In this article, the rectilinear pipe flow of a polymeric liquid under a time‐dependent pressure drop is considered. The fluctuating component of the pressure drop is assumed to be of small amplitude and can adequately be represented by a weakly stationary stochastic process. Perturbation solutions are developed for two integral constitutive equations; both have strain‐dependent memory kernels. The first is the recently proposed nonaffine network model and the second is a simplified version of the B‐KBZ constitutive equation. It is shown that both models predict a positive flow rate enhancement. This flow enhancement is a decreasing function of the frequency of the fluctuating pressure drop when the latter is represented by a sinusoidal function of time. Based on the limited data published by Barnes, Townsend, and Walters, it is concluded that both models considered here cannot describe the frequency dependence of the flow enhancement. In this regard, it would appear that an integral model with a strain‐rate dependence memory kernel is best suited for this flow field despite a number of theoretical defects that can be leveled against it.