We present a numerical method for simulating 2D flow through a channel
with deformable walls. The fluid is assumed to be incompressible and
viscous. We consider the highly viscous regime, where fluid dynamics are
described by the Stokes equations, and the less viscous regime described
by the Navier-Stokes equations. The model is formulated as an immersed
boundary problem, with the channel defined by compliant walls that are
immersed in a larger computational fluid domain. The channel traverses
through the computational domain, and the walls do not form a closed
region. When the walls deviate from their equilibrium position, they
exert singular forces on the underlying fluid. We compute the numerical
solution to the model equations using the immersed interface method,
which preserves sharp jumps in the solution and its derivatives. The
immersed interface method typically requires a closed immersed
interface, a condition that is not met by the present configuration.
Thus, a contribution of the present work is the extension of the
immersed interface method to immersed boundary problems with open
interfaces. Numerical results indicate that this new method converges
with second-order accuracy in both space and time, and can sharply
capture discontinuities in the fluid solution.