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The flow of an elasticoviscous fluid between a rotating and a stationary disk with uniform suction at the stationary disk is studied. The equations of motion are solved by a regular perturbation method for small suction Reynolds number and by the Runge‐Kutta numerical integration method. The results from the two methods have been compared. The effects of the elasticity of the fluid and the suction parameter on the velocity components, torque on the disks, and the pressure coefficient have been studied.