An incompressible viscous flow is contained between two parallel plates separated from each other by distance b. The flow is caused by the movement of the bottom plate which has a velocity U, while the upper plate is fixed. The flow has zero pressure gradient in the x-direction. The velocity of the moving plate is U =7 m/s and b= 1 m. a) Determine the flow rate passing between the two plates per unit width (the width is perpendicular to the drawing plane). b) Calculate the magnitude of the rotation. c) Calculate the volumetric dilatation rate.

Problem Consider a cylindrical pipe of length L and diameter D = 2R. The angle that the axis of the pipe forms with the vertical direction is a. Assume that when the fluid enters the pipe its velocity is uniform (i.e., it has the same value over the entire cross-section of the pipe) and equal to U in the axial direction. In the radial and angular directions, the velocity is zero. So, it is: 2 =0: v(r, 0, 2) = Ue, (1.1) Here v is the fluid velocity and e, is a vector of unit magnitude parallel to the coordinate axis z; furthermore, we have assumed that the pipe inlet is located at z = 0. Near the entrance of the pipe, the velocity profile varies in the axial direction. But after a certain entrance length, the profile becomes fully developed, no longer changing with z. The evolution of the velocity profile is sketched in Fig. 1, where, for clarity, the pipe inclination is not shown. The entrance length is denoted by L.. For z > L., the fluid velocity is no longer a function of the axial…

a. a long rectangle of constant area A containing an irregu- larly shaped obstacle (figure 2). The upstream flow into the tube is u1 = (U, 0). Determine the flow at point 2, far downstream of the obstacle. b. now write down steady conservation of momentum in integral form, taking a simple control volume including points (1) and (2). Include pressure forces and horizontal drag from the particle but ignore viscosity. c. thus, determine the value of the drag force for an arbitrary irregular shaped obstacle in an inviscid, incompressible, irrotational fluid flow.

4. Consider a viscous flow inside an annular pipe. The inner surface of the pipe is located at r = R₁ and is stationary. The outer surface is located at r = Ro and moves to the right with speed Upipe. The pressure inside the pipe is constant. The fluid inside the pipe has dynamic viscosity μ. Assume that the flow is steady, incompressible, fully-developed, axisymmetric, non-rotating, and has negligible body forces. Starting from the differential form of the conservation of mass and momentum equations: a. Derive an equation for the velocity inside the pipe in terms of Upipe, R₁, Ro, and r. b. Determine the average velocity inside the pipe in terms of Upipe, R₁, and Ro. C. Determine the shear stress acting on the inner annulus in terms of Upipe, Ri, Ro, and u. Upipe er R₁ fluid