Take Home Final
2.
Flow past Sphere (25 points)
(aka 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.
Very far from the sphere,
at infinity, assume there is a uniform flow,
u
, in the positive zdirection given by
u
=
z
^
u
1
, where u
1
is a constant.
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).
Box it.
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 leading order flow
u
(r,
θ
) must be incompressible.
In addition, show that if the plasma density at
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 Fall '09
 Fluid Dynamics, Plasma Physics, Coordinate system, Polar coordinate system

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