1- Calculate the displacement thickness and momentum thickness for the following velocity profiles in the boundary layer: u 9. b- U 8;0. 15 a- = 2 U. 110

a. If the velocity distribution for the laminar boundary layer over a flat plate is given by :- ** (²) ² - 21 + A₂ x U A₁ + A₂ x Determine the form of the velocity profile by using the necessary boundary conditions. After that by using the Von-Karman integral momentum equation find an expression in terms of the Reynolds number to evaluate 1- Boundary Layer Thickness Force 2-Wall Shear Stress 3-Drag 5- Displacement Thickness + A₁ x 4- Local and Average Skin Friction Coefficients 6- Momentum Thickness 7- Energy Thickness

Consider the boundary layer over a flat plate at 45° angle as shown. The exact flow field in this configuration is described by the Falkner-Skan similarity solution with n = 1/3 (see Figure 10.8 of the textbook, the Falkner-Skan profile chart). The objective is to find the approximate solution to this problem using the Thwaites method and calculate its error. Ve 11/4 Assume that for this approximate solution the free stream velocity is U₂(x) = ax" where a is an unknown constants and n = 1/3. Use the Thwaites method to find the momentum 0/x and 8*/x displacement thicknesses as well as the friction coefficient cf = 0.5, as functions of Re₂ = Uer/v, where is the shear stress at the wall. (No need to interpolate the Thwaites method table values; you can pick the nearest numbers.) Using the Falkner-Skan profile chart approximate the friction coefficient c; (by estimating the slope of the corresponding velocity profile at the wall). How does this value compare with your prediction in part…

Rain water is flowing through an inclined plane under gravity as shown in figure. A thin layer of thickness h is formed over the plate. An upward wind flow that exerts a shear stress on the upper surface (y = h) of the liquid layer. The wind flow tends to push the rain layer upward while gravity pulls the layer downward Write continuity and and Navier-Stokes equations to model this flow field. (Assume the flow is fully developed) Obtain expressions for the liquid velocity profile, the shear stress distribution, the volume flow rate, and the the expression for the thickness h of the liquid layer for which there is no net volume flow along the roof

Consider a steady laminar flow through the annular space formed by two coaxial tubes aligned with the z-axis, as shown below. The flow is along the axis of the tubes and is maintained by the constant pressure gradient. The symbol a is the radius of the inner tube and b is the radius of the outer tube, and assume that b = 3a. dp/dz 9. a (a) Start with Navier-Stokes equation and assume that the body force does not affect the flow. Derive the axial velocity at any radius R (a < R < b) uz(R) in terms of dp/dz, R, a and µ (fluid dynamic viscosity). (b) Find the radius at which the maximum velocity is reached (c) Find the volume flow rate (d) Find the stress distribution