MATH76109/21/2018

We consider a problem similar to that in Homework02, but now for a viscous fluid, and solve it using a finite volumes in Bearclaw and finite elements in FreeFEM.

Problem.

Wave scattering by an elastic sphere submerged in an incompressible, viscous fluid. Inside the sphere the elasticity equations are valid

${\displaystyle \rho {\partial }^2𝒖/\partial t^2=(\lambda +2\mu )\nabla (\nabla \cdot 𝒖)-\mu \nabla \times (\nabla \times 𝒖)}$

(1)

with same notation and values as in Homework02. The fluid motion is however now described by the Navier-Stokes equations

${\displaystyle {𝒖}_t+(𝒖\cdot \nabla )𝒖=-\nabla p+\nu {\nabla }^2𝒖.}$

Figure 1. Schematic of sphere scattering problem

${\displaystyle }$

1. Solve the coupled fluid-structure interaction problem using a projection method for the Navier-Stokes equation using finite volumes (Bearclaw).

2. As above, using finite elements (FreeFEM).