Journal of Applied and Computational Mechanics

Heat and Mass Transfer of Natural Convective Flow with Slanted ‎Magnetic Field via Fractional Operators

Document Type : Research Paper

Authors

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

2 Department of Mathematics, Cankaya University, Ankara, 06790, Turkey

3 Institute of Space Sciences, Magurele, Bucharest, 077125, Romania

4 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung City, 40402, Taiwan

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

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

Abstract

This article explores the MHD natural convective viscous and incompressible fluid flow along with radiation and chemical reaction. The flow is confined to a moving tilted plate under slanted magnetic field with variable temperature in a porous medium. Non-dimensional parameter along Laplace transformation and inversion algorithm are used to investigate the solution of system of dimensionless governing equations. Fractional differential operators namely, Caputo (C), Caputo-Fabrizio (CF) and Atangana-Baleanu in Caputo sense (ABC) are used to compare graphical behavior of for velocity, temperature and concentration for emerging parameters. On comparison, it is observed that fractional order model is better in explaining the memory effect as compared to classical model. Velocity showing increasing behavior for fractional parameter a whereas there is a decline in temperature, and concentration profiles for a. Fluid velocity goes through a decay due to rise in the values of M, Sc and j. However, velocity shows a reverse profile for augmented inputs of Kp , Gr and S. Tabular comparison is made for velocity and Nusselt number and Sherwood number for fractional models.

Keywords

Main Subjects

[1] Jha, B.K., MHD free-convection and mass-transform flow through a porous medium. Astrophysics and Space science, 2, 1991, 283 – 289.
[2] Kamel, M.H., Unsteady MHD convection through porous medium with combined heat and mass transfer with heat source/sink. Energy Conversion and Management, 4, 2001, 393 – 405.
[3] Ibrahim, F.S., Hassanien, I.A., Bakr, A.A., Unsteady magnetohydrodynamic micropolar fluid flow and heat transfer over a vertical porous plate through a porous medium in the presence of thermal and mass diffusion with a constant heat source. Canadian Journal of Physics, 10, 2004, 775 – 790.
[4] Eldabe, N.T.M., Elbashbeshy, E.M.A., Hasanin, W.S.A., Elsaid, E.M., Unsteady motion of MHD viscous incompressible fluid with heat and mass transfer through porous medium near a moving vertical plate. International Journal of Energy and Technology, 3, 2011, 1 – 11.
[5] Doungmo Goufo, E. F., Mathematical analysis of peculiar behavior by chaotic, fractional and strange multiwing attractors. International Journal of Bifurcation and Chaos, 28, 2018, 1850125.
[6] Goufo E. F., Application of the Caputo-Fabrizio Fractional Derivative without Singular Kernel to Korteweg-de Vries-Burgers Equation. Mathematical Modelling and Analysis, 21, 2016, 188 – 198.
[7] Goufo, E.F., On chaotic models with hidden attractors in fractional calculus above power law. Chaos, Solitons and Fractals, 127, 2019, 24 – 30.
[8] Owusu, K.F. Goufo, E. F, Mugisha, S., Modelling intracellular delay and therapy interruptions within Ghanaian HIV population. Advances in Difference Equations, 1, 2020, 1 – 9.
[9] 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, 5, 9, 2020, 10016 – 10030.
[10] 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.
[11] 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. 
[12] Song, D.Y., Jiang, T.Q., Study on the constitutive equation with fractional derivative for the viscoelastic fluids modified Jeffreys model and its application. Rheologica Acta, 5, 37, 1998, 512 – 517.
[13] Riaz, M.B., Siddiqui, M., Saeed, S.T.,Atangana, A., MHD Oldroyd-B Fluid with Slip Condition in view of Local and Nonlocal Kernels, Journal of Applied and Computational Mechanics, 6(SI), 2020,  1540 – 1551.
[14] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, 1998.
[15] Wenchang, T., Wenxiao, P., Mingyu, X., A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. International Journal of Non-Linear Mechanics, 5, 38, 2003, 645 – 650.
[16] Riaz, M.B., Iftikhar, N., A comparative study of heat transfer analysis of MHD Maxwell fluid in view of local and nonlocal differential operators. Chaos, Solitons and Fractals, 132, 2020, 109556.
[17] 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.
[18] 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.  
[19] 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. Wiley, 2020, 253 – 279.    
[20] Qi, H., Xu, M., Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel. Mechanics Research Communications, 2, 34, 2007, 210 – 212.
[21] 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, https://doi.org/10.1142/S0218348X20400332.
[22] 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.
[23] 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, 3, 133, 2018, 1 – 13.
[24] Abro, K.A., Khan, I., Gomez-Aguilar, J.F., A mathematical analysis of a circular pipe in rate type fluid via Hankel transform. The European Physical Journal Plus, 10, 133, 2018, 397.
[25] Umemura, A., Law, C.K., Natural-convection boundary-layer flow over a heated plate with arbitrary inclination. Journal of Fluid Mechanics, 219, 1990, 571 – 584.
[26] Pattnaik, J.R., Dash, G.C., Singh, S., Radiation and mass transfer effects on MHD flow through porous medium past an exponentially accelerated inclined plate with variable temperature. Ain Shams Engineering Journal, 1, 8, 2017, 67 – 75.
[27] Nandkeolyar, R., Das, M., MHD free convective radiative flow past a flat plate with ramped temperature in the presence of an inclined magnetic field. Computational and Applied Mathematics, 34, 1, 2015, 109 – 123.
[28] Endalew, M.F., Nayak, A., Thermal radiation and inclined magnetic field effects on MHD flow past a linearly accelerated inclined plate in a porous medium with variable temperature. Heat Transfer Asian Research, 1, 48, 2019, 42 – 61.
[29] Uddin, Z., Kumar, M., Unsteady free convection in a fluid past an inclined plate immersed in a porous medium. Computer Modelling and New Technologies, 3, 14, 2010, 41 – 47.
[30] Ali, F., Khan, I., Shafie, S., Conjugate effects of heat and mass transfer on MHD free convection flow over an inclined plate embedded in a porous medium. PLoS One, 6, 8, 2013, 65223.
[31] Makinde, O.D., Heat and mass transfer by MHD mixed convection stagnation point flow toward a vertical plate embedded in a highly porous medium with radiation and internal heat generation. Meccanica, 5, 47, 2012, 1173 – 1184.
[32] Khalid, A., Khan, I., Khan, A., Shafie, S., Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium. Engineering Science and Technology, an International Journal, 3, 18, 2015, 309 – 317.
[33] Iftikhar, N., Husnine, S.M., Riaz, M.B., Heat and mass transfer in MHD Maxwell fluid over an infinite vertical plate. Journal of Prime Research in Mathematics, 15, 2019, 63 – 80.
[34]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, 4, 55, 2016, 3267 – 3275.
[35] Abro, K.A., A Fractional and Analytic Investigation of Thermo-Diffusion Process on Free Convection Flow: An Application to Surface Modification Technology. The European Physical Journal Plus, 1, 31, 2020, 135.
[36] Abro, K.A., Atangana, A., A comparative study of convective fluid motion in rotating cavity via Atangana–Baleanu and Caputo–Fabrizio fractal–fractional differentiation. The European Physical Journal Plus, 2, 135, 2020, 226
[37] Shah, N.A., Elnaqeeb, T., Wang, S., Effects of Dufour and fractional derivative on unsteady natural convection flow over an infinite vertical plate with constant heat and mass fluxes. Computational and Applied Mathematics, 4, 37, 2018, 4931-4943.
[38] Khan, A., Abro, K. A., Tassaddiq, A., Khan, I., Atangana Baleanu and Caputo Fabrizio analysis of fractional derivatives for heat and mass transfer of second grade fluids over a vertical plate: a comparative study. Entropy, 8, 19, 2017, 279.
[39] Tassaddiq, A., MHD flow of a fractional second grade fluid over an inclined heated plate. Chaos Solitons & Fractals, 1, 123, 2019, 341 – 346.
[40] Shah, N. A., Ahmed, N., Elnaqeeb, T., Rashidi, M. M., Magnetohydrodynamic free convection flows with thermal memory over a moving vertical plate in porous medium. Journal of Applied and Computational Mechanics, 1, 5, 2019, 150-161.
[41] Sheikh, N.A., Ali, F., Khan, I., Saqib, M., A modern approach of Caputo Fabrizi time-fractional derivative to MHD free convection flow of generalized second grade fluid in a porous medium. Neural Computing and Applications, 6, 2018, 1865 – 1875.
[42] 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.
[43] Ahmad, M., Imran, M.A., Aleem, M., Khan, I., A comparative study and analysis of natural convection flow of MHD non-Newtonian fluid in the presence of heat source and first-order chemical reaction. Journal of Thermal Analysis and Calorimetry, 5, 137, 2019 , 1783 – 1796.
[44] Caputo, M., Fabrizio, M., A New Definition of Fractional Derivative Without Singular Kernel, Progress in Fractional Differentiation and Applications, 2, 1, 2015, 1 – 11.
[45] Atangana, A., On the new fractional derivative and application to nonlinear Fisher's reaction–diffusion equation. Applied Mathematics and Computation, 273, 2016, 948 – 956.
[46] Atangana, A., Koca, I., New direction in fractional differentiation. Mathematics in Natural Science, 1, 2017, 18–25.
[47] 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.
[48] Akgül, A., A novel method for a fractional derivative with non-local and non-singular kernel. Chaos Solitons & Fractals, 114, 2018, 478 – 482.
[49] Stehfest, H. A., Numerical inversion of Laplace transforms. Communications of the ACM, 13, 1970, 9 – 47.
[50] Tzou, D. Y., Macro to Microscale Heat Transfer: The Lagging Behaviour, Washington: Taylor and Francis, 1970. 
Journal of Applied and Computational Mechanics
Volume 7, Issue 1
January 2021
Pages 189-212