5. Consider a steady, laminar flow of an incompressible fluid through a channel of two infinitely

long flat plates (in x and z directions), with the lower plate (at y -b) stationary and

the upper plate (at y = b) moving at a constant velocity of Vo. A known, constant pressure

gradient p' is applied in the a direction. Neglect the effect of gravity. Let the dynamic viscosity

of the fluid be μ and density be p.

y.

X

Infinitely long and wide plates

2b

Fixed plate

Vo

(a) Write down the component of the momentum equation you will use to solve for the

velocity profile u(y).

(b) It is argued that the flow is fully developed. Is it true? Explain (briefly) why or why

not?

(c) Is it safe to assume that the flow is two-dimensional? Why? What does it mean in terms

of simplifying the equation?

(d) Using appropriate assumptions, simplify the momentum equation clearly indicating which

terms drop out and why. Clearly indicate the boundary conditions needed to solve the

equation.

(e) Solve for u in terms of μ, b, Vo, and p'.

(f) Find the and expression for the power needed to move the top plate. How does this

power change if the viscosity doubles? How does it change if the top wall moves twice

as fast?

(g) Under what conditions (in terms of u, b, Vo, and p') would the net mass flow rate be

zero? Briefly explain why the flow rate is zero? Sketch how the velocity profile would

look like under this condition.