Fluid Dynamics | Part IB, 2002

A fluid flow has velocity given in Cartesian co-ordinates as u=(kty,0,0)\mathbf{u}=(k t y, 0,0) where kk is a constant and tt is time. Show that the flow is incompressible. Find a stream function and determine an equation for the streamlines at time tt.

At t=0t=0 the points along the straight line segment x=0,0ya,z=0x=0,0 \leqslant y \leqslant a, z=0 are marked with dye. Show that at any later time the marked points continue to form a segment of a straight line. Determine the length of this line segment at time tt and the angle that it makes with the xx-axis.

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