Computer Algebra Recipes

5.2. DIFFUSION AND LAPLACE'S EQUATION MODELS 241
kept at T =0andT = 1, respectively, while the circular surface is kept at
T = 1. Determine the steady-state temperature pro¯le inside the cylinder and
plot the contours of equal temperature.
Problem 5-27: Elliptic cross section
An in¯nitely long bar of elliptic cross section has its curved surface, x2 +4 y2 =1,
kept at T =0 ± . Determine the temporal evolution of T in the cylinder if initially
T =100 ± everywhere inside and the di®usion constant d=1. Animate the result.
5.2.5 Hugo Prepares for His Job Interview
I'm notorious for giving a bad interview. I'm an actor and I can't
help but feel I'm boring when I'm on as myself.
Rock Hudson, American movie actor (1925{1985)
Hugo has been invited by International Hydrodynamics Inc. to interview for a
research position with their company, which specializes in designing streamlined
hulls for surface vessels as well as underwater craft. As part of his preparation,
Hugo decides to create some ¯les for the interview that demonstrate his skills
at simulating liquid °ow around a variety of rigid geometrical shapes. As a
\warm-up" exercise, he recalls from his undergraduate days a problem involving
uniform incompressible °uid °ow around a sphere, a problem that can be
solved analytically. Hugo remembers that the °uid can be characterized by a
velocity potential U that satis¯es Laplace's equation. The velocity ~ V of a °uid
element is then given by ~ V = ¡rU.
Since a spherical boundary of, say, radius a is involved, Hugo chooses to use
spherical coordinates (r; μ; Á) with the origin at the center of the sphere and the
physicist's convention that μ is measured from the positive z-axis and Á from
the positive x-axis. Then, the Cartesian and spherical polar coordinates are
connected by the relations x = r sin μ cos Á, y = r sin μ sin Á, andz = r cos μ.
The range of r is from 0 to 1, μ from 0 to ¼, andÁfrom0to2¼. At a distance r far from the sphere, the °uid °ow is assumed to be uniform
and directed along the z-axis.Theasymptotic(larger) form of the velocity is
~V = V0 ^ez where V0 is the undisturbed °uid speed. Thus, since
~V = ¡ @U
@x ^ex ¡ @U
@y ^ey ¡ @U
@z ^ez = V0 ^ez; (5.11)
then, on equating vector components and integrating, the asymptotic velocity
potential is given (to within an arbitrary constant) by U = ¡V0 z = ¡V0 r cos μ.
This will serve as one of the boundary conditions on the solution. The other
boundary condition is at the surface of the sphere. If the surface is idealized to
be absolutely rigid, it will acquire no momentum from the °uid. This implies
that the normal component of the °uid velocity vector must vanish at r = a.
Writing the gradient operator in spherical polar coordinates, this means that
the velocity potential must satisfy the boundary condition @U=@rjr=a =0.