The problem of flow of an elasticoviscous fluid characterized by a four‐constant Oldroyd model is treated when such a fluid is confined between a pair of infinite coaxial parallel disks. The governing system of nonlinear coupled differential equations for the stresses and the velocity field is expressed by the corresponding finite difference analogs and solved by using the SOR method. For the case of one disk held at rest, the solutions are provided for Reynolds numbers 10 and 100, while for the case of counterrotations of the disks, they are also provided for the intermediate value of 20. The graphical representations of the velocity functions bring out some very interesting differences in the behavior of a Newtonian, slightly elastic and an elastic liquid.