Spontaneous motion in a wind tunnel Suppose that a hollow truncated cone is placed in a wind tunnel with a steady wind speed $V$. The cone is placed in such a way that it's base of area $A_1$ faces the wind (rather than the other side of area $A_2$ for which $A_1>A_2$). Here's a contradiction:
Supposing that $V_2$ is the speed for which wind appears from that end of the cone of area $A_2$, using the conservation of mass law, we get $A_1V=A_2V_2$ and hence $V_2>V$. Now using the relation $\sum F=\dot{m}\Delta V$ (which results from the more general relation $\sum F=\frac{\partial}{\partial t}\int_{C.V}V.\rho (dv)+\int_{C.S}V.\rho (VdA)$) we see that a net thrust is generated which pushes the cone forward and against the wind, but this is impossible.
What's wrong with this argument? I've been giving it some thought but I can't figure out a way to solve it.
 A: Air entering the cone at $A_1$ is not traveling at speed $V$, but slower, due to the pressure within the cone.
If you want to see why, let $A_2$ shrink to a very small size.
A: About moving against the wind. This is not entirely impossible. Put a windmill on a boat and let it power the propeller. This worked in an experiment done on a lake in the Netherlands. An explanation was given by a professor in theoretical physics, based on the (non-linear) so-called windmill formula. The wind speed has to be in a certain range.
A: Your first assumption ($V_2 > V$) is correct, but you neglect the base areas when cutting out a slice of the tunnel for your integration. The forward-facing base sees the pressure of stagnating air at it's center, accelerating towards the edge to whatever speed is reached when air flows around the corner from the forward base to the mantle of the cone. The rear-facing and bigger area sees the pressure of separated flow (weak suction) over the full area. You need to account for the pressure difference as well, not just momentum, because your control areas include the bases of the cone.
If you move the control areas away from the cone, air expands behind the cone and the speed drops to it's old value (neglecting any viscous effects).
