Mathematical derivation of the Blasius "hypothesis" Consider the conservation equations that govern an incompressible flow around a horizontal flat plate:
$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\hspace{50pt}(1)$$
$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu\frac{\partial^2u}{\partial y^2}\hspace{20pt}(2)$$
where $u=u(x,y)$ and $v(x,y)$ are, respectively, the $x$ and $y$ components of the vector velocity of the flow, and $\nu$ is the kinematic viscosity (a constant).
I'm currently learning about the fluid mechanics of boundary layers, and my professor said that the exact solution of the above equations is obtained by considering a similarity hypothesis (Blasius hypothesis):
$$\frac{u(x,y)}{U_\infty}=F(\eta)\,\,\,\,\,\,\,\,\,\,\,\,(3)$$
where $U_{\infty}$ is the approximation velocity of the flow (a constant) and $F=F(\eta)$ is some function of $\eta=y\sqrt{\frac{U_{\infty}}{\nu x}}$.
I asked the professor if $(3)$ can be derived mathematically. He said no, it is just an experimental result. But I still believe that it can be derived mathematically. Maybe it just doesn't appear in regular books. After some google search, I think that $(3)$ can be obtained through self-similarity method. My problem is that I never learnt about that method. Do you know any good books about that self-similarity? Can you show (3)? Do you know anything that could help me? 
 A: This is an incomplete answer. A good reference is Boundary Layer Theory, by Schlichting, 8th Edition Springer, ISBN 3540662707, or earlier editions. Page 167 in the 8th Edition.
Many solutions are shown there. The argument begins by defining the stream function $\psi(x,y)$ as the flux of $(u,v)$ across any curve that starts at the origin and ends at point $(x,y)$. This is independent of path because of the divergence zero. Then the second pde becomes $$\psi_y\psi_{xy}-\psi_x\psi_{yy} = \nu\psi_{yyy}.$$ A scaling argument is then made, where we take a new $y$
variable so to speak, $\eta = y/b(x)$, where $b(x)$ is supposed to represent the thickness of the boundary layer at position $x$. So far that is the only dimensional argument, but in the next step Schlichting puts in effect $\psi(x,y) = b(x)F(x,\eta)$, which I think must be motivated by dimensional consideration that I don't get. It next requires several pages to push this through to the solutions. Schlichting includes the case of more general external flows than your $(U_\infty,0)$.
