Question

can you solve the last question (e)

Q1. Consider a steady, two-dimensional, incompressible flow field has the velocity potential a) b) c) 2 (x-7)(x+y) Determine the velocity components and verify that continuity is satisfied. [4 marks] Verify that the flow is irrotational. [2 marks] Determine the corresponding stream function. [4 marks) Now, consider a steady, two-dimensional, incompressible flow defined by velocity components u = ax + b&v=-ay + cx, where a, b and care constants. Neglect gravity. d) e) Show that continuity is satisfied. [2 marks] Solve the two dimensional Navier-Stokes equations for the pressure field P(x,y) if P(0,0) = 0. [8 marks]

Answer 1

e) A two dimensional, steady in compressible velocity flow es given by ū=ui + vj where u = ax + b V = - ay tcx The two dimensional Naveerastokes with neglecting gravity term for steady ssible flow field es given as, equation en compre- x component (P₂) s [ u au e al du ala + žu ag² + v لر } az aa a x² y component (P₂) a az a ly + + u axt dy $(we en vente an al Calculating derivatives, u= ax + b Y = ay tca au au av av С 11 a ox dy al az azu a²v MPV a²u dy2 =o = o o da? ax? уу? The equation of a component & y components become as,

x component (Pe). an s [(ax+b) a + (- ay tez):o) : evo + 2 (0+0] -alx s [a²x + ab لا ax Pa - sha²x + ab] de Pa = -8 [a²x² + abx] + f(y) te, 2 but p(0,0) = 0 o=o+ f(y) + C, f(y) to, co .. = -9 (**+ abx] a ل y component (P2) - : ((0&th) ctr-az + (x) (-2)] = 2 x t u (0) 9 [aca acx t + bec + a²y - aca] -dly ag s{a²y + be] =-2 / fys S-3 (a²y + bc) dy

a² : Py = -S Lary + bcy ] + f(a) + but pro,o=0 11 - loto] + f(x) + (2 f(x) + z = 0 b c I Py=-9 + cy 2 components of pressure field are given as, px 11 -s Total + abx] L py c - [ 22 7 + bcy ICS anne