satisfying
The Hele-Shaw problem is: find satisfying
² =0 where
x
\
(t)
with
[]-+=-2V, =-
where
x(t)
where represents the temperature in a bounded region, ,
V×n is the normal velocity of (t) into the
liquid region n is
the unit normal pointing pointing into the liquid region
[.]-+
denotes the difference in the limits of the quantity in the brackets
as x tends to (t) from each side and is the
sum of the principal curvatures (t) (with the convention
that is positive when the solid region is convex).
This is a very difficult problem to solve. By relaxing this problem and
considering the Cahn-Hilliard equation
ut-=0, x
× ¼ ×
×=-²² u+
'(u), x
where (.) is a double well potential function. It is possible
to formally show that in the limit as goes to 0 the problem
converges to the solution of the Hele-Shaw problem.
Below is a numerical simulation with the white lines representing the
level set (t) as time evolves
