Fluid Dynamics 3
2011/12
Sheet 6
Homework to be handed in 25th November: questions 2,3,4
1.
(i) Show that in spherical polars, the potential of a uniform stream and of a dipole are (a)
φ
=
Ur
cos
θ
, and (b)
φ
=

μ
cos
θ
r
2
, respectively.
(ii) By using the expression for the Laplacian in spherical coordinates, confirm that each
of these potentials satisfy Laplace’s equation.
(iii) Calculate
u
in spherical coordinates for each of the potentials.
(iv) Verify that
φ
=
Ur
cos
θ
+
A
cos
θ
r
2
satisfies the boundary condition
n
· ∇
φ
= 0 on the surface of a sphere
r
=
R
for a
suitably chosen
A
.
2. A sphere is moving with velocity
U
through an ideal, incompressible fluid, which is
at rest
at infinity.
(i) Compute
φ
= (
A
· ∇
)
1
r
,
where
A
is a constant vector, and show that
4
φ
= 0.
(ii) Placing
x
= 0 at the instantaneous centre of the sphere, compute the normal velocity
u
n
≡
u
·
n
on the surface. Determine
A
.
(iii) Using Bernoulli’s equation for unsteady flow, find the pressure on the surface of the
sphere.
Compare to what we found in the lecture for a sphere in a uniform stream.
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 Fall '11
 Eggers
 Kinetic Energy, φ, uniform stream, bubble radius

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