Finite-element approximation of viscoelastic fluid flow with slip boundary condition

We present a theoretical study of a creeping, steady-state, isothermal flow of a viscoelastic fluid obeying an Oldroyd-type constitutive law with slip boundary condition. The slip boundary condition is appropriate for problems that involve free boundaries and other examples where the usual no-slip condition u = 0 is not valid, such as fiber spinning and microfluidics. First, we study the Newtonian problem with slip boundary condition where the viscoelastic stress is added into the list of unknowns. In addition, the normal viscoelastic stress component associated with the slip boundary condition is introduced. In order to balance its effects, a second inf-sup condition is proven. To treat the discrete case, we assume that the continuous solution of the non-Newtonian problem exists and is small and smooth. Approximating the extra stress, velocity, pressure, and normal viscoelastic stress component via P1 discontinuous, P2 continuous, P1 continuous, and P0 discontinuous elements, respectively, yields a stable finite-element scheme. Finally, via a fixed point argument, we establish the existence of an approximate solution and derive error estimates.