Journal of Non-Newtonian Fluid Mechanics
The Immersed Boundary (IB) method has been widely used to solve fluid-structure interaction problems, including those where the structure interacts with polymeric fluids. In this paper, we examine the convergence of one such scheme for a well known two-dimensional benchmark flow for the Oldroyd-B constitutive model, and we show that the traditional IB-based scheme fails to adequately capture the polymeric stress near to embedded boundaries. We analyze the reason for such failure, and we argue that this feature is not specific to the case study chosen, but a general feature of such methods due to lack of convergence in velocity gradients near interfaces. In order to remedy this problem, we build a different scheme for the Oldroyd-B system using the Immersed Boundary Smooth Extension (IBSE) scheme, which provides convergent viscous stresses near boundaries. We show that this modified scheme produces convergent polymeric stresses through the whole domain, including on embedded boundaries, and produces solutions in good agreement with known benchmarks.
Complex fluids, Complex geometry, High-order, Immersed boundary, Oldroyd-B, Partial differential equations
Stein, David B.; Guy, Robert D.; and Thomases, Becca, "Convergent Solutions of Stokes Oldroyd-B Boundary Value Problems using the Immersed Boundary Smooth Extension (IBSE) Method" (2019). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.