Mathematical and physical requirements for successful computations with viscoelastic fluid models

Abstract

For general applicability in arbitrary geometries, viscoelastic fluid models must have regular viscometric functions. Classical quasi-linear Maxwell and Jeffreys (or Oldroyd) models do not satisfy this requirement. Their plane-elongation viscosity is infinite for a finite value of the elongation. The recently developed non-linear Leonov and Giesekus models have regular viscometric functions. The type and boundary conditions of the system of equations are analyzed. M-type models (retardation time equal to zero, or extra viscosity η s = 0) have characteristics that are not the same at those of the equations of motion and the constitutive equations separately. A simplified 2-D problem is analysed, and it appears that one stress component satisfies an elliptic equation and that it is not allowed to specify this stress component independently at the inflow boundary. J-type models (η s > 0) have characteristics, which are the same for the total system and for the decomposed system and there is no problem with boundary conditions. A few computational results are presented based on an algorithm that solves the equations of motion with an estimate of the stress tensor, and (re)-computes the stresses by integration along the characteristics. For an arbitrary, given velocity field the stability of integration of the stresses along the characteristics (the particle trajectories) is studied. This shows the inferiority of the quasi-linear models compared with the non-linear ones.