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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 = n Əx dy²¹ др = 0, ду ди Ju + = 0. əx ду (a) Solve the lubrication model equations to show that the velocity profile is 1 dp (y² - hy) + U (1 - 12/). u(x, y) = 2n dx [6] (b) Show that the pressure in the fluid can be determined by the Reynolds equation d dp dh - ( 16³ dr) = MU dx 6 dx dx Subsequently, show that the pressure is given by d.s p - P₁ = 6nU (²₁ dx h² (s) ho So That where ho and po are constants. [6] 1³ (5)); (c) Now assume a lubrication system with a simple slider bearing of shape y = h(x) = h₁ ax, where a = h₁-h₂ and h₂ L = h(L). Assuming a fully sub- i.e. that the pressure at the entry and at the exit is equal to po, merged bearing, 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 = UL²(1 + h) (²) 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]