I do not need help with part a, but please do part b in step by step detail, I have been struggling to learn similarity solutions. Thank you!

1. In this problem we will consider the buoyancy driven flow of a viscous liquid in a plume far above an object that creates a source of heat Q per unit length in the y direction. The fluid velocity, pressure and temperature satisfy the Bou equations. v=00=pTg+2vvT=2T where T is the deviation of the temperature from its value far away. In the momentum equation we include a gravity term Tg that describes the variation of the gravity due to the fact that the fluid density changes with temperature. Here =(1/)/T. Elsewhere in the equations we assume approximately constant density. The heat source extends infinitely far in the y direction and the velocity and temperature are independent of y. Assume that the convective heat flux (per unit length in the y direction) across any horizontal plane is equal to Q. Assume that the horizontal extent of the plume xp is much smaller than the characteristic vertical position zp. Assume that T=0,p=0,v=0 as x and T and vz are symmetric about x=0. a) Introduce a stream function and use the boundary conditions in the wake to show that the momentum equation in the wake can be used to express T in terms of derivatives of . In this way, you can reduce the problem to a single unknown . b) Show that there is a similarity solution to this problem in which a stream function takes the form =z8f() for a suitable choice of and the similarity variable by showing that this problem can be reduced to a solution of an ordinary differential equation with appropriate boundary conditions and an integral constraint. You do not need to solve the ordinary differential equation