Consider the two-dimensional vector field u = -3x²i+6x (y+9x²t) j (x > 0). (a) Show that u could represent the velocity vector field of an incompressible fluid flow. (b) (i) By solving appropriate differential equations, show that the pathlines of this flow are described by the equations 1 2 A x= Y + 3(t + A)' 3(t + A) 2(t + A)² + B(t+A)², where A and B are arbitrary constants. Hint: Note that t 1 = - (t + A)5 (t + A)4 A (t + A)5* (ii) For the pathline that passes through the point (1, 3) at time t = 0, eliminate t between the equations given in part (b)(i) to obtain an explicit equation of the form y = y(x) for the pathline. (c) (i) Write down the equations describing the stream function for the velocity vector field u. Hence find the stream function for this flow, and the equations of the streamlines. (ii) Find in particular the equation of the streamline at time t = 0 that passes through the point (1,3), in the form y = y(x). (iii) Sketch on a single diagram the pathline whose equation was found in part (b)(ii) and the streamline whose equation was found in part (c)(ii). Indicate in each case the asymptotic behaviour of the curve and the direction of flow along it.