In the present investigation, a streamfunction-vorticity form for Boussinesq-Stokes liquids (with suspended particles) is suitably usedto examine the problem of 2-D unsteady incompressible flow in a square cavity with moving top and bottom wall. A new algorithm isused for this form in order to compute the numerical solutions for high Reynolds numbers up to Re=2500. This algorithm is conductedas a combination of the multi-time-stepping temporal differential transform and the spatial finite difference methods. Convergenceof the time-series solution is ensured by multi-time-stepping method. The classical benchmark results of the Newtonian liquid arerecovered as a limiting case and the decelerating influence of the suspended particle on the Newtonian liquids’ flow field is clearlyelaborated.