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Navier-Stokes (Low-Re Spherical)

by alvi123
Tags: lowre, navierstokes, spherical
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alvi123
#1
Mar2-10, 03:34 PM
P: 1
My first time posting..

I am looking for guidance in how to solve this question : "Show that the pressure is a HARMONIC FUNCTION ([tex]\nabla^2 P = 0[/tex] I did that), and that the following solution [tex]P = P_{\infty}[/tex] - [tex]\mu\nabla\cdot[/tex] (U/r) where [tex]P_{\infty}[/tex] is the pressure far away from the sphere, is harmonic and satisfies the appropriate boundary conditions. "

Background : This is trying to find the velocity, pressure, vorticity and streamfunction of SLOW FLOW around a sphere. Assumptions are incompressible, steady flow and no slip. Boundary conditions are that when r = a [tex]V_{\theta} = 0[/tex] [tex]V_{r} = 0[/tex] where a is the radius of the sphere and U is the flow speed in the Z direction.

I thought i did ALL the work when i found the pressure, velocity, stream function and vorticity already, but when i solved for the pressure...my supervisor said i didnt answer the question. To cut a 6 page derviation short, i will just give the results : [tex]\Psi = A/r + Br + Cr^{2}[/tex] , [tex]P_{\infty} - P = (3Ua\mu cos\vartheta)/2[/tex]
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