Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform

doi:10.1007/s10483-016-2021-8

Abstract

Abstract:

This paper introduces a new model for the Fourier law of heat conduction with the time-fractional order to the generalized Maxwell fluid. The flow is influenced by magnetic field, radiation heat, and heat source. A fractional calculus approach is used to establish the constitutive relationship coupling model of a viscoelastic fluid. We use the Laplace transform and solve ordinary differential equations with a matrix form to obtain the velocity and temperature in the Laplace domain. To obtain solutions from the Laplace space back to the original space, the numerical inversion of the Laplace transform is used. According to the results and graphs, a new theory can be constructed. Comparisons of the associated parameters and the corresponding flow and heat transfer characteristics are presented and analyzed in detail.

Key words: radiation heat, Laplace transform, Maxwell fluid, fractional derivative, heat source

2010 MSC Number:

Yaqing LIU, Boling GUO. Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform. Applied Mathematics and Mechanics (English Edition), 2016, 37(2): 137-150.

TrendMD

References

[1] Fetecau, C., Athar, M., and Fetecau, C. Unsteady flow of a generalized Maxwell fluid with frac-tional derivative due to a constantly accelerating plate. Computers and Mathematics with Appli-cations, 57, 596-603(2009)
[2] Hayat, T. and Sajid, M. Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid. International Journal of Engineering Science, 45, 393-401(2007)
[3] Tan, W. C. and Masuoka, T. Stability analysis of a Maxwell fluid in a porous medium heated from below. Physics Letters A, 360, 454-460(2007)
[4] Podlubny, I. Fractional Differential Equations, Academic Press, New York (1999)
[5] Guo, B. L., Pu, X. K., and Huang, F. H. Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing (2011)
[6] Bagley, R. L. and Torvik, P. J. A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27, 201-210(1983)
[7] Friedrich, C. Relaxation and retardation functions of the Maxwell model with fractional deriva-tives. Rheologica Acta, 30, 151-158(1991)
[8] Song, D. Y. and Jiang, T. Q. Study on the constitutive equation with fractional derivative for the viscoelastic fluids modified Jeffreys model and its application. Rheologica Acta, 27, 512-517(1998)
[9] Qi, H. T. and Xu, M. Y. Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B model. Acta Mechanica Sinica, 23, 463-469(2007)
[10] Qi, H. T. and Xu, M. Y. Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative. Applied Mathematical Modeling, 33, 4184-4191(2009)
[11] Fetecau, C., Prasad, S. C., and Rajagopal, K. R. A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid. Applied Mathematical Modeling, 31, 647-654(2007)
[12] Fetecau, C., Fetecau, C., Kamran, M., and Vieru, D. Exact solutions for the flow of a generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate. Journal of Non-Newtonian Fluid Mechanics, 156, 189-201(2009)
[13] Vieru, D., Fetecau, C., and Fetecau, C. Flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate. Applied Mathematics and Computation, 201, 834-842(2008)
[14] Fetecau, C., Athar, M., and Fetecau, C. Unsteady flow of a generalized Maxwell fluid with frac-tional derivative due to a constantly accelerating plate. Computers and Mathematics with Appli-cations, 57, 596-603(2009)
[15] Khan, M., Hayat, T., and Asghar, S. Exact solution for MHD flow of a generalized Oldroyd-B fluid with modified Darcy's law. International Journal of Engineering Science, 44, 333-339(2006)
[16] Zheng, L. C., Liu, Y. Q., and Zhang, X. X. A new model for plastic-viscoelastic magnetohydrody-namic (MHD) flow with radiation thermal transfer. International Journal of Nonlinear Sciences and Numerical Simulation, 14, 435-441(2013)
[17] Zheng, L. C., Liu, Y. Q., and Zhang, X. X. Exact solutions for MHD flow of generalized Oldroyd-B fluid due to an infinite accelerating plate. Mathematical and Computer Modelling, 54, 780-788(2011)
[18] Shen, F., Tan, W. C., Zhao, Y. H., and Masuoka, T. Decay of vortex velocity and diffusion of temperature in a generalized second grade fluid. Applied Mathematical Modeling, 25, 1151-1159(2004)
[19] Shen, F., Tan, W. C., Zhao, Y. H., and Masuoka, T. The Reyleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model. Nonlinear Analysis: Real World Applications, 7, 1072-1080(2006)
[20] Ezzat, M. A. Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer. Physica B, 405, 4188-4194(2010)
[21] Qi, H. T. and Liu, J. G. Time-fractional radial diffusion in hollow geometries. Meccanica, 45, 577-583(2010)
[22] Jiang, X. Y. and Qi, H. T. Thermal wave model of binoheat transfer with modified Riemann-Liouville fractional derivative. Journal of Physics A: Mathematical and Theoretical, 45, 831-842(2012)
[23] Xu, H. Y. and Jiang, X. Y. Time fractional dual-phase-lag heat conduction equation. Chinese Physics B, 24, 034401(2015)
[24] Jiang, X. Y., Xu, M. Y., and Qi, H. T. The fractional diffusion model with an absorption term and modified Fick's law for non-local transport processes. Nonlinear Analysis: Real World Applications, 11, 262-269(2010)
[25] Tan, W. C. and Masuoka, T. Stokes' first problem for a second grade fluid in a porous half-space with heated boundary. International Journal of Non-Linear Mechanics, 40, 515-522(2005)
[26] EI-Aziz, M. A. Radiation effect on the flow and heat transfer over an unsteady stretching sheet. International Communications in Heat and Mass Transfer, 36, 521-524(2009)
[27] Cortell, R. Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Physics Letters A, 371, 631-636(2008)
[28] Ezzat, M., El-Bary, A. A., and Ezzat, S. Combined heat and mass transfer for unsteady MHD flow of perfect conducting micropolar fluid with thermal relaxation. Energy Conversion and Man-agement, 52, 934-945(2011)
[29] Ezzat, M. and El-Karamany, A. S. Magnetothermoelasticity with two relaxation times in conduct-ing medium with variable electrical and thermal conductivity. Applied Mathematics and Compu-tation, 142, 449-467(2003)
[30] Ezzat, M. A., Othman, M. I., and El-Karamany, A. S. State space approach to generalized ther-moviscoelasticity with two relaxation times. International Journal of Engineering Science, 40, 283-302(2002)
[31] Ezzat, M. A., El-Karamany, A. S., and Samaan, A. A. The dependence of the modulus of elas-ticity on reference temperature in generalized thermoelasticity with thermal relaxation. Applied Mathematics and Computation, 147, 169-189(2004)
[32] Ezzat, M. A. The relaxation effects of the volume properties of electrically conducting viscoelastic material. Materials Science and Engineering B, 130, 11-23(2006)
[33] Ezzat, M. A. and El-Karamany, A. S. The relaxation effects of the volume properties of viscoelastic material in generalized thermoelasticity. International Journal of Engineering Science, 41, 2281-2298(2003)
[34] El-Karamany, A. S. and Ezzat, M. A. Thermal shock problem in generalized thermo-viscoelasticty under four theories. International Journal of Engineering Science, 42, 649-671(2004)
[35] Honig, G. and Hirdes, U. A method for the numerical inversion of Laplace transforms. Journal of Computational and Applied Mathematics, 10, 113-132(1984)