  The steady flow of three viscoelastic fluids (Oldroyd-B, FENE-P, and Owens
model for blood) in a two-dimensional channel, partly bound by a deformable,
finite thickness neo-Hookean solid, is computed. The limiting Weissenberg
number beyond which computations fail to converge is found to increase with
increasing dimensionless solid elasticity parameter {\Gamma}, following the
trend Owens > FENE- P > Oldroyd-B. The highly shear thinning nature of Owens
model leads to the elastic solid always collapsing into the channel, for the
wide range of values of {\Gamma} considered here. In the case of the FENE-P and
Oldroyd-B models, however, the fluid-solid interface can be either within the
channel, or bulge outwards, depending on the value of {\Gamma}. This behaviour
differs considerably from predictions of earlier models that treat the
deformable solid as a zero-thickness membrane, in which case the membrane
always lies within the channel. The capacity of the solid wall to support both
pressure and shear stress, in contrast to the zero-thickness membrane that only
responds to pressure, is responsible for the observed difference. Compar- ison
of the stress and velocity fields in the channel for the three viscoelastic
fluids, with the predictions for a Newtonian fluid, reveals that shear thinning
rather than elasticity is the key source of the observed differences in
behaviour.
