Problem 17
This problem is to calculate the flow pattern around a sphere.
Consider a sphere of
radius a, situated at the origin of a spherical coordinate system.
At infinity, there is a
uniform flow,
u
, in the positive zdirection given by
u
=
z
^
u
0
.
In general, there is an
axisymmetric flow
u
(r,
θ
) everywhere such that the normal flow at the surface of the
sphere is zero (the tangential flow at the surface can be nonzero, ie, “freeslip” boundary
conditions).
Find
u
(r,
θ
) if u
0
<< c
s
, ie, the flow is subsonic.
Do this as follows:
1.
Write down all the unmagnetized ideal MHD equations (ie, the ideal fluid
equations)
2.
Scale the momentum equation and show that the system is isobaric to lowest
order, ie, p
0
= constant.
3.
Use this result in the remaining equations and show that the lowest order flow
must be incompressible.
In addition, show that if the plasma density at infinity is
constant, then the density has to be constant everywhere.
You must prove this by
first showing that density must be constant on each streamline.
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 Fall '09
 Fluid Dynamics, Plasma Physics, Geographic coordinate system, Coordinate system, Polar coordinate system, lowest order flow

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