# Time-dependent velocity fields

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• $\mathbf v = u\,\mathbf i + v\cos (kx-\alpha t)\, \mathbf j$.

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• $\mathbf v(x,y,t)=\cos(y+t)\,\mathbf i+\sin(x-\frac12 t)\,\mathbf j$.

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• First, define the functions $f,g$ and $h$ as follows:
\begin{eqnarray*} f(x,t)&=&\pi (0.25\sin(2\pi / 10 t) x^2 + (1 - 2 (0.25) \sin(2\pi / 10 t)) x),\\ g(x,t)&=&\pi (0.25\sin(2\pi / 10 t) x^2 + (1 - 2 (0.25) \sin(2\pi / 10 t)) x),\\ h(x,t)&=&2 (0.25) \sin(2\pi / 10 t) x + 1 - 2 (0.25) \sin(2\pi / 10 t). \end{eqnarray*}

Thus the velocity field is defined as
$$\mathbf v = -\pi 0.5 \sin(g(x,t)) \cos(\pi y)\,\mathbf i + \pi 0.5 \cos(g(x,t)) \sin(\pi y) h(x,t)\,\mathbf j.$$

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