Question

Consider the steady flow between two flat plates. The plates are 10 cm apart and the width of the channel is 10 cm. Water is the working fluid. The flow velocity varies linearly from zero at the bottom plate to 10 m/s at the top plate. (This general class of flows is known as Couette Flow). Determine the total mass flow and momentum per unit time crossing an imaginary plane across the channel.

Answer 1

Rate of change of velocity with distance from the bottom plate, r = (10 m/s)/10cm = (10 m/s)/(0.1 cm)= 100 m/s/m

Therefore velocity at a distance 'x' cm from the bottom plate, v = 0+r*x = r*x

Differential area of a strip of thickness dx at this distance x from the bottom is dA=w*dx

where w = 10 cm =0.10 m, is width of the channel

Therefore volume of water flowing through this strip per unit time, dV = v*dA = r*x*w*dx

Mass of water flowing through this strip per unit time,

Where is density of water and

Distance between the plates, d=w = 10 cm = 0.10 m

Therefore mass of water flowing through the channel per unit time,

Putting values:

mass flowing per unit time is 50 kg/s

momentum flowing through the differential strip per unit time,

momentum flowing through the channel per unit time:

Putting values:

Momentum flowing per unit time is 333.33 kgm/s