On non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid

In this work we develop a mathematical model to predict the velocity profile for an unidirectional, incompressible and steady flow of an Oldroyd 6-constant fluid. The fluid is electrically conducting by a transverse magnetic field. The developed governing equation is non-linear. This equation is solved analytically to obtain the general solution. The governing non-linear equation is also solved numerically subject to appropriate boundary conditions (three cases of typical plane shearing flows) by an iterative technique with the finite-difference discretizations. A parametric study of the physical parameters involved in the problems such as the applied magnetic field and the material constants is conducted. The obtained results are illustrated graphically to show salient features of the solutions. Numerical results show that the applied magnetic field tends to reduce the flow velocity. Depending on the choice of the material parameters, the fluid exhibits shear-thickening or shear-thinning behaviours.