Summary: We formulate the problem under an infinitely long wavelength approximation, a negligible Reynolds number and a small magnetic Reynolds number. We decide on a perturbation method of solution. The viscosity parameter is chosen as a perturbation parameter. The governing equations are developed up to first-order in the viscosity parameter . The zero-order system yields the classical Poiseuille flow when the Hartmann number tends to zero.
For the first-order system, we simplify a complicated group of products of Bessel functions by approximating the polynomials. The results show that the increasing magnetic field increases the pressure rise. In addition, the pressure rise increases as the viscosity parameter decreases at zero flow rate. Moreover, it is independent of the Hartmann number and viscosity parameter at certain values of the flow rate. We make comparisons with other studies.