Thesis (Ph.D. (Applied Mathematics)) -- University of Limpopo, 2020 / The flow of electrically conducting fluids in the presence of a magnetic field has wide applications in science, engineering and technology. Examples of the applications include industrial processes such as the cooling of reactors, extrusion of plastics, purification of crude oil, medical applications, aerodynamics and many more. The induced magnetic field usually act as a flow control mechanism, especially under intense heat. In this study a couple stress fluid in a channel will be used as the working fluid. Channel flow and heat transfer characteristics of couple stress fluids find applications in processes such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plates in a bath, tribology of thrust bearings and lubrication of engine rod bearings. One major characteristic that distinguishes the couple stress fluid from other non-Newtonian fluids is the inclusion of size-dependent microstructure that is of mechanical significance. As such, the couple stress constitutive model is capable of describing the couple stresses, the effect of body couples and the nonsymmetric tensors manifested in several real fluids of technological importance. A fully developed laminar magnetohydrodynamic (MHD) flow of an incompressible couple stress fluid through a vertical channel due to a steady-periodic temperature on the channel plates is investigated. Specifically, the effects of couple stresses and internal heat generation on MHD natural convection flow with steady-periodic heat input, the impact of magnetic field induction on the buoyancy-induced oscillatory flow of couple stress fluid with varying heating and a mixed convective two dimensional flow of unsteady MHD couple stress fluid through a channel field with porous medium are studied. Analytical methods and the semi-analytic Adomian decomposition method will be used to solve the resulting non-linear differential equations governing the flow systems. Useful results for velocity, temperature, skin friction and Nusselt number are obtained and discussed quantitatively. The effects of the various flow governing parameters on the flow field are investigated.