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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 J²u dy² əx др = 0, ду du Ju + = 0. əx ду (a) Solve the lubrication model equations to show that the velocity profile is 1 dp u(x, y) = (y² — hy) + U (1 − 1). 2n dx [6] (b) Show that the pressure in the fluid can be determined by the Reynolds equation h³ dp dh 法(2) = nu d.x 6 dr dx Subsequently, show that the pressure is given by dx ds P-Po = 6nU C - ho So h2² (3)). where ho and po are constants. [6] h² (s) h³(s) with a simple slider bearing of shape = (c) Now assume a lubrication system y=h(x) = h₁ ax, where a ₁2 and ₂ 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 L 2h1h₂ ho 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-h Pmax Po= 2UL ´h² (1 + ²) (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] = = n =