27th April 10:13
Conformal Map in Three Dimensions
I am dealing with a system consisting of a narrow strip of but
infinitely long strip embedded in a infinite coplanar insulating plane.
The strip is hold at a temperature lower than the enviroment and I am
calculating the time dependend heat transfer by solving the two (space)
dimensional heat transfer equation of the form
dT/dt = d^2T/dx^2 + d^2/dz^2
where z is the coordinate perpendicular to the strip and x is the
coordinate along the width
I am mapping the Cartesian coordinates onto a conformal space (H,G) by
applying the transformation equations
x=cos(G)*cosh(H) and z=sin(G)*sinh(H)
Finally I am using F=H/(1-H) and the infinite region in (X,Y) space is
mapped onto a closed region 0<=F<=1 and 0<=G<=pi/2.
Next I would like solve the problem for strip of finite length, embedded
in an infinite insulating plane. In that case I have to consider heat
transfer in three space dimensions X, Y and Z. I am looking for a
conformal map similar to the one above, to map the coordinates x, y, z
onto a closed region of a cube or cuboid with coordinates (G, D, F), so
that 0<=F<=1, 0<=G<=pi/2, 0<=D<=pi/2
Any help is much appreciated.
All the best,