An long, thin plate is suspended in a still fluid. It is infinitely large in the x- and y-directions and infinitely thin in the z-direction, and sits in the plane defined by z = 0. At time t = 0, the plate begins to move with a constant velocity in the x-direction, given by U0i. We would like to know what the velocity profile looks like above the plate (for positive values of z) as it develops in time.
a) What are the appropriate assumptions to make for this flow? Justify them.
b) Write out the continuity equation in Cartesian coordinates and simplify it using your assumptions from (a). What does this tell you about w?
c) Write out the x-component of the Navier-Stokes equation and simplify it as much as you can, using your assumptions from (a). Justify each term you strike out and write out the final (simplified) equation. Do not attempt to solve it.
d) Your equation from (c) should be second-order in space and first-order in time and contain only u (not v or w). This means we need two boundary conditions and one initial condition for u. What are they?