Consider the slipper-pad bearing of Prob. 10–23.

(a) Generate a characteristic scale for v, the y-component of velocity. 

(b) Perform an order-of-magnitude analysis to compare the inertial terms to the pressure and viscous terms in the x-momentum equation. Show that when the gap is small (h₀ ≪ L) and the Reynolds number is small (Re = ρVh₀/μ ≪ 1), the creeping flow approximation is appropriate. 

(c) Show that when h₀ ≪ L, the creeping flow equations may still be appropriate even if the Reynolds number (Re = ρVh₀/μ) is not less than 1. Explain.

Data from Problem 23

A slipper-pad bearing (Fig. P10–23) is often encountered in lubrication problems. Oil flows between two blocks; the upper one is stationary, and the lower one is moving in this case. The drawing is not to scale; in actuality, h ≪ L. The thin gap between the blocks converges with increasing x. Specifically, gap height h decreases linearly from h₀ at x = 0 to h_L at x = L. Typically, the gap height length scale h₀ is much smaller than the axial length scale L. This problem is more complicated than simple Couette flow between parallel plates because of the changing gap height. In particular, axial velocity component u is a function of both x and y, and pressure P varies nonlinearly from P = P₀ at x = 0 to P = P_L at x = L. (∂P/∂x is not constant). Gravity forces are negligible in this flow field, which we approximate as two-dimensional, steady, and laminar. In fact, since h is so small and oil is so viscous, the creeping flow approximations are used in the analysis of such lubrication problems. Let the characteristic length scale associated with x be L, and let that associated with y be h₀ (x ∼ L and y ∼ h₀). Let u ∼ V. Assuming creeping flow, generate a characteristic scale for pressure difference ΔP = P – P₀ in terms of L, h₀, μ, and V.

FIGURE P10–23

Transcribed Image Text:

T ho u(x, y) V X L- h(x) μ hL