I have a pressure flow problem where I'm trying to understand the velocity profile of a fluid in an annular space between a stationary exterior cylinder and a rotating, longitudinally advancing cylinder at its center.(adsbygoogle = window.adsbygoogle || []).push({});

So the boundary conditions a zero velocity at the exterior surface and a constant angular and longitudinal velocity at the interior surface.

I begin by simplifying the usual form of Navier-Stokes in cylindrical coordinates to the following three equations, knowing that acceleration and velocity in the radial direction are zero:

[1] -ρ(u_θ^2)/r=μ*(2/r^2)((∂u_θ)/∂θ)-(∂u_θ)/∂θ+ρ*g_r

[2] ρ((∂u_θ)/∂t+(u_θ/r)((∂u_θ)/∂θ)+u_z*((∂u_θ)/∂z))=μ[(∂^2*u_θ)/(∂r^2 )+(1/r^2)*((∂^2 u_θ)/(∂θ^2))+(∂^2 u_θ)/(∂z^2 )]-(1/r)(∂p/∂θ)+ρ*g_θ

[3] ρ((∂u_z)/∂t+(u_θ/r)(∂u_z)/∂θ+u_z*(∂u_z)/∂z)=μ[(∂^2*u_z)/(∂r^2)+(1/r^2)(∂u_z)/(∂θ^2)+(∂^2*u_z)/(∂z^2)]-∂p/∂z+ρ*g_z

How do I solve for u_θ and u_z??

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# Seeking closed form solution of Navier-Stokes for a fluid in an annular space.

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