Dufour and Soret Effects on Unsteady Heat and Mass Transfer for Powell-Eyring Fluid Flow over an Expanding Permeable Sheet

Document Type : Research Paper


1 Department of Mathematics, The University of Burdwan, Burdwan-713104, West Bengal, India

2 Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi–221005, Uttar Pradesh, India


In the present analysis, the Dufour and Soret effects on unsteady heat-mass transfer of a viscous incompressible Powell-Eyring fluids flow past an expanding/shrinking permeable sheet are reported. The fluid boundary layer develops over the variable sheet with suction/injection to the non-uniform free stream velocity. Under the symmetry group of transformations, the governing equations along with three independent variables, are converted into a system of PDEs with two independent variables. Finally, by employing the order-reduction technique the PDEs are transformed into ODEs, which are then solved numerically. The results are presented graphically and analyzed. The main advantage of this technique is that without any prior knowledge, one can easily find the scaling transformations, expanding velocity, suction/injection velocity, and free-stream velocity. From computed numerical results many important findings are obtained. Most importantly, thermal and concentration overshoots are found for larger values of Dufour and Soret numbers, respectively. Also, thermal and concentration crossing over found for different values of Soret and Dufour numbers, respectively.


Main Subjects

[1] Ma, K.H.P. and Hui, W.H., Similarity solutions of the two-dimensional unsteady boundary-layer equations, Journal of Fluid Mechanics, 216, 1990, 537-559.
[2] Dholey S. and Gupta, A.S., Unsteady separated stagnation-point flow of an incompressible viscous fluid on the surface of a moving porous plate, Physics of Fluids, 25, 2013, 023601.
[3] Layek, G.C., An Introduction to Dynamical System and Chaos, Springer, India, 2015.
[4] Cantwell, B.J., Introduction to Symmetry Analysis, Cambridge University Press, Cambridge, New York, 2002.
[5] Bluman, G.W. and Kumei, S., Symmetries and Differential Equations, Springer, New York, 1989.
[6] Jalil, M., Asghar, S. and Imran, S.M., Self-similar solutions for the flow and heat transfer of Powell-Eyring fluid over a moving surface in a parallel free stream, International Journal of Heat and Mass Transfer, 65, 2013, 73-79.
[7] Jalil, M. and Asghar, S., Flow and heat transfer of Powell-Eyring fluid over a stretching surface: a Lie group analysis, Journal of Fluids Engineering, 135, 2013, 121201.
[8] Akgül, M.B. and Pakdemirli, M., Lie group analysis of a non-Newtonian fluid flow over a porous surface, Scientia Iranica, 19, 2012, 1534–1540.
[9] Yurusoy, M., Unsteady boundary layer flow and symmetry analysis of a Carreau fluid, Mathematics & Statistics, 1, 2016, 004.
[10] Layek, G.C. and Sunita, On the nature of multitude scalings in decaying isotropic turbulence, International Journal of Non-Linear Mechanics, 95, 2017, 143-150.
[11] Layek G.C., Mandal B., Bhattacharyya K. and Banerjee A., Lie symmetry analysis of boundary layer stagnation-point flow and heat transfer of non-Newtonian power-law fluids over a nonlinearly shrinking/stretching sheet with thermal radiation, International Journal of Nonlinear Sciences and Numerical Simulation, 19(3-4), 2018, 415-426.
[12] Patel, M. and Timol, M.G., Numerical treatment of MHD Powell-Eyring fluid flow using the method of satisfaction of asymptotic boundary conditions, Applied Numerical Mathematics, 59(10), 2009, 2584-2592.
[13] Javed, T., Ali, N., Abbas, Z. and Sajid, M., Flow of an Eyring-Powell non-Newtonian fluid over a stretching sheet, Chemical Engineering Communications, 200(3), 2013, 327-336.
[14] Malik, M.Y., Hussain, A. and Nadeem, S., Boundary layer flow of an Eyring-Powell model fluid due to a stretching cylinder with variable viscosity, Scientia Iranica, 20, 2013, 313-321.
[15] Hamid M., Usman M., Khan Z.H, Ahmad R., and Wang W., Dual solutions and stability analysis of flow and heat transfer of Casson fluid over a stretching sheet, Physics Letters A, 383, 2019, 2400-2408.
[16] Hamid M., Usman M., Zubair T., Haq R. U., and Wang, W., Shape effects of  nanoparticles on rotating flow of nanofluid along a stretching surface with variable thermal conductivity: A Galerkin approach, International Journal of Heat and Mass Transfer, 124, 2018, 706-714. 
[17] Soomro F.A., Usman M., Haq R.U., and Wang W., Thermal and velocity slip effects on MHD mixed convection flow of Williamson nanofluid along a vertical surface: Modified Legendre wavelets approach, Physica E: Low-dimensional Systems and Nanostructures, 104, 2018, 130-137.
[18] Usman, M., Hamid, M., Zubair, T., Ul Haq, R., and Wang, W., /Water hybrid nanofluid through a permeable surface in the presence of nonlinear radiation and variable thermal conductivity via LSM, International Journal of Heat and Mass Transfer, 126, 2018, 1347-1356.
[19] Usman, M., Mohyud Din S. T., Zubair T., Hamid M., and Wang, W., Fluid flow and heat transfer investigation of blood with nanoparticles through porous vessels in the presence of magnetic field, Journal of Algorithms & Computational Technology, 2019, https://doi.org/10.1177/1748301818788661.
[20] Usman M., Hamid M., Haq R. U., and Wang W., Heat and fluid flow of water and ethylene-glycol based Cu- nanoparticles between two parallel squeezing porous disks: LSGM approach, International Journal of Heat and Mass Transfer, 123, 2018, 888-895.
[21] Usman M., Haq R. U., Hamid M., and Wang W., Least square study of heat transfer of water based Cu and Ag nanoparticles along a converging/diverging channel, Journal of Molecular Liquids, 249, 2018, 856-867.
[22] Prasad, K.V., Datti, P.S. and Raju, B.T., Momentum and heat transfer of a non-Newtonian Eyring-Powell fluid over a non-Isothermal stretching sheet, International Journal of Mathematical Archive, 4(1), 2013, 230-241.
[23] Hayat, T., Farooq, M., Alsaedi, A. and Iqbal, Z., Melting heat transfer in the stagnation point flow of Powell Eyring fluid, Journal of Thermophysics and Heat Transfer, 27, 2013, 761-766.
[24] Khader, M.M. and Megahed, M.A., Numerical studies for flow and heat transfer of the Powell-Eyring fluid thin film over an unsteady stretching sheet with internal heat generation using the Chebyshev finite difference method, Journal of Applied Mechanics and Technical Physics, 54, 2013, 440–450.
[25] Hayat, T., Asad, S., Mustafa, M. and Alsaedi, A., Radiation effects on the flow of Powell- Eyringfluid past an unsteady inclined stretching sheet with non-uniform heat source/sink, PLoS ONE, 9(7), 2015, e103214.
[26] Reddy, A.M.R., Reddy, J.V.R., Sandeep, N. and Sugunamma, V. Unsteady flow of nonlinear radiative Powell-Eyring fluid past an inclined stretching sheet with first order chemical reaction, Middle-East Journal of Scientific Research, 24(7), 2018, 2366-2374.
[27] Hayat, T., Ali, S., Farooq, M.A. and Alsaedi, A., On comparison of series and numerical solutions for flow of Eyring-Powell fluid with Newtonian heating and internal heat generation/absorption, PLoS ONE, 10, 2015, 1-13.
[28] Hayat, T., Gull, N., Farooq, M. and Ahmad, B., Thermal radiation effect in MHD flow of Powell-Eyringnanofluid induced by a stretching cylinder, Journal of Aerospace Engineering, 29(1), 2015, 03.
[29] Ara, A., Khan, N.A., Khan, H. and Sultan, F., Radiation effect on boundary layer flow of an Eyring-Powell fluid over an exponentially shrinking sheet, Ain-Shams Engineering Journal, 5, 2014, 1337-1342.
[30] Nadeem, S. and Saleem, S., Mixed convection flow of Eyring–Powell fluid along a rotating cone, Results in Physics, 4, 2014, 54-62.
[31] Panigrahi, S., Reza, M. and Mishra, A.K., MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid past nonlinear stretching surface, Applied Mathematics and Mechanics (English Edition), 35(12), 2014, 1525-1540.
[32] Panigrahi, S., Reza, M. and Mishra, A.K., Mixed convective flow of a Powell-Eyring fluid over a non-linear stretching surface with thermal diffusion and diffusion thermo, Procedia Engineering, 127, 2015, 645-651.
[33] Malik, M.Y., Khan, I., Hussain, A. and Salahuddin, T., Mixed convection flow of MHD Eyring-Powell nanofluid over a stretching sheet: A numerical study, AIP Advances, 5, 2015, 117-118.
[34] Megahed, M.A., Flow and heat transfer of Powell-Eyring fluid due to an exponential stretching sheet with heat flux and variable thermal conductivity, Zeitschrift fur Naturforschung A, 70, 2015, 163-169.
[35] Krishna, P.M., Sandeep, N., Reddy, J.V.R. and Sugunamma, V., Dual solutions for unsteady flow of Powell-Eyring fluid past an inclined stretching sheet, Journal of Naval Architecture and Marine Engineering, 13, 2016, 89-99.
[36] Rehman, K.U., Malik, M.Y., Salahuddin, T. and Naseer, M., Dual stratified mixed convection flow of Eyring-Powell fluid over an inclined stretching cylinder with heat generation/absorption effect, AIP Advances, 6, 2016, 075112.
[37] Zaib, A., Bhattacharyya, K. and Shafie, S., Unsteady boundary layer flow and heat transfer over an exponentially shrinking sheet with suction in a copper-water nanofluid, Journal of Central South University, 22, 2015, 4856-4863.
[38] Ye, W.B., Melting process in a rectangular thermal storage cavity heated from vertical walls, Journal of Thermal Analysis and Calorimetry, 123(1), 2015, 873-880.
[39] Ye, W.B., Finite volume analysis the thermal behavior of electrode non-uniformity, Heat and Mass Transfer, 53(3), 2016, 1123-1132.
[40] Ye, W.B., Design method and modeling verification for the uniform air flow distribution in the duct ventilation, Applied Thermal Engineering, 110, 2017, 573-583.
[41] Eckert, E.R.G. and Drake, R.M., Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 2002.
[42] Khan, N.A. and Sultan, F., On the double diffusive convection flow of Eyring-Powell fluid due to cone through a porous medium with Soret and Dufour effects, AIP Advances, 5, 2015, 057140.
[43] Hayat, T., Muhammad, T., Shehzad, S.A. and Alsaedi, A., Soret and Dufour effects in three-dimensional flow over an exponentially stretching surface with porous medium, chemical reaction and heat source/sink, International Journal of Numerical Methods for Heat & Fluid Flow, 25(4), 2015, 762-781.
[44] Powell, R.E. and Eyring, H., Mechanisms for the relaxation theory of viscosity, Nature, 154, 1944, 427-428.
[45] Brewster, Q.M., Thermal Radiative Transfer Properties, John Wiley and Sons, Chichester, 1972.
[46] Khan, I., Qasim, M. and Shafie, S., Flow of an Erying-Powell fluid over a stretching sheet in presence of chemical reaction, Thermal Science, 20(6), 2016, 1903-1912.