Navier-Stokes (Low-Re Spherical)

[alvi123]

Navier-Stokes (Low-Re Flow past a Sphere)

My first time posting..

I am looking for guidance in how to solve this question : "Show that the pressure is a HARMONIC FUNCTION ($$\nabla^2 P = 0$$ I did that), and that the following solution $$P = P_{\infty}$$ - $$\mu\nabla\cdot$$ (U/r) where $$P_{\infty}$$ 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 $$V_{\theta} = 0$$ $$V_{r} = 0$$ 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 : $$\Psi = A/r + Br + Cr^{2}$$ , $$P_{\infty} - P = (3Ua\mu cos\vartheta)/2$$

 

[Valerie Park]

r^{2} and V_{\theta} = (Ur/a)sin\varthetaI have been trying to solve this for weeks and still can't seem to get the answer, so any help and advice would be appreciated. Thank you very much in advance!