What Equations Govern This Hypothetical Fluid Flow?

[member 428835]

Hi PF!

I was reading lecture notes from a university and I stumbled on this situation:

We have a hypothetical 2D inviscid, steady, uniform and parallel, potential flow, described by
the velocity vector $\vec{v} = <u,w>$, with $u= U[z]$ and $w= W[z]$. It is moving parallel to a stationary plate that lies along the $x$ axis. By “parallel,” it is meant that flow is everywhere parallel to the plate.
By “uniform”, it is meant that the flow is spatially uniform. By “steady”, it is meant that the flow
is time invariant.

When solving for a velocity profile, my first thought was to consult the Navier-Stokes Equations for ractangular coordinates and go from there. What I was thinking was steady implies $\partial_t = 0$, inviscid implies $\mu = 0$ (or would this be $\nabla \times \vec{v} = 0$?). Parallel implies $v_y = 0$ where $y$ is orthogonal to the base plate (I think $u$ and $w$ move in the plate direction and from side-to-side respectively (any ideas here)? Potential implies the existence of some scalar $\phi : \vec{v} = \nabla \phi$. Uniform implies $\partial_x v = \partial_z v = 0$.

Can someone confirm this?

If I'm right, the equation of motion would simply be $\nabla ^2 \vec{v} = 0 \implies \nabla^2 u = 0$ and $\nabla^2 w = 0$. Do you agree or disagree?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?