A line force of magnitude $F$ is applied in the positive $x$-direction to an unbounded fluid, generating a thin two-dimensional jet along the positive $x$-axis. The fluid is at rest at $y=\pm \infty$ and there is negligible motion in $x<0$. Write down the pressure gradient within the boundary layer. Deduce that the function $M(x)$ defined by

$$M(x)=\int_{-\infty}^{\infty} \rho u^{2}(x, y) d y$$

is independent of $x$ for $x>0$. Interpret this result, and explain why $M=F$. Use scaling arguments to deduce that there is a similarity solution having stream function

$$\psi=(F \nu x / \rho)^{1 / 3} f(\eta) \quad \text { where } \quad \eta=y\left(F / \rho \nu^{2} x^{2}\right)^{1 / 3}$$

Hence show that $f$ satisfies

$$3 f^{\prime \prime \prime}+f^{\prime 2}+f f^{\prime \prime}=0 .$$

Show that a solution of $(*)$ is

$$f(\eta)=A \tanh (A \eta / 6)$$

where $A$ is a constant to be determined by requiring that $M$ is independent of $x$. Find the volume flux, $Q(x)$, in the jet. Briefly indicate why $Q(x)$ increases as $x$ increases.

[Hint: You may use $\left.\int_{-\infty}^{\infty} \operatorname{sech}^{4}(x) d x=4 / 3 .\right]$