# Navier-Stokes (Low-Re Spherical)

1. Mar 2, 2010

### 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 : $$\Psi = A/r + Br + Cr^{2}$$ , $$P_{\infty} - P = (3Ua\mu cos\vartheta)/2$$

Last edited: Mar 2, 2010