2. (a) [12 marks] A fluid has velocity field u = (2xt, -yt, -zt). Show that the surface defined by F(x, y, z, t) =consta

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2 A 12 Marks A Fluid Has Velocity Field U 2xt Yt Zt Show That The Surface Defined By F X Y Z T Consta 1 (117.44 KiB) Viewed 18 times

2. (a) [12 marks] A fluid has velocity field u = (2xt, -yt, -zt). Show that the surface defined by F(x, y, z, t) =constant moves with the fluid, that is, it is always composed of the same fluid particles, where F(1,4, z,t) = r’e2° + (y² +222)e". (b) [18 marks] () If a complex potential w(z) satisfies w2 = U2 (22 +c?), for strictly positive real constants U and c, show that the stream function satisfies U2 (U2,2 + 42) y2 = 42 [4632 +U? (x2 +c)]. (ii) Show that this stream function represents the possible flow of a fluid in the upper half-plane (y > 0), with a solid boundary along y = 0. (ii) Show that the flow at large distances from the origin is Ui, where i is the unit vector in the x-direction. (iv) Determine the stagnation points in the flow and show that there is a singularity at the point (0,c). (v) Consider the behaviour of along the y-axis to deduce the geometry of the flow domain