The unsteady mixed convection flow within a vertical concentric cylindrical annulus with time-dependent moving walls is investigated. Fluid is suctioned/injected through the cylinder walls of the annulus. The effects of wall movement on flow and heat transfer characteristics inside the cylindrical annulus are sought. An exact solution of Navier-Stokes and energy equations is obtained in this problem, for the fi rst time. Here, the heat transfer occurs from the walls of the hot cylinder with constant temperature to the moving fluid with a cooling role. It is interesting to note that the results indicate that the time-dependency of cylinder wall movements has no effect on the temperature pro le. The results also indicate that usage of an inverse of time for velocity movement causes a signifi cant increase in velocity, stress tensor, and Nusselt number in the vicinity of the moving wall. Also, increasing the mixed convection and suction/injection parameters increases the Nusselt number and decreases the stress tensor on the inner and outer cylinders, respectively; no matter what velocity function is chosen for the moving wall. Therefore, using the velocity function of the wall movement and the change in other nondimensional governing parameters, flow and heat transfer can be controlled.