Question 1 You are working as an engineer in a bearing systems design company. The flow of lubricant inside a hydrodynamic bearing (µ = 0.001 kg m¯¹ s¯¹) can be approximated as a parallel, steady, two-dimensional, incompressible flow between two parallel plates. The top plate, representing the moving part of the bearing, travels at a constant speed, U, while the bottom plate remains stationary (Figure Q1). The plates are separated by a distance of 2h = 1 cm and are W = 20 cm wide. Their length is L = 10 cm. By applying the above approximations to the Navier-Stokes equations and assuming that end effects can be neglected, the horizontal velocity profile can be shown to be 1 dP u(y) = 2μ dx -y² + Ay + B y= +h Ꮖ 2h=1 cm 1 x1 y = −h moving plate stationary plate 2 X2 L = 10 cm Figure Q1: Flow in a hydrodynamic bearing. The plates extend a width, W = 20 cm, into the page. (a) By considering the appropriate boundary conditions, show that the constants take the following forms: U U 1 dP A = and B = 2h 2 -h² 2μ dx Thus, write out the expression for the velocity profile, u(y) in terms of U, h, μ and dP/dx. (b) Recalling that we are told that the flow is parallel, write an expression for the ver- tical component of the velocity, v. By considering the relevant velocity gradients, determine whether the flow field is rotational or irrotational. (c) By considering relevant velocity gradients, determine whether the fluid elements get deformed. If so, what types of deformation do they undergo? (d) Compute the shear stress distribution and evaluate its values at the lower and upper walls. (e) Write down expressions for the tangential forces, Ft, exerted in the x-direction by the fluid on the lower and upper plates as a function of the stresses computed in part (d). (f) What pressure gradient is required to ensure that the tangential force exerted on the upper plate, Ft,upper, is zero? Express your answer in terms of the variables, U, μ and h. x (g) In fact, through measurements in the company's lab, the fluid is found to be exerting a tangential force of Ft,upper = +0.1 × 10-3 N on the moving plate and Ft,lower +0.9 × 10-3 N on the stationary plate. Determine the speed U of the moving plate and the pressure gradient, dP/dx. = (h) In a second measurement campaign in the lab, you record the pressure at station 1 to be P₁ = 100Pa. At a distance, L, downstream of this point is a second station 2, such that x2 = x1 + L. Using the pressure gradient, dP/dx, you calculated in part (g), compute the pressure, P2, at station 2. Thus, determine the normal force, Fn, exerted on the lower plate by the fluid.