This work investigates the flow of a third grade fluid in a rotating frame of reference. The fluid is incompressible and magnetohydrodynamic (MHD). The flow is bounded between two porous plates, the lower of which is shrinking linearly. Mathematical modelling of the considered flow leads to a nonlinear problem. The solution of this nonlinear problem is computed by the homotopy analysis method (HAM). Graphs are presented to demonstrate the effect of several emerging parameters, which clearly describe the flow characteristics.