Algorithm Development and Analysis for Non-Newtonian Fluids Interacting with Elastic and Poroelastic Structures

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This research concerns the development and rigorous analysis of stable and efficient numerical schemes for non-Newtonian fluid - structure interactions (FSI). The study will be focused on partitioning schemes of an implicit type for the coupled model systems that allow each subproblem to be solved independently using existing local solvers in a fixed/moving domain setting. Stability and accuracy properties of numerical methods will be investigated using non-Newtonian fluid and poroelastic/elastic structure models. The theoretical investigation will provide a solid foundation and guidance to the further development of numerical algorithms. Because of the many important biological and engineering processes involving non-Newtonian fluid flows, there is a great demand for mathematical support in these applications. This research will provide an underlying mathematical foundation for non-Newtonian flows in a multiphysical setting. The technical goal of this project is to develop algorithms and analyze numerical schemes for two coupled systems: (a) quasi-Newtoinan fluid - poroelastic structure and (b) viscoelastic fluid - elastic structure. For system (a) the PI will focus on the development and analysis of a nonlinear operator, where a solution to the operator equation yields subsystem solutions satisfying interface conditions of the whole coupled system. Advantages of this approach over the previously developed optimization approach are the flexibility to choose a nonlinear solver and that no extra coding effort is needed as the adjoint system is no longer involved in a solution process. Since it is expected that the linearized operator is not self-adjoint, the linear operator equation should be solved by an iterative solver for a non-self-adjoint problem. When a partitioned scheme is considered for simulating viscoelastic FSI, extra difficulty is encountered due to the lack of information on the stress along the moving boundary and movement of inlet and outlet boundaries along the interface of two subsystems. The PI will investigate various choices for a stress boundary value and their effect on a solution of FSI. Another issue with the viscoelastic FSI is the size of the fluid problem to be solved in each time step (or in each internal iteration), which may require an operator splitting based on a temporal discretization scheme such as a fractional time step method. To tackle this, the PI will investigate various time stepping schemes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.