Strand II Mini-Course: Immersed Boundary Problems
(January 12 to February 15)
Instructor
Anita Layton
Description
When unsteady fluid flows interact with dynamic boundaries, the
resulting fluid dynamics defy easy representation and conventional
analysis. This class of problems spans many disciplines, as the short
list below should give a sense of its vast scope:
-
all of biofluid dynamics, from flow in a beating heart to
interactions of molecular motors with intracellular water;
-
many areas of chemical engineering, e.g., polymeric suspensions;
-
the design of sails and parachutes.
Traditionally, boundary fitted grids have been used to model complex
geometries. However, in addition to challenges arising in grid generation,
this approach prevents the use of simple orthogonal grids, thereby
foregoing the numerical efficiency and accuracy associated with simple
orthogonal grids.
To study flow patterns around heart valves, Charles Peskin (1972)
introduced the immersed boundary method, which models complex geometries
by representing the necessary shape through forces within the grid. In
this mini-course, we will first study the immersed boundary method as a
_mathematical formulation_, which employs a mixture of Eulerian and
Lagrangian variables. These variables are related by interaction
equations in which the Dirac delta function plays an essential role. We
will then examine the computational challenges presented by the
immersed boundary method as a _numerical scheme_. These challenges
include the tight-coupling between boundary and fluid, the differing
and interacting length and time scales, and the complexity of unsteady
motions.
Prerequisites
This course is open to everyone. I will assume a certain level of
comfort with basic fluid mechanics. (Only basic knowledge is necessary,
e.g., you should know what is means for a fluid to be incompressible, or
at least be able to quickly look it up.) I will also assume familiarity
with numerical ODE methods. Numerical PDE methods will be needed in a
small part of the course, but I will make that self-contained and will
give a brief introduction.
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Duke University
Last modified October 29, 2004