Conservation of Mechanical Energy

If we assume that the flow out of the containers is frictionless and at least quasi-steady, then along any streamline in the flow, the so-called Bernoulli's equation of fluid dynamics applies:

Here, p is simply the local fluid pressure, is the mass density of the fluid, v is the local fluid speed, g is the acceleration due to gravity, and y is a vertical coordinate measured from some convenient reference point. Bernoulli's equation simply states that the change in kinetic energy along any streamline is due to the work done by gravity and pressure. This is a way of stating that the mechanical energy of a small fluid element is conserved when there is no friction. The assumptions necessary to apply this equation never hold exactly, but its results are frequently sufficient for estimates or comparison purposes. And comparison is exactly what we have in mind here.

Let us choose the origin for our coordinate system to be at the opening of the drain at the bottom of the tank, as shown in a sketch of the cone. Consider a streamline which begins at the free surface of the liquid and extends all the way through the drain opening. Let's call the upper end of the streamline, the point on the free surface, a, and the lower end b. Bernoulli's equation written between these two points yields:

But, and both and are just atmospheric pressure, which is the same at the top surface of the liquid and at the outlet and will thus have no net effect. So we are left with only

since the mass density is constant and thus divides out. Solving for the drain velocity, gives

Thus we have a tentative model for the drain velocity, which was obtained from a statement of the conservation of mechanical energy of the fluid moving without friction along a streamline. You probably recognize this formula as the one which describes the velocity of an object in free fall.