2) Consider a steady, two-dimensional, incompressible laminar boundary layer flow over a surface with free-stream velocity U(x), which may vary with stream-wise distance from leading edge, x in an arbitrary manner. Assuming negligible body forces and constant fluid viscosity, the continuity and momentum equations reduce to: dv 0 (1) ax +v = 1 dP + pdx (2) where, x and y are the directions along and perpendicular to the surface respectively, u, v are the local velocity components in the x and y directions respectively, p is fluid density, and is viscosity. a. Write the appropriate boundary conditions necessary for solution of Eq. (2) [10 points] b. Show that pressure in the boundary layer can be expressed by: P(x) = Po + (U - U(x)) (3) where U, and P. are respectively the velocity and pressure at x=0 [15 pts] c. Integrating the momentum equation across the boundary layer gives the following approximate expression (termed the momentum integral equation or M.I.E.): " dl dr + U dx (4) Write the appropriate M.I.E. for a boundary layer with zero pressure gradient. [10 pts] d. Consider the laminar boundary layer with zero pressure gradient and assume the approximate velocity profile, u/U = a + b(y/8) + c(y/8) + d(y/8) where a, b, c and d are constants, (5) i. Use appropriate boundary conditions to obtain the constants a, b, c and d, and determine appropriate expression for the velocity profile of Eq. (5) [25 pts]. ii. Use the velocity profile obtained in (i) and the momentum integral equation you obtained in (c) to derive the following expression for the boundary layer thickness: [40 pts] where A is a constant. 8=A{ux/(pU)} ? (6)