4. Consider fully developed laminar flow in the annular space formed by two concentric cylinders with a pressure gradient, ap/ox, and the inner cylinder stationary. Let r = Rand r₁ = kR. Assuming that the velocity profile is given by u = Rt @[₁-(A) + R R² ap In R 1-k² In(1/k)

An incompressible viscous flow is contained between two parallel plates separated from each other by distance b. as shown in Figure 1. The flow is caused by the movement of the upper plate which has a velocity U, while the bottom plate is fixed. If U =7 m/s and b= 1 cm, and there is no pressure gradient in the flow direction. A.) Start with Navier-Stokes equations and determine the velocity at the point x = 3 cm and y= 0.41 cm. The value of the velocity is.B.) Calculate the magnitude of the vorticity at the same point. The magnitude value of vorticity. C.) Calculate the rate of angular deformation at the same point. The angular deformation value

(a) A wing section with a chord of c and a span of b is mounted at zero angle of attack in a wind tunnel. A pitot probe is used to measure the velocity profile in the viscous region downstream of the wing section as shown in the figure. The measured velocity profile is u(z) = U∞ - (U/2) cos[TZ/(2w)] for -w ≤ z ≤w. Here, w = 0.02c. Assuming a constant pressure p = po along the streamlines (dashed lines in the figure) and across the wake where the velocity was measured, calculate the friction drag coefficient Cp, of the wing section. U Streamlines u = U_ -- cos 22 2W² +w Viscous wake = -W (b) Consider a thin flat plate at zero angle of attack in an airflow at P∞ = 1.225 kg/m³, T∞ = 288 K and μ∞ 1.7894 x 10-5 kg/m/s. The length of the plate is 2 m and the span is 0.5 m. Assume the boundary layers on the plate are laminar throughout (on the upper and lower surfaces both) where LBL(x)/x = 0.664/√√/Rex applies. The freestream velocity is 100 m/s. Calculate the friction drag (Df) of the first…

Consider a steady, two-dimensional flow field in the xy-plane whose x-component of velocity is given by                                                                      u = a + b(x − c)2 where a, b, and c are constants with appropriate dimensions. Of what form does the y-component of velocity need to be in order for the flow field to be incompressible? In other words, generate an expression for ? as a function of x, y, and the constants of the given equation such that the flow is incompressible

Consider a steady, incompressible, y=+h ↑y hydrodynamically fully-developed flow between two infinite flat plates as shown. The flow can be described by the following equation: y=-h 2U K -y² +C¸y+C, 2µ where u is the dynamic viscosity, u is the axial velocity (i.e., the velocity in x-direction), and K, Cı, and C2 are constants. (a) Obtain the velocity profile u(y) in terms of h, U, K, µ̟ and y. (b) For what value of K is the shear stress zero at y=h/2. Consider h=0.01 mm, U=0.1 mm/s, and u=0.001 Pa.s

4. Consider fully developed laminar flow in the annular space formed by two concentric cylinders with a pressure gradient, op/ax, and the inner cylinder stationary. Let r = Rand r = kR. Assuming that the velocity profile is given by u: [¹-( 7 )² + +(ma 1 R R² ap 4μ &x 1-k² In- In(1/k) R a) Obtain an expression for the location of the maximum velocity as a function of k. b) Compare the limiting case, k→ 0,with the corresponding expression for flow in a circular pipe. c) Determine the volume flow rate. d) Find an expression for the average velocity. e) Compare the limiting cases for the volume flow rate and average velocity, k →0, with the corresponding expression for flow in a circular pipe.

Consider a two-dimensional flow which varies in time and is defined by the velocity field, u = 1 and v = 2yt. Do the fluid elements experience angular rotation? Thus, state whether the flow field is rotational or irrotational.