Rauf, A., Mahsud, Y. (2019). Electro-magneto-hydrodynamics Flows of Burgers' Fluids in Cylindrical Domains with Time Exponential Memory. Journal of Applied and Computational Mechanics, 5(4), 577-591. doi: 10.22055/jacm.2018.26478.1336

Abdul Rauf; Yasir Mahsud. "Electro-magneto-hydrodynamics Flows of Burgers' Fluids in Cylindrical Domains with Time Exponential Memory". Journal of Applied and Computational Mechanics, 5, 4, 2019, 577-591. doi: 10.22055/jacm.2018.26478.1336

Rauf, A., Mahsud, Y. (2019). 'Electro-magneto-hydrodynamics Flows of Burgers' Fluids in Cylindrical Domains with Time Exponential Memory', Journal of Applied and Computational Mechanics, 5(4), pp. 577-591. doi: 10.22055/jacm.2018.26478.1336

Rauf, A., Mahsud, Y. Electro-magneto-hydrodynamics Flows of Burgers' Fluids in Cylindrical Domains with Time Exponential Memory. Journal of Applied and Computational Mechanics, 2019; 5(4): 577-591. doi: 10.22055/jacm.2018.26478.1336

Electro-magneto-hydrodynamics Flows of Burgers' Fluids in Cylindrical Domains with Time Exponential Memory

^{1}Department of Computer Science and Engineering, Air University Multan, Abdali Road, Multan, 60000, Pakistan

^{2}Abdus Salam School of Mathematical Sciences, GC University Lahore, 68-B, New Muslim Town, Lahore, 54600, Pakistan

Abstract

This paper investigates the axial unsteady flow of a generalized Burgers’ fluid with fractional constitutive equation in a circular micro-tube, in presence of a time-dependent pressure gradient and an electric field parallel to flow direction and a magnetic field perpendicular on the flow direction. The mathematical model used in this work is based on a time-nonlocal constitutive equation for shear stress with time-fractional Caputo-Fabrizio derivatives; therefore, the histories of the velocity gradient will influence the shear stress and fluid motion. Thermal transport is considered in the hypothesis that the temperature of the cylindrical surface is constant. Analytical solutions for the fractional differential momentum equation and energy equation are obtained by employing the Laplace transform with respect to the time variable t and the finite Hankel transform with respect to the radial coordinate r. It is important to note that the analytical solutions for many particular models such as, ordinary/fractional Burgers fluids, ordinary/fractional Oldryd-B fluids, ordinary/fractional Maxwell fluids and Newtonian fluids, can be obtained from the solutions for the generalized fractional Burgers' fluid by particularizing the material coefficients and fractional parameters. By using the obtained analytical solutions and the Mathcad software, we have carried out numerical calculations in order to analyze the influence of the memory parameters and magnetic parameter on the fluid velocity and temperature. Numerical results are presented in graphical illustrations. It is found that ordinary generalized Burgers’ fluids flow faster than the fractional generalized Burgers’ fluids.

[1] Huang, J., He, G. and Liu, C., Analysis of general second-order fluid flow in double cylinder rheometer. Science in China Series A: Mathematics, 40(2) (1997) 183-190.

[2] Tan, W., Xian, F. and Wei, L., An exact solution of unsteady Couette flow of generalized second grade fluid. Chinese Science Bulletin, 47(21) (2002) 1783-1785.

[3] Xu, M. and Tan, W., Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion. Science in China Series A: Mathematics, 44(11) (2001) 1387-1399.

[4] Hayat, T., Nadeem, S. and Asghar, S., Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model. Applied Mathematics and Computation, 151(1) (2004) 153-161.

[5] Khan, M., Maqbool, K. and Hayat, T., Influence of Hall current on the flows of a generalized Oldroyd-B fluid in a porous space. Acta Mechanica, 184(1-4) (2006) 1-13.

[6] Qi, H. and Jin, H., Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders. Acta Mechanica Sinica, 22(4) (2006) 301-305.

[7] Chakraborty, R., Dey, R. and Chakraborty, S., Thermal characteristics of electromagnetohydrodynamic flows in narrow channels with viscous dissipation and Joule heating under constant wall heat flux. International Journal of Heat and Mass Transfer, 67 (2013) 1151-1162.

[8] Jian, Y., Si, D., Chang, L. and Liu, Q., Transient rotating electromagnetohydrodynamic micropumps between two infinite microparallel plates. Chemical Engineering Science, 134 (2015) 12-22.

[9] Sinha, A. and Shit, G.C., Electromagnetohydrodynamic flow of blood and heat transfer in a capillary with thermal radiation. Journal of Magnetism and Magnetic Materials, 378 (2015) 143-151.

[10] Wang, L., Jian, Y., Liu, Q., Li, F. and Chang, L., Electromagnetohydrodynamic flow and heat transfer of third grade fluids between two micro-parallel plates. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 494 (2016) 87-94.

[11] Jyothi, K.L., Devaki, P. and Sreenadh, S., Pulsatile flow of a Jeffrey fluid in a circular tube having internal porous lining. International Journal of Mathematical Archive, 4(5) (2013) 75-82.

[12] Ghosh, A.K. and Sana, P., On hydromagnetic channel flow of an Oldroyd-B fluid induced by rectified sine pulses. Computational & Applied Mathematics, 28(3) (2009) 365-395.

[13] Elshehawey, E.F., Eldabe, N.T., Elghazy, E.M. and Ebaid, A., Peristaltic transport in an asymmetric channel through a porous medium. Applied Mathematics and Computation, 182(1) (2006) 140-150.

[14] Hayat, T., Khan, S. B., Khan, M., Exact solution for rotating flows of a generalized Burgers’s fluid in a porous space. Applied Mathematical Modelling, 32 (2008) 749-760.

[15] Das, S. and Chakraborty, S., Analytical solutions for velocity, temperature and concentration distribution in electroosmotic microchannel flows of a non-Newtonian bio-fluid. Analytica Chimica Acta, 559(1) (2009) 15-24.

[16] Zhao, C. and Yang, C., Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels. Applied Mathematics and Computation, 211(2) (2009) 502-509.

[17] Chakraborty, S., Electroosmotically driven capillary transport of typical non-Newtonian biofluids in rectangular microchannels. Analytica Chimica Acta, 605(2) (2007) 175-184.

[18] Zhao, C. and Yang, C., Nonlinear Smoluchowski velocity for electroosmosis of Power‐law fluids over a surface with arbitrary zeta potentials. Electrophoresis, 31(5) (2010) 973-979.

[19] Kang, Y., Yang, C. and Huang, X., Electroosmotic flow in a capillary annulus with high zeta potentials. Journal of Colloid and Interface Science, 253(2) (2002) 285-294.

[20] Chakraborty, R., Dey, R. and Chakraborty, S., Thermal characteristics of electromagnetohydrodynamic flows in narrow channels with viscous dissipation and Joule heating under constant wall heat flux. International Journal of Heat and Mass Transfer, 67 (2013) 1151-1162.

[21] Jian, Y., Si, D., Chang, L. and Liu, Q., Transient rotating electromagnetohydrodynamic micropumps between two infinite microparallel plates. Chemical Engineering Science, 134 (2015) 12-22.

[22] Sinha, A. and Shit, G.C., Electromagnetohydrodynamic flow of blood and heat transfer in a capillary with thermal radiation. Journal of Magnetism and Magnetic Materials, 378 (2015) 143-151.

[23] Buren, M., Jian, Y., Chang, L., Li, F. and Liu, Q., Combined electromagnetohydrodynamic flow in a microparallel channel with slightly corrugated walls. Fluid Dynamics Research, 49(2) (2017) 25517.

[24] Escandón, J.P., Bautista, O.E., Santiago, F. and Méndez, F., Asymptotic analysis of non-Newtonian fluid flow in a microchannel under a combination of EO and MHD micropumps. Defect and Diffusion Forum, 348 (2014) 147-152.

[25] Chakraborty, S. and Paul, D., Microchannel flow control through a combined electromagnetohydrodynamic transport. Journal of Physics D: Applied Physics, 39(24) (2006) 5364.

[26] Khuzhayorov, B., Auriault, J.L. and Royer, P., Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media. International Journal of Engineering Science, 38(5) (2000) 487-504.

[27] Escandón, J., Santiago, F., Bautista, O. and Méndez, F., Hydrodynamics and thermal analysis of a mixed electromagnetohydrodynamic-pressure driven flow for Phan–Thien–Tanner fluids in a microchannel. International Journal of Thermal Sciences, 86 (2004) 246-257.

[28] Hayat, T., Khan, M. and Asghar, S., On the MHD flow of fractional generalized Burgers’ fluid with modified Darcy’s law. Acta Mechanica Sinica, 23(3) (2007) 257-261.

[29] Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2) (2015) 1-13.

[30] Jian, Y., Transient MHD heat transfer and entropy generation in a microparallel channel combined with pressure and electroosmotic effects. International Journal of Heat and Mass Transfer, 89 (2015) 193-205.

[31] Maynes, D. and Webb, B.W., The effect of viscous dissipation in thermally fully-developed electro-osmotic heat transfer in microchannels. International Journal of Heat and Mass Transfer, 47(5) (2004) 987-999.

[32] Abdulhameed, M., Vieru, D. and Roslan, R., Modeling electro-magneto-hydrodynamic thermo-fluidic transport of biofluids with new trend of fractional derivative without singular kernel. Physica A: Statistical Mechanics and its Applications, 484 (2017) 233-252.

[33] Bhatti, M.M., Zeeshan, A., Ijaz, N., Bég, O.A. and Kadir, A., Mathematical modelling of nonlinear thermal radiation effects on EMHD peristaltic pumping of viscoelastic dusty fluid through a porous medium duct. Engineering Science and Technology, An International Journal, 20(3) (2017) 1129-1139.