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A simple 2D shear velocity field: v (x-direction) = v (x-dir)(y,t), v (y-dir) = 0, a barotropic flow with uniform density. Does this flow involve expansion, contraction, rotation and or deformation? How does the motion of the fluid look like in the vicinity of an arbitrary point x(0) - streamlines and particle paths? and what is the resulting volume force.

Concavity and convexity of the structure of the velocity field are important and four possible cases are possible - which ones? and in which direction is the x-momentum transferred in each case?

How much energy density per unit time must be given to the system to sustain the staionarity of the flow?

By which other mean can a steady state be achieved when the flow is given by: v(x-dir)=v(x-dir)(y,t), v(y)=v(y-dir)(y) ?

Can anyone out there give me some guidance?