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 z-direction given by u = z ^ u0 . 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, “free-slip” boundary conditions). Find u (r, θ ) if u0 << 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, p0 = 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. 4.
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