Unsteady laminar hydromagnetic fluid–particle flow and heat transfer in channels and circular pipes

The problem of unsteady laminar flow and heat transfer of a particulate suspension in an electrically conducting fluid through channels and circular pipes in the presence of a uniform transverse magnetic field is formulated using a two-phase continuum model. Two different applied pressure gradient (oscillating and ramp) cases are considered. The general governing equations of motions (which include such effects as particulate phase stresses, magnetic force, and finite particle-phase volume fraction) are non-dimensionalized and solved in closed form in terms of Fourier cosine and Bessel functions and the energy equations for both phases are solved numerically since they are non-linear and are difficult to solve analytically. Numerical solutions based on the finite-difference methodology are obtained and graphical results for the fluid-phase volumetric flow rate, the particle-phase volumetric flow rate, the fluid-phase skin-friction coefficient and the particle-phase skin-friction coefficient as well as the wall heat transfer for plane and axisymmetric flows are presented and discussed. In addition, these numerical results are validated by favorable comparisons with the closed-form solutions. A comprehensive parametric study is performed to show the effects of the Hartmann magnetic number, the particle loading, the viscosity ratio, and the temperature inverse Stokes number on the solutions.