The diagram shows a load-bearing pad resting on a moving belt with lubricating oil supplied at a gauge pressure of 12 kPa at one end of the pad. The oil flows through the space between the two surfaces, emerging at atmospheric pressure. The pad is 120 mm long and the gap between the two surfaces is 0.18 mm. The belt speed is 5 m/s and the dynamic viscosity of the oil is 0.5 Pa-s. Assuming that the flow may be taken as purely unidirectional, determine (per unit span of pad):

(a) the load the pad will support;

(b) the rate at which oil must be supplied;

(c) the tangential force acting on the base of the pad. State the direction of this force.

 

 

Hints:

(a) Load on pad is balanced by normal force acting on the base of the pad, where the normal force is due to pressure. Recall from hydrostatics: resultant normal force due to pressure = mean (average) pressure x area. Here, we assume a linear pressure distribution along the flow, from 12 kPa at one end to 0 kPa at the other end. Hence the mean pressure is 6 kPa. Multiplying this mean pressure by the area (0.12 x 1 m2), you will get the result.

(b) Let us revisit the problem of Poiseuille-Couette flow as presented in the notes. The general solution is the same:

(c) The tangential force acting on the base of the pad is due to shear stress of fluid at the upper boundary. From the above velocity profile,

This is shear stress in the fluid, so, by reaction, tangential stress acting on the base of the pad τ???????????? = -τ????????(ℎ). Multiplying this stress by the area, you will get the tangential force. It is easy to figure the direction of this force.
 
 

Transcribed Image Text:

Oil in G Seals Load Oil in G Seals Load