<br>
-<i>This program was contributed by Luca Heltai.
-</i>
+<i>This program was contributed by Luca Heltai (thanks to Michael
+Gratton for pointing out what the exact solution should have been in
+the threedimensional case). </i>
<a name="Intro"></a>
-<h1> Warning! </h1>
-
-The three dimensional solution generated by this example program is
-wrong. There is a factor of $1/2$ that appears, and we haven't figured
-out where it comes from yet. For the moment this issue is "solved" by
-multiplying the wind function in the parameter file by two.
-
-If you think you have spotted the mistake, please let us know.
-
<h1>Introduction</h1>
<h3> Irrotational flow </h3>
will then be attached to the triangulation to allow mesh refinement
that respects the continuous geometry behind the discrete initial
mesh.
+
+For a sphere of radius $a$ translating at a velocity of $U$ in the $x$ direction, the potential reads
+\[
+\phi = -\frac{1}{2}U \left(\frac{a}{r}\right)3 r \cos\theta
+
+\] see, e.g. J.N. Newman, \emph{Marine Hydrodynamics}, 1977,
+pp. 127. For unit speed and radius, and restricting $(x,y,z)$ to lie
+on the surface of the sphere,
+\[ \phi = -x/2.\] In the test problem,
+the flow is $(1,1,1)$, so the appropriate exact solution on the
+surface of the sphere is the superposition of the above solution with
+the analogous solution along the $y$ and $z$ axes, or \[ \phi =
+\frac{1}{2}(x + y + z) \]
\ No newline at end of file
# Separate vector valued expressions by ';' as ',' is used internally by the
# function parser.
- set Function expression = x+y+z # default: 0
+ set Function expression = .5*(x+y+z) # default: 0
# The name of the variables as they will be used in the function, separated
# by ','.
# Separate vector valued expressions by ';' as ',' is used internally by the
# function parser.
- set Function expression = 2; 2; 2 # default: 0; 0; 0
+ set Function expression = 1; 1; 1 # default: 0; 0; 0
# The name of the variables as they will be used in the function, separated
# by ','.
# Separate vector valued expressions by ';' as ',' is used internally by the
# function parser.
- set Function expression = x+y+z # default: 0
+ set Function expression = .5*(x+y+z)
# The name of the variables as they will be used in the function, separated
# by ','.
# Separate vector valued expressions by ';' as ',' is used internally by the
# function parser.
- set Function expression = 2; 2; 2 # default: 0; 0; 0
+ set Function expression = 1; 1; 1
# The name of the variables as they will be used in the function, separated
# by ','.