The present work studies the unsteady, viscous, and incompressible laminar flow and heat transfer over a shrinking permeable cylinder. The unsteady nonlinear Navier–Stokes and energy equations are reduced, using similarity transformations, to a system of nonlinear ordinary differential equations. The boundary conditions associated with the governing equations are the time dependent surface temperature and flow conditions. The method of solution is based on a combination of the implicit Runge–Kutta method and the shooting method. The present study predicts two solutions for both the flow and heat transfer fields, and a unique solution at a specific critical unsteadiness parameter. An analysis of the results, for a specific suction parameter, suggests that the corresponding unique unsteadiness parameter does not depend on the Prandtl number. However, the unique rate of heat transfer is increasing as the Prandtl number increases. In addition, our results confirm that the unique value of heat transfer rate increases as the suction parameter increases, regardless the value of the Prandtl number.