A solution is presented to the boundary value problem for the transient development of the velocity profile in a viscoelastic fluid subjected to the sudden onset of shearing in an idealized viscometer. A differential Oldroyd type of constitutive equation is used to describe the mechanical behavior of the fluid. The results indicate that the elastic nature of the fluid significantly increases the length of time required for the steady‐state velocity profile to be established. A solution is also presented for the drainage of a viscoelastic fluid off a vertical plate. The predicted initial elastic overshoot or rapid drainage of the fluid is in qualitative agreement with reported experimental results.