<p>This paper is devoted to the investigation of weak solutions for initial-boundary value problems describing alpha-models of viscoelastic fluid motion with memory. The memory effect is incorporated along the trajectories of fluid particles, which are determined by the velocity field. Since the velocity field lacks the smoothness required to uniquely determine trajectories for every initial condition, we formulate the problems in terms of regular Lagrangian flows. Employing a topological approximation method, we prove the existence of weak solutions for these alpha-models. Furthermore, we establish the convergence of the solutions of an alpha-model to the solution of the classical model in the limit as the parameter alpha tends to zero.</p>