keywords: Third grade fluid, Joule heating, Magnetohydrodynamics, adomian decomposition method
We investigated the flow of a third grade fluid through a cylindrical pipe in the presence of a magnetic field and joule heating, with the aim of finding approximate analytic solutions to the dimensionless velocity and temperature of the fluid. The Adomian decomposition method was applied to the dimensionless momentum and energy equations for the Reynolds’ viscosity model case. In the absence of magnetic effect and joule heating we found a difference of at most 〖10〗^(-1) between the adomian decomposition solution and the perturbation solution of Jayeoba and Okoya. Graphs depicting the velocity and temperature distributions for various values of the thermo-physical parameters were plotted and analyzed, and it is observed that the magnetic field parameter decreases the velocity of the fluid and increases the temperature while the joule heating parameter reverses the effect of the heat generation parameter.
Adomian GA 1992. A review of the decomposition method and some results for non-linear equation. Maths. & Computer Modelling, 13(7): 17-43. Adomian GA 1994. Solving Frontier Problems of Physics: The Decomposition Method, Volume 60, Dordrecht, Kluwer Academic Publishers, p. 370. Aiyesimi YM, Okedayo GT & Lawal OW2012a. MHD flow of a third grade fluid with heat transfer and slip boundary condition down an inclined plane. Maths. Theory & Modeling, 2(9): 108-120. Aiyesimi YM, Abah SO & Okedayo GT 2012b. Radiative Effects on the Unsteady Double Diffusive MHD Boundary Layer Flow over a Stretching Vertical Plate. Am. J. Scient. Res., 65: 51-61. Alfvn H 1942. Existence of Electromagnetic-hydrodynamic Wave. Nature, 150: 405-406. Branover H & Gershon P 1976. MHD Turbulence Study. Report BGUN-RDA-100-176. Ben-Gurion University. De T, Costa S &Sandberg D 2004. Mathematical model of a smoldering log. Combustion and Flame, 139: 227-238. Gbadeyan JA, Idowu AS, Okedayo GT, Ahmed LO & Lawal OW 2014. Effect of suction on thin film flow of a third grade fluid in a porous medium down an inclined plane with heat transfer. Int. J. Scient. & Engr. Res., 5(4): 748-754. Hartmann JI 1937. Hydrodynamics, theory of laminar flow of an electrically conducting liquid in a homogeneous magnetic field. Int. Kongelige Danske Videnskabernes Selskab Matematisk-Fysiske Meddelelser, 15: 1-27. Hayat T, Shafiq A & Alsaedi A 2014. Effect of joule heating and thermal radiation in flow of third grade fluid over radiative surface. Public Library of Sci. (PLoS One), 9(1): e83153. Holroyd RJ 1979. An experimental study of the effect of wall conductivity, non-uniform magnetic field and variable-area ducts on liquid metal flow at high hartmann number, Part 1: Ducts with non-conducting wall. J. Fluid Mechanics, 93: 609-630. Holroyd RJ 1980. MHD flow in a rectangular duct with pairs of conducting and non-conducting walls in the presence of a non-uniform magnetic field. J. Fluid Mechanics, 96: 335-3. Jayeoba OJ & Okoya SS 2012. Approximate analytical solutions for pipe flow of a third grade fluid with variable models of viscosities and heat generation/absorption. J. Nig. Maths. Soc., 31: 207-227. Makinde OD 2009. Hermite- pade approach to thermal radiation effect on inherent irreversibility in a variable viscosity channel flow. Computers & Maths. Applic., 58: 2330-2338. Massoudi M & Christe I 1995. Effect of variable viscosity and viscous dissipation on the flow of third grade fluid in a pipe. Int. J. Non-linear Mechanics, 30(5): 687-699. Okoya SS 2011. Disappearance of criticality for reactive third grade fluid with reynold’s model viscosity in a flat channel. Int. J. Non-linear Mechanics, 46(9): 1110-11. Olajuwon BI 2009. Flow and natural convection heat transfer in a power law fluid past a vertical plate with heat generation. Int. J. Non-linear Sci., 7(1): 50-56. Pakdemirli M & Yilbas BS 2006. Entropy generation for pipe flow of a third grade fluid with Vogel model viscosity. Int. J. Non-linear Mechanics, 41(3): 432-437. Raftari B, Parvaneh F & Vajravelu K 2013. Homotopy analysis of the magnetohydrodynamic flow and heat transfer of a second grade fluid in a porous channel. Energy, 59: 625-632. Siddiqui AM, Hameed M, Siddiqui BM &Ghori QK 2005. Use of adomoian decomposition method in the study of parrallel plate flow of a third grade fluid. Int. J. Non-linear Mechanics, 40: 807-820. Smith P 1971. Some asymtoticextremum principle for magnetohydrodynamic pipe flow. Appl. Sci. Resou., 24: 452-466. Yurusoy M & Pakdemirli M 2002. Approximate analytical solution for the flow of a third grade fluid in a pipe. Int. J. Non-linear Mechanics, 37(2): 187-195.