- 2 Incompressible Newtonian Fluid Of Viscosity N Occupies The Space Between A Solid Sur Face Of Shape Y H X And A Pl 1 (163.52 KiB) Viewed 22 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 əx dy²¹ 0, ду Ju Ju + = 0. əx ду (a) Solve the lubrication model equations to show that the velocity profile is u(x, y) 1 dp 2n dx (y² — hy) + U (1 - 12/1). [6] (b) Show that the pressure in the fluid can be determined by the Reynolds equation h³ dp dh de (b'de) = nU- dx 6 dx dx Subsequently, show that the pressure is given by dx ds P-Po = 6nU (19²(1) - 10 / 12(2) (S ho where ho and po are constants. h³(s) h₁-h₂ ₁2 and ha = - (c) Now assume a lubrication system with a simple slider bearing of shape 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. 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 h₂ h₁ Pmax Po= MUL h² (1+₂)(h)' 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. =