Mathematical and physical requirements for successful computations with viscoelastic fluid models

For general applicability in arbitrary geometries, viscoelastic fluid modelsmust have regular viscometric functions. Classical quasi-linear Maxwell andJeffreys (or Oldroyd) models do not satisfy this requirement. Their plane-elongational viscosity is infinite for a finite value of the elongation. Therecently developed non-linear Leonov and Giesekus models have regularviscometric functions. The type and boundary conditions of the system ofequations are analyzed. M-type models (retardation time equal to zero, orextra viscosity is zero) have characteristics that are not the same at those ofthe equations of motion and the constitutive equations separately. A simplified 2-D problem is analysed, and it appears that one stress componentsatisfies an elliptic equation and that it is not allowe! d to specify this stresscomponent independently at the inflow boundary. J-type models (extra viscosity islarger than zero) have characteristics, which are the same for the total system and for thedecomposed system and there is no problem with boundary conditions.A few computational results are presented based on an algorithm thatsolves the equations of motion with an estimate of the stress tensor, and(re-)computes the stresses by integration along the characteristics. For anarbitrary, given velocity field the stability of integration of the stresses alongthe characteristics (the particle trajectories) is studied. This shows theinferiority of the quasi-linear models compared with the non-linear ones.