The goal of this paper is to examine the evaluation of
interfacial stresses using a standard, finite difference based,
immersed boundary method (IMBM). This
calculation is not trivial for two fundamental reasons. First, the
immersed boundary is represented by a localized boundary force which is
distributed to the underlying fluid grid by a discretized delta
function. Second, this discretized delta function is used to
spatially averaged no-slip condition at the immersed boundary.
approximations can cause errors in interpolating stresses near the
To identify suitable methods for evaluating stresses, we investigate
three model flow problems at very low Reynolds numbers. We
compare the results of the immersed boundary calculations to those
achieved by the
boundary element method (BEM).
The stress on an immersed boundary may be calculated
either by direct evaluation of the fluid stress (FS) tensor or, for
the stress jump, by direct evaluation of the locally distributed
boundary force (wall stress or WS). Our first model problem is Poiseuille
channel flow. Using an analytical solution of
the immersed boundary formulation in this simple case,
we demonstrate that FS calculations should be evaluated at a
distance of approximately one grid spacing inward from the immersed
For a curved immersed boundary we present a procedure for selecting
representative interfacial fluid stresses using the concepts from
the Poiseuille flow test problem. For the final two model problems, steady state flow over a bump
in a channel and unsteady peristaltic pumping, we present an
'exclusion filtering' technique for accurately measuring stresses. Using
this technique, these studies show that the immersed boundary method
can provide reliable approximations to interfacial stresses.
Mathematics Subject Classification: Primary: 65M06; Secondary: 76Z05.