An incompressible Newtonian fluid with density p and viscosity flows steadily in a gap between

two infinitely long parallel plates as shown in Figure 1. The plates are separated by a known

distance h. The left plate at x = 0 is fixed. The right plate moves with a constant upward velocity

Vo. The gravitational acceleration vector g is given by g = -gĵ. There is no pressure gradient in

the flow. Treat the plates as no-slip, impermeable walls.

List relevant assumptions and boundary conditions.

Simplify and solve the continuity equation.

1.

2.

3.

4.

5.

6.

7.

N

Simplify and solve the Navier-Stokes equations to find the velocity profile.

Find the expression for the shear stress profile in the flow.

Find the volumetric flow rate per unit depth q.

Find the velocity Vo at which q = 0.

Sketch the velocity profile for the flow with q = 0.

stationary wall

fluid

ρ,μ

h

Vo

moving wall

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