9 HW 9

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. A stationary Newtonian fluid occupies the half space above a horizontal doubly-infinite plate. Take the $y$ coordinate of your Cartesian coordinate system to be normally up wards from the plate. At time $t=0$, the plate starts accelerating into the positive $x$ direction with a constant acceleration $\dot U$, so that its velocity is $\dot Ut$. Assuming that $u$ does not depend on $x$ and $z$ because the plate is infinite in both directions, that $w$ remains zero because of symmetry, and that the (kinetic) pressure at infinite $y$ is a constant, simplify the Navier-Stokes to give equations for the kinetic pressure and velocity fields. Solve the equation for the pressure field. Reduce the equation for the velocity field to a single partial differential and find its initial and boundary conditions. Then use dimensional analysis to argue that the solution must have a similarity form. Derive the ordinary differential equation that the nondimensional velocity profile $f(y/\sqrt{4\nu t})$ must satisfy for this flow.