MHD Oldroyd-B Fluid with Slip Condition in view of Local and ‎Nonlocal Kernels

Document Type : Research Paper


1 Department of Mathematics, University of Management and Technology, Lahore, 54000, Pakistan

2 Institute for Groundwater Studies (IGS), University of the Free State, Bloemfontein, 9301, South Africa

3 Department of Science & Humanities, National University of Computer and Emerging Sciences, Lahore Campus, 54000, Pakistan

4 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan


We examine the velocity field of an incompressible Oldroyd-B fluid over a horizontal plate of continual length in a permeable medium with magnetohydrodynamics effect. Firstly, the results for the dimensionless classical model (governing equation) have been studied analytically then the study is extended for different fractional operators. The relations to determine the velocity fields of this problem are found by Laplace transformation and different numerical inversion algorithms. The impact of physical parameters on velocity profiles is analyzed graphically for integer and non-integer models. Non-integer operators are used to analyzing the impact of fractional parameters on the fluid curves of the fluid.


Main Subjects

[1] Zaman, H., Ahmad, Z., Ayub, M., A note on the unsteady incompressible MHD fluid flow with slip conditions and porous walls, ISRN Mathematical Physics, 1, 2013, 1 – 10.
[2] Ghosh, A.K., Sana, P., On hydromagnetics flow of an Oldroyd-B fluid near a pulsating plate, Acta Astronautica, 64, 2009, 272 – 280.
[3] Nadeem, S., Mehmood, R., Akbar, N.S., Non-orthogonal stagnation point flow of a nano non-Newtonian fluid towards a stretching surface with heat transfer, International Journal of Heat and Mass Transfer, 57, 2013, 679 – 689.
[4] Sharmilaa, K., Kaleeswari, S., Dufour effects on unsteady free convection and mass transfer through a porous medium in a slip regime with heat source/sink, International Journal of Scientific Engineering and Applied Science, 1, 2015, 307 – 320.
[5] Waters, N.D., King, M.J., Unsteady flow of an elastico-viscous liquid, Rheologica Acta, 9, 1970, 345 – 355.
[6] Riaz, M.B., Zafar, A.A., Vieru, D., On flows of generalized second grade fluids generated by an oscillating flat plate, Secția Matematică. Mecanică Teoretică. Fizică, 1, 2015, 1 – 9.
[7] Ali, F., Khan, I., Shafie, S., Closed form solutions for unsteady free convection flow of a second grade fluid over an oscillating vertical plate, PLoS ONE, 9(2), 2014, e85099.
[8] Choi, J.J., Rusak, Z., Tichy, J.A., Maxwell fluid suction flow in a channel, Journal of Non-Newtonian Fluid Mechanics, 85, 1999, 165 – 187.
[9] Omowaye, A.J., Animasaun, I.L., Upper-convected Maxwell fluid flow with variable thermo-physical properties over a melting surface situated in hot environment subject to thermal stratification, Journal of Applied Fluid Mechanics, 9(4), 2016, 1777 – 1790.
[10] Fetecau, C., Fetecau, C., Starting solutions for some unsteady unidirectional flow of a second grade fluid, International Journal of Engineering Science, 43, 2005, 781 – 789.
[11] Maxwell, J.C., on the dynamical theory of gases, Philosophical Transactions of the Royal Society of London, 157, 1866, 49 – 88.
[12] Oldroyd, G.J., On the formulation of rheological equations of state, Proceedings of the Royal Society A, 200, 1950, 523 – 541.
[13] Oldroyd, G.J., The motion of an elastic-viscous liquid contained between coaxial cylinders, The Quarterly Journal of Mechanics and Applied Mathematics, 4, 1951, 271 – 282.
[14] Guillope, C., Saut, J.C., Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, Mathematical Modeling and Numerical Analysis, 24, 1990, 369 – 401.
[15] Hayat, T., Siddiqui, A.M., Asghar, S., Some simple flows of an Oldroyd-B fluid, International Journal of Engineering Science, 39, 2001, 135 – 147.
[16] Hayat, T., Khan, M., Ayub, M., Exact solutions of flow problems of an Oldroyd-B fluid, Applied Mathematics and Computation, 151, 2004, 105 – 119.
[17] Siddiqui, A.M., Haroon, T., Zahid, M., Shahzad, A., Effect of slip condition on unsteady flows of an Oldroyd-B uid between parallel plates, World Applied Sciences Journal, 13(11), 2011, 2282 – 2287.
[18]Riaz, M.B., Imran, M.A., Shabbir, K., Analytic solutions of Oldroyd-B fluids with fractional derivatives in a circular duct that applies a constant couple, Alexendria Engineering Journal, 55(4), 2016, 3267 – 3275.
[19] Navier, C.L.M.H., Mmoires de l'Acadmie (royale) des sciences de l'Institut (imperial) de France, 1, 1823, 414 – 416.
[20] Pit, R., Hervet, H., Leger, L, Friction and slip of a simple liquid at a solid surface, Tribology Letters, 7, 1999, 147-152.
[21] Blake, T.D., Slip between a liquid and a solid: DM Tolstoi's (1952) theory reconsidered, Colloids and Surfaces, 47,1990, 135 – 145.
[22] A. Germant, On fractional differentials, Philosophical Magazine, 25, 1938, 540 – 549.
[23] Smith, W., De Vries, H., Rheological models containing fractional derivatives, Rheological Acta, 9, 1970, 525 – 534.
[24] Sarwar, S., Rashidi, M.M., Approximate solution of two-term fractional-order diffusion, wave-diffusion, and telegraph models arising in mathematical physics using optimal homotopy asymptotic method, Waves in Random and Complex Media, 26(3), 2016, 365 – 382.
[25] Koeller, R.C., Application of fractional calculus to the theory of viscoelasticity, Journal of Applied Mechanics, 51(2), 1984, 299 – 307.
[26] Mainardi, F., Fractional calculus and waves in linear viscoelasticity, Imperial College Press World Scientific, London 2010.
[27] Miller, K.S., Ross, B., An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons Inc. New York, 1993.
[28] Podlubny, I., Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation, Fractional Calculus and Applied Analysis, 4, 2002, 367 – 386.
[29] Zafar, A.A., Riaz, M.B., Shah, N.A., Imran, M.A., Influence of non-integer derivatives on unsteady unidirectional motions of an Oldroyd-B fluid with generalized boundary condition, The European Physical Journal Plus, 133(3), 2018, 1 – 13.
[30] Riaz, M.B., Atangana, A., Abdeljawad, T., Local and non-local differential operators: A comparative study of heat and mass transfer in MHD Oldroyd-B fluid with ramped wall temperature, Fractal, 2020,
[31] Khan, I., Saeed, S.T., Riaz, M.B., Abro, K.A., Husnine, S.M., Nissar, K.S., Influence in a Darcy’s Medium with Heat Production and Radiation on MHD Convection Flow via Modern Fractional Approach, Journal of Materials Research and Technology, 9(5), 2020, 10016 – 10030.
[32] Riaz, M.B., Saeed, S.T., Baleanu, D., Ghalib, M., Computational results with non-singular & non-local kernel flow of viscous fluid in vertical permeable medium with variant temperature, Frontier in Physics, 8, 2020, 275.
[33] Riaz, M.B., Saeed, S.T., Baleanu, D., Role of Magnetic field on the Dynamical Analysis of Second Grade Fluid: An Optimal Solution subject to Non-integer Differentiable Operators, Journal of Applied and Computational Mechanics, 6(SI), 2020, 1475 – 1489.
[34] Riaz, M.B., Saeed, S.T., Comprehensive analysis of integer order, Caputo-fabrizio and Atangana-Baleanu fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition, Discrete and Continuous Dynamical Systems, 2020, Accepted.
[35] Caputo, M., Fabrizio, M., A New Definition of Fractional Derivative Without Singular Kernel, Progress in Fractional Differentiation and Applications, 2(1), 2016, 1 – 11.
[36] Atangana, A., On the new fractional derivative and application to nonlinear Fisher's reaction–diffusion equation, Applied Mathematics and Computation, 273, 2016, 948 – 956.
[37] Atangana, A., Koca, I., New direction in fractional differentiation, Mathematics in Natural Science, 1, 2017, 18–25.
[38] Saeed, S.T., Riaz, M.B., Baleanu, D., Abro, K.A., A Mathematical Study of Natural Convection Flow Through a Channel with non-singular Kernels: An Application to Transport Phenomena, Alexandria Engineering Journal, 59, 2020, 2269 – 2281.
[39] Atangana, A., Baleanu, D., New fractional derivatives with nonlocal and nonsingular kernel: Theory and Application to heat transfer model, Journal of Engineering Mechanics, 9(5), 2016, 1 – 5.
[40] Riaz, M.B., Atanganaa, A., Saeed, S.T., MHD free convection flow over a vertical plate with ramped wall temperature and chemical reaction in view of non-singular kernel, Publisher: Wiley, 2020, 253 – 279.
[41] Imran, M.A., Aleem, M., Riaz, M.B., Ali, R., Khan, I., A comprehensive report on convective flow of fractional (ABC) and (CF) MHD viscous fluid subject to generalized boundary conditions, Chaos, Solitons & Fractals, 118, 2018, 274 – 289.
[42] Riaz, M.B., Atangana, A., Iftikhar, N., Heat and mass transfer in Maxwell fluid in view of local and non-local differential operators, Journal of Thermal Analysis and Calorimetry, 2020,
[43] Riaz, M.B., Iftikhar, N., A comparative study of heat transfer analysis of MHD Maxwell fluid in view of local and non-local differential operators, Chaos Solitons & Fractals, 132, 2020, 109556.
[44] Fatecau, C., Zafar, A.A., Vieru, D., Awrejcewicz, J., Hydromagnetic flow over a moving plate of second grade fluids with time fractional derivatives having non-singular kernel, Chaos Solitons & Fractals, 130, 2020, 109454.
[45] Akgül, A., A novel method for a fractional derivative with non-local and non-singular kernel, Chaos Solitons & Fractals, 114, 2018, 478 – 482.
[46] Akgül, E.K., Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(2), 2019, 023108.
[47] Khan, I., Imran, M., Fakhar, K., New exact solutions for an Oldroyd-B fluid in a porous medium, International Journal of Mathematics and Mathematical Sciences, 2011, 2011, Article ID 408132.
[48] Stehfest, H. A., Numerical inversion of Laplace transforms, Communications of the ACM, 13, 1970, 9 – 47.
[49] Tzou, D. Y., Macro to Microscale Heat Transfer: The Lagging Behaviour, Washington: Taylor and Francis, 1970.