# Common approximations for fluid outflow beyond Torricelli's law

Last month, a puzzle here in physics.stackexchange asked to calculate the dynamics of the outflow in a loose (over rails, no friction) tank using Torricelli's law as a simplification. The simplification turned out to be problematic; one would prefer some solution where the height $h(0)=h_0$ starts decreasing with the usual condition $h'(0)=0$.

A first try to generalize is to use the potential energy stored in the tank to provide kinetic energy both to the CoM of the fluid and to the outflow. Consider for instance two horizontal equal nozzles, so that the CoM only goes down vertically. While this situation drives to a manageable second order differential equation, I am not sure if it is good enough. Particularly it seems that when we close the nozzle(s), so that the CM stops, this kinetic energy should change to be stored as internal energy, does it? We could imagine some configuration where we open the nozzle an interval $\Delta t$, close it, open it at the next interval, &c, so that we force this conversion to internal, not CoM, energy.

So, is there some generic condition where we can grant that all the energy goes either to the outflow or to the fluid CoM? If not, there is some usual parameter measuring how much of the energy goes to CoM and how much goes to internal (vorticity or so)?

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Reynolds number? –  user346 Jan 27 '11 at 15:29
@space_cadet I am worried mainly about laminar cyclic currents and some macroscopic vortex. And perhaps viscosity has a role to create some kinetic energy invisible to the CoM. –  arivero Jan 28 '11 at 12:56
respect to the puzzle, probably the trick is use a continuous funtion to open the nozzle (but you are invited to offer your own solutions there in the question physics.stackexchange.com/questions/1683/… ). The question here is more general, how we control and parametrise the distribution of energies in a fluid? –  arivero Jan 29 '11 at 17:39