8, 2019, 119.
[38] Saleem, S., Awais, M., Nadeem, S., Sandeep, N. and Mustafa, M.T., Theoretical analysis of upper-convected Maxwell fluid flow with Cattaneo–Christov heat flux model, Chinese Journal of Physics, 55(4), 2017, 1615-1625.
[39] Daftardar-Gejji, V. and Jafari, H., An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, 316(2), 2006, 753-763.
[40] Daftardar-Gejji, V. and Bhalekar, S., Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method, Computers & Mathematics with Applications, 59(5), 2010, 1801-1809.
[41] Daftardar-Gejji, V. and Bhalekar, S., Solving fractional diffusion-wave equations using a new iterative method, Fractional Calculus and Applied Analysis, 11(2), 2008, 193-202.
[42] Bhalekar, S. and Daftardar-Gejji, V., Solving fractional-order logistic equation using a new iterative method, International Journal of Differential Equations, 2012, 2012, Article ID 975829.
[43] Bhalekar, S. and Daftardar-Gejji, V., Convergence of the new iterative method, International Journal of Differential Equations, 2011, 2011, 989065.
[44] Hemeda, A.A., New iterative method: application to nth-order integro-differential equations, International Mathematical Forum, 7(47), 2012, 2317-2332.
[45] Yaseen, M. and Samraiz, M., A modified new iterative method for solving linear and nonlinear Klein-Gordon Equations, Applied Mathematical Sciences, 6, 2012, 2979-2987.
[46] Siddiqui, A.M., Ahmed, M. and Ghori, Q.K., Couette and Poiseuille flows for non-Newtonian fluids, International Journal of Nonlinear Sciences and Numerical Simulation, 7(1), 2006, 15-26.
[47] Siddiqui, A.M., Zeb, A., Ghori, Q.K. and Benharbit, A.M., Homotopy perturbation method for heat transfer flow of a third-grade fluid between parallel plates, Chaos, Solitons& Fractals, 36(1), 2008, 182-192.
[48] Islam, S. and Zhou, C.Y., Exact solutions for two dimensional flows of couple stress fluids, Zeitschrift für Angewandte Mathematik und Physik, 58(6), 2007, 1035-1048.
[49] Islam, S., Ali, I., Ran, X.J., Shah, A. and Siddiqui, A.M., Effects of couple stresses on Couette and Poiseuille flow, International Journal of Non-linear Sciences and Numerical Simulations, 10(1), 2009, 99-112.
[50] EL-Dabe, N.T. and El-Mohandis, S.M., Effect of couple stresses on pulsatile hydromagnetic Poiseuille flow, Fluid Dynamics Research, 15(5), 1995, 313.
[51] Eldabe, N.T.M., Hassan, A.A. and Mohamed, M.A., Effect of couple stresses on the MHD of a non-Newtonian unsteady flow between two parallel porous plates, Zeitschrift für Naturforschung A, 58(4), 2003, 204-210.
[52] Aksoy, Y. and Pakdemirli, M., Approximate analytical solutions for flow of a third-grade fluid through a parallel-plate channel filled with a porous medium, Transport in Porous Media, 83(2), 2010, 375-395.
[53] Massoudi, M. and Christie, I., Effects of variable viscosity and viscous dissipation on the flow of a third-grade fluid in a pipe, International Journal of Non-Linear Mechanics, 30(5), 1995, 687-699.
[54] Chinyoka, T. and Makinde, O.D., Analysis of transient Generalized Couette flow of a reactive variable viscosity third-grade liquid with asymmetric convective cooling, Mathematical and Computer Modelling, 54(1-2), 2011, 160-174.
[55] Reynolds, O., IV. On the theory of lubrication and its application to Mr. Beauchamp tower’s experiments, including an experimental determination of the viscosity of olive oil, Philosophical Transactions of the Royal Society of London, (177), 1886, 157-234.
[56] Szeri, A.Z. and Rajagopal, K.R., Flow of a non-Newtonian fluid between heated parallel plates, International Journal of Non-Linear Mechanics, 20(2), 1985, 91-101.
[57] Marinca, V., Herişanu, N. and Nemeş, I., Optimal homotopy asymptotic method with application to thin film flow, Open Physics, 6(3), 2008, 648-653.
[58] Daftardar-Gejji, V. and Jafari, H., An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, 316(2), 2006, 753-763.
[59] Papanastasiou, T., Georgiou, G. and Alexandrou, A.N., Viscous fluid flow, CRC press, 1999.
)