Narrow Results

We propose a new reduced model for gravity-driven free-surface flows of shallow viscoelastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for viscoelastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. In addition to the classical layer depth and velocity in shallow models, our system describes also the evolution of two components of the stress. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver.

Steady flows of slightly compressible viscoelastic fluids of Oldroyd's type with zero boundary conditions are considered on a bounded two-dimensional domain with an isolated corner point. We prove the existence and the uniqueness of the solution for small data in weighted Sobolev spaces , where the index ξ characterizes the power growth of the solution near the angular point. The proof follows from an analysis of a linearized problem through the fixed point theory. We use a method of decomposition for such linearized equations: the velocity field u is split into a non-homogeneous incompressible part v and a compressible part ∇_φ.

The authors consider here some Oldroyd models of non-Newtonian flows consisting of a strong coupling between incompressible Navier-Stokes equations and some transport equations. It is proved that there exist global weak solutions for general initial conditions. The existence proof relies upon showing the propagation in time of the compactness of solutions.