Fluid Dynamics and Stream Function

Q2 b) Consider the velocity field V = Ax/(x²+y²)i + Ay/(x²+y²)j in the xy plane, where A = 10 m²/s, and x and y are measured in meters.
i) Show this is an incompressible flow field. ii) Derive an expression for the fluid acceleration. iii) Evaluate the acceleration along the x axis, the y axis, and along a line defined by y = x. (14 marks)

Question 1 ( 15 markah )
a) Define and explain briefly the following :
i) velocity potential, f (x,y) ( 4 markah ) ii) stream function, y (x,y) ( 4 markah )

b) If the expression for stream function is described by y (x,y) = x3 – 3 xy2 , determine whether flow is rotational or irrotational. If the flow is irrotational, then determine the velocity potential, f and velocity vector, V. ( 7 markah )

Question 1
b) The velocity vector in a certain two-dimensional flow field is given by the equation;
V (x,y) = (x2 – y2 + x) i – (2xy + y) j

i) show that the continuity equation is satisfied ii) is this an irrotational flow field? Justify your answer with the necessary calculations. iii) determine the corresponding stream function iv) sketch several streamlines passing through the origin (12 marks)

Question 1
a) i) Define ‘potential flow’. ii) Describe the governing equation for ‘potential flow’ (8 marks)

b) Consider the flow field given by V = xy2i – 1/3(y3)j + xyk

i) Derive an expression for the fluid acceleration. ii) Evaluate the velocity along the x axis. iii) Evaluate the acceleration along the y axis. iv) Determine the number of dimensions of the flow. (12 marks)

Q2 In a two-dimensional incompressible flow the fluid velocity components are given by u = x - 4y and v = - y - 4x.

(i) Show that the flow satisfies the continuity equation. (ii) Obtain the expression for the stream function. (iii) Obtain the expression for the velocity potential. (iv) Show that the flow is potential. (15 marks)