The immersed boundary method for simulating gravitational settling and fluid shear-induced deformation of elastic structures

In this work, we study several fluid-structure interaction problems in two dimensions using the "Immersed Boundary (IB) method". The IB method is a versatile and robust approach for simulating the interaction of complex, elastic structures with an incompressible fluid flow. The method has been used to simulate a wide range of biological and non-biological fluid-structure interaction problems. The particular choice of problems we study is motivated by the dynamics of biofilms which are characterized by flow-induced deformation of complex structures as well as free flowing suspended biofilm particles. In this work we consider massless as well as massive structures. For the massive structures, mass is incorporated by distributing point masses at a collection of IB points on and inside the structure. Our mathematical model for massive IB structures uses a Boussinesq approximation which is valid for $\Delta\rho \leq \rho_f$, where $\rho_f$ is the density of the fluid and $\Delta\rho$ is the density difference between the structure and fluid. We consider gravitational settling of massive particles and aim to capture hydrodynamic interactions with bounding walls as well as other particles. In all cases, we perform extensive numerical simulations and comparisons to other published experimental and numerical results in the literature. Then we consider the deflection of a flexible cantilever beam in response to a shear flow. Our numerical results are discussed in the context of analytical results from the linear beam theory. This model is extended to handle porosity in the solid, which is a characteristic feature of biofilm layers.