Immersed Interface Methods for Incompresssible
Navier-Stokes Equations
Randall J. LeVeque
University of Washington
Consider an incompressible fluid in which a thin elastic membrane is
immersed. The motion of the membrane is then coupled with the motion of
the fluid and it is necessary to solve the incompressible Navier-Stokes
equations with a singular forcing term along the membrane. This must be
coupled with boundary conditions requiring that the elastic and fluid
forces balance across membrane, and that the membrane moves with the local
fluid velocity. Peskin's immersed boundary method is a popular approach
to solving such problems, especially in biofluid dynamics applications.
Rather than attempting to follow the deforming geometry with a moving
grid, the fluid dynamics equations are solved on a uniform Cartesian
grid, with a forcing term given by spreading the singular force to the
grid using discrete delta functions of finite width. This method is
robust but typically only first order accurate due to this spreading.
I will discuss an approach to achieving better accuracy by using the
elastic force to impose jump conditions directly on the pressure in the
process of solving the elliptic equation that arises in a projection
method for the Navier-Stokes equations. Membranes with mass can also
be handled, in which case there is an additional inertial force that
must be included in the force balance.