The velocity distribution for the shear flow shown between two plates of gap a is giv 1 dp (y² – ay) +-y U u(у) 2µ dx а What is the pressure gradient that will result in a zero shear stress at the lower wall

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

(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…

c. For a given velocity field calculate the constants a, b, and c such that the flow field is irrotational. V = (0.657 + 1.73x + 0.948y + az)i + (2.61 + cx + 1.91y + bz)j+(-2.73x - 3.66y – 3.64z)k