On the elongational viscosity of viscoelastic slip flows in hyperbolic confined geometries

We study theoretically the elongational viscosity (or Trouton ratio, in dimensionless form) for steady viscoelastic flows in confined and symmetric hyperbolic tubes considering Navier-type slip along the wall(s). Both the planar and the cylindrical axisymmetric geometrical configurations are addressed. Under the classic lubrication approximation, and for a variety of constitutive models such as Phan-Thien and Tanner, Giesekus, and Finite Extensibility Nonlinear Elastic with the Peterlin approximation models, the same general analytical formula for the Trouton ratio is derived as for the Oldroyd-B model, in terms of the velocity at the midplane/axis of symmetry and the Deborah number only. Assuming that the velocity field is approximated by the Newtonian lubrication profile, based on our previous study in the absence of slip, we show that a constant extensional strain rate can be achieved in the limits of zero or infinite slip. For finite slip, a slight modification of the geometry is required to achieve a constant strain rate. In these cases, the formula for the steady state Trouton ratio reduces to that for transient homogeneous elongation. We also provide analytical formulae for the modification (decrease) for both the extensional strain rate and the Hencky strain achieved in the confined geometries because of introducing wall slip.