Fluid Dynamics | Part IB, 2003

An unsteady fluid flow has velocity field given in Cartesian coordinates (x,y,z)(x, y, z) by u=(1,xt,0)\mathbf{u}=(1, x t, 0), where tt denotes time. Dye is released into the fluid from the origin continuously. Find the position at time tt of the dye particle that was released at time ss and hence show that the dye streak lies along the curve

y=12tx216x3y=\frac{1}{2} t x^{2}-\frac{1}{6} x^{3}

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