Drag on the sphere

We now get to the point of the problem, find the drag on the sphere.  

Form drag

We need to integrate the normal stress, resolved onto the flow direction all over the sphere.  The normal stress is -p + 2 μ .  It must be resolved onto the flow direction which takes Cos[θ].  The area element for a sphere is Sin[θ] dθ dφ.  The dφ provides a 2 π and the integral becomes:

2 π Integrate[-p + 2 μ D[u[r],r]Cos[θ] Sin[θ] ,{θ,0,π}].

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Skin drag

Now we need to integrate the shear stress, resolved onto the flow direction all over the sphere.  The shear stress is .  It must be resolved onto the flow direction which takes Sin[θ].  The area element for a sphere is Sin[θ] dθ dφ.  The dφ provides a 2 π and the integral becomes:

2 π Integrate[-p + 2 μ D[u[r],r]Cos[θ] Sin[θ] ,{θ,0,π}].

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