Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 1 / 6C (Fluid Dynamics)

The divergence is ${\partial }_x(kty)+{\partial }_{y}0+{\partial }_{z}0=0$, so this is an incompressible flow. With the planar convention $(u,v)=({\psi }_y,-{\psi }_x)$, one stream function isAt a fixed nonzero $kt$, the streamlines have $y=\mathrm{constant}$ and $z=\mathrm{constant}$, since the instantaneous velocity is purely axial. If $kt=0$, every point is instantaneously stationary and there is no distinguished streamline direction.

Follow a marked particle initially at $(0,y_0,0)$. Its trajectory satisfies $\dot{y}=\dot{z}=0$ and $\dot{x}=kt{y}_0$, soThese points form a straight segment on $x=(k{t}^2/2)y$, $z=0$, not an instantaneous horizontal streamline. Its length isThe directed angle of its vector from the origin to the marked endpoint isFor $k>0$ and $t>0$, this is $\arctan [2/(k{t}^2)]$; it equals $\pi /2$ at $t=0$. If an unoriented acute angle to the axis is intended, use $\arctan [2/(|k|t^2)]$, with $\pi /2$ when $kt=0$. This distinction handles a negative value of the given constant as well. If $a=0$, the marked set is a single point: the length formula gives zero and no line direction is defined.