Flow-through problem for a Jeffreys type viscoelastic fluid

We investigate a nonlinear boundary value problem describing the steady-state flow of an incompressible viscoelastic fluid of Jeffreys type through a 3D (or 2D) bounded domain. On the flow domain boundary, inhomogeneous Dirichlet conditions are stated for the flow velocity. Both the weak and strong formulations of this problem are considered. Using the vanishing artificial viscosity method and an existence result from the Leray–Schauder theory of topological degree, we construct a weak solution. Notably, the existence theorem, proved in the framework of the weak formulation, does not necessitate smallness assumptions on forcing and boundary data. It is established that the weak solutions set is sequentially weakly closed. We also obtain sufficient conditions for the existence and coincidence of strong solutions. Moreover, we analyzed the convergence of the constructed solutions when the stress relaxation and retardation times simultaneously tend to zero.