Vorticity and the flow of a river [closed]

Q. a) If there's an obstacle at the bottom of a river why does one observe an indentation on the surface of the river at the position of the obstacle?

Attempted solution: The water near the bottom of the river is forced to flow quicker over the obstacle which in turn results in the pressure decreasing at this position. Since the atmospheric pressure on the surface of the river remains constant, there is a greater net downward pressure on the river at the position of the obstacle and thus an indentation is observed.

Q. b) If a ball is floating on the surface of the river and is rotating with the local vorticity, why does the rotation slow down when the ball flows into the indent in the river?

Attempted Solution: Since vorticity is given by $$\vec \omega = \nabla \times\vec v,$$ and we assume a 2d flow (i.e no z-direction), $$\vec\omega = {\left(\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \right)}\vec z.$$ Initially $\vec v_y$ = 0 (that is before the indentation), so $|\vec \omega| = \frac{\partial v_x}{\partial y}$ and in the indentation $|\vec \omega|= {\left(\left(\frac{\partial v_y}{\partial x}\right)^2 + \left(\frac{\partial v_x}{\partial y}\right)^2 \right)^{1/2}}.$ For the vorticity to be conserved, the rotation must slow down since we have two contributing terms for when the ball is in the indent.

Is this along the right track of thinking?

closed as off-topic by John Rennie, Jon Custer, Gert, heather, DilithiumMatrixNov 2 '16 at 21:12

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