A steady two-dimensional jet is generated in an infinite, incompressible fluid of density $\rho$ and kinematic viscosity $\nu$ by a point source of momentum with momentum flux in the $x$ direction $F$ per unit length located at the origin.

Using boundary layer theory, analyse the motion in the jet and show that the $x$-component of the velocity is given by

$$u=U(x) f^{\prime}(\eta)$$

where

$$\eta=y / \delta(x), \quad \delta(x)=\left(\rho \nu^{2} x^{2} / F\right)^{1 / 3} \text { and } U(x)=\left(F^{2} / \rho^{2} \nu x\right)^{1 / 3}$$

Show that $f$ satisfies the differential equation

$$f^{\prime \prime \prime}+\frac{1}{3}\left(f f^{\prime \prime}+f^{\prime^{2}}\right)=0 .$$

Write down the appropriate boundary conditions for this equation. [You need not solve the equation.]