Viscoelastic flow in a collapsible channel

A numerical method based on a fluid-structure interaction formulation is used to understand the role of viscoelasticity on flow in a two-dimensional collapsible channel. Three different viscoelastic fluid models have been considered - the Oldroyd-B, the FENE-P and Owens model for blood [R. G. Owens, A new microstructure-based constitutive model for human blood, J. Non-Newtonian Fluid Mech. 140 (2006), 57-70]. Initially the collapsible wall is considered as a zero thickness membrane model. Subsequently the collapsible wall is modelled as an incompressible neo-Hookean solid. Experiments in micro collapsible channels have also been performed. At present, there are no models in the literature thatsimultaneously account for the elastic nature of the collapsible wall and the non-Newtonian rheology of the flowing fluid. In this study, for the first time, a viscoelastic fluid-structure interaction model has been developed that accounts for a viscoelastic fluid and a finite thickness elastic wall, and the resulting governing equations are solved with a sophisticated finite element method. The rheological behaviour of the viscoelastic fluids is described in terms of a conformation tensor model. The mesh equation and transport equations are discretized by using the DEVSS-TG/SUPG mixed finite element method. The computational method developed in this work is validated by comparing with the available analytical and numerical results. While considering viscoelastic flow in a two-dimensional collapsible channel with a zero-thickness membrane, a distinct difference has been observed in the collapse wall profile for the Oldroyd-B, FENE-P and Owens model as compared to a Newtonian fluid at low values of membrane tension. The shape change of the collapsible wall depends on the Weissenberg number (Wi) for the Oldroyd-B and FENE-P fluids. The shape change in Owens model is essentially due to its shear thinning property. There is a limiting Weissenberg number beyond which computations fail, which increases with mesh refinement and decreases with decrease in membrane tension. One of the major outcomes of the zero-thickness membrane model study is that the significant differences that arise amongst the different viscoelastic fluids in the predicted value of the tangential shear stress on the membrane surface, has no influence on the shape of the deformable membrane, because of the boundary condition adopted in the model. Essentially it is assumed that the shape of the membrane is governed only by the normal stresses acting on it. In order to use a more realistic model for the collapsible wall, the zero-thickness membrane model has been replaced by a deformable finite thickness elastic solid which accounts for the effect of shear stress on membrane shape. The limiting Weissenberg number beyond which computations fail to converge is found to be sensitive to the choice of viscoelastic model and depends on a dimensionless solid elasticity parameter. The shape of the fluid-solid interface and the stress and velocity fields in the channel, for the three viscoelastic fluids, are compared with predictions for a Newtonian fluid, and the observed differences are related to individual fluid rheological behaviour. Predictions with a finite thickness elastic solid model for the deformable wall differ considerably from those in which it is modelled as a zero-thickness membrane. Experiments have been carried out in a micro-collapsible channel made of polydimethysiloxane (PDMS) that mimics the numerically simulated geometry. The experiments show that the channel width perpendicular to the flow must be significant in order for wall effects to be negligible (an assumption that is made in the 2D simulation). As a consequence, the commercial software ANSYS has been used to develop a full 3D model of the channel which captures the deformation of the flexible membrane in the absence of flow. The elastic properties of PDMS have been extracted by comparing the load-displacement curves obtained from the FEM simulations with the experiments. Preliminary comparison has been made between simulations and experiments for the flow of a Newtonian fluid in the micro-collapsible channel