A thin 30-cm × 30-cm flat plate is pulled at - Mechanical Engineering

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A thin 30-cm × 30-cm flat plate is pulled at 3 m/s horizontally through a 3.6-mm-thick oil layer sandwiched between two plates, one stationary and the other moving at a constant velocity of 0.3 m/s, as shown in Fig. P2–81. The dynamic viscosity of the oil is 0.027 Pa·s. Assuming the velocity in each oil layer to vary linearly, (a) plot the velocity profile and find the location where the oil velocity is zero and (b) determine the force that needs to be applied on the plate to maintain this motion.

FIGURE P2–81

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Step 1 of 5

(a)

Calculate the surface area of the plate.

Here, A is the surface area.

Consider the portion of the oil that is below the plate.

Calculate the viscous force acting in the plate in this portion.

Here, is the viscous force, is the coefficient of viscosity, V is the velocity of the plate, is the velocity of the oil, is the change in velocity in the lower portion, and is the thickness of the oil layer in the lower portion.

Substitute for , for A, for V, for , and 0.0026 m for .
Step 2 of 5

Calculate the location where the velocity of the oil is zero.

Substitute for , for A, for , 0 for , and for .

In the upper portion the velocity of the oil is zero at the surface of the fixed wall.

Hence, the location where the oil velocity is zero in the lower potion is from the plate.
Step 3 of 5

Draw the velocity profile of the fluid:
Step 4 of 5

(b)

Consider the portion of the oil that is above the plate.

Calculate the viscous force acting in the plate in this portion.

Here, is the viscous force, is the coefficient of viscosity, is the change in velocity in the upper portion, and is the thickness of the oil layer in the upper portion.

Substitute for , for A, for , and 0.001 m for .
Step 5 of 5

Calculate the force that needs to be applied to maintain this motion.

Here, F is the total force applied on the plate.

Substitute 7.29 N for and for .

Therefore, the force that needs to be applied to maintain this motion is .