Question

Two circular plates each of radius R are separated by a distance of b. The space between the plates is filled with a fluid. The upper plate moves downward at a constant rate V resulting in fluid being squeezed out between the plates. Determine the mass flow rate and maximum velocity under the following conditions: a) The exit velocity is uniform; b). The exit velocity is parabolic.

Answer 1

Two circular plates each of radius R are separated by a distance of b. The space between the plates is filled with a fluid.

The upper plate moves downward at a constant rate Vz resulting in fluid being squeezed out between the plates with velocity Vr.

Applying the equation of continuity in cylindrical coordinates

or

or

r-component of the Navier–Stokes equation in cylindrical co-ordinate systems

From Eq (19.5)

Multiply by r on both sides

or

or

From Eq(19.10)

or

From Eq (19.13) and Eq (19.11)

or

Boundary conditions

at

From Eq (19.15)

At z=0

and

or

velocity profile

mass flow rate at r or