- 2 Incompressible Newtonian Fluid Of Viscosity N Occupies The Space Between A Solid Sur Face Of Shape Y H X And A P 1 (161.65 KiB) Viewed 16 times
2. Incompressible, Newtonian fluid of viscosity n occupies the space between a solid sur- face of shape y = h(x) and a plane rigid boundary y = 0 of length L, which moves in the x-direction with velocity U. The Navier-Stokes equations then reduce to the well-known lubrication model др J² u dy = n 𐐀х = 0, ду ди Ju əx + = 0. ду (a) Solve the lubrication model equations to show that the velocity profile is u(x, y) = 1 dp (y² - hy) + U (1 − ²). 2n dx [6] (b) Show that the pressure in the fluid can be determined by the Reynolds equation d dp dh = nu dx 6 dx dx Subsequently, show that the pressure is given by dx ds P-Po = 6nU (S. 1) - he hate)) ho where ho and po are constants. T² h² (s) [6] " h³(s) = = (c) Now assume a lubrication system with a simple slider bearing of shape hi-h₂ and h₂ y = h(x) = h₁ ax, where a h(L). Assuming a fully sub- merged bearing, i.e. that the pressure at the entry and at the exit is equal to po, show that ho = 2h1h₂ h₁ + h₂ Show also that P-Po= 6nUL- (h₁ - h)(h-h₂) h² (h²h) Compare the pressure in the lubrication system when: (i) h₁ > h₂ (ii) h₁ <h₂. Which of two options should you choose when designing a lubrication system and why? [9] (d) When h₁> h2, determine the location in the system where the pressure takes its maximum value. You can assume that n = 1Pa.s, U = 1m/s, L = 10m, h₁ = 2m, h2 = 1m and po = 0Pa. Then using the following expression for the maximum value of the pressure (without deriving it) 3 1 - h₂ hi Pmax Po = 2 MUL h² (1+²)(1/²) sketch the pressure in the lubrication system. Briefly explain how you could vary h₁ and h₂ in order to achieve a larger pressure. [4]