Bernoulli's Principle

Before we delve right into the physics behind Bernoulli's principle, it's best to review the continuity equation. For an ideal fluid, if the area through which a fluid flows decreases, its speed increases which is elaborated above. However, remember that gases, like air, are also fluids so the same fluid mechanics that apply to liquids like water roughly apply to gases. Keep this in mind because it's an important cornerstone behind one of the biggest applications of Bernoulli's Principle: airplane wings.

Bernoulli's principle states that if the speed of a fluid increases, its internal pressure or potential energy decreases or both do. The explanation for why the potential energy could decrease is quite simple and is just a matter of conservation of mechanical energy.

For reference, if the mechanical energy of a system is conserved, the sum of its potential(commonly gravitational potential) energies and kinetic energy is constant. That means if you increase the speed of a fluid, thereby increasing its kinetic energy, its potential energy must decrease to keep mechanical energy conserved.

Similarly, the decrease in pressure can be explained simply through Newton's 2nd Law. If we take a column of fluid moving horizontally through a pipe and the fluid goes from a place of high pressure to a place of low pressure, there will be more pressure behind the fluid than in front, which means the net force on the fluid is in the direction of its motion. This means the fluid will accelerate as it goes from high pressure to low pressure which therefore proves that if the speed of a fluid increases, its internal pressure will decrease.

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If we combine this with the concept behind fluid flow continuity, we get another important distinction. If you see an increase in the velocity of a fluid, that means the area it flows through must be smaller by the continuity equation. Since an increase in fluid velocity also means lower pressure, this means that increasing the area will increase the pressure and vice versa if you decrease either. This may seem counterintuitive but if this weren't the case, then the velocity would not increase upon a lowering of pressure.

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In summary, Bernoulli's principle states that the velocity of a fluid is inversely proportional to both the area the fluid passes through and the energy applied to the fluid through pressure.

Let's say we have a pipe(like the one shown below) with an ideal incompressible fluid flowing through it. If we assume the conditions are perfect, the fluid's energy is constant as frictional forces aren't really a factor at all. Since the energy is constant, we can mathematically state the following to be true:

This final equation is very similar to the equation for the conservation of mechanical energy, except you have pressure instead of work and density instead of mass. The reasons for this are given above mathematically. Conceptually, this also makes sense because the density and pressure of fluid are more useful to us than the work done on a fluid and its mass. Density allows us to effectively distinguish different fluids and how they interact without worrying about their volumes. This also means the terms that aren't pressures are not the formulae for gravitational potential energy and kinetic energy. The quantity given by ρgh is just the potential energy density of the fluid or its potential energy per unit volume. Similarly, the term similar to kinetic energy is the kinetic energy density or the kinetic energy per unit volume.

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As seen, this equation proves Bernoulli's principle above as the sum of kinetic energy density, potential energy density, and pressure is constant. If any one of them decreases, one of the others has to increase. Usually, the height variation of a fluid pipe is fixed so the potential energy density won't inherently change which is why changes in fluid velocity usually cause changes in strictly fluid pressure.

Torricelli's Equation is a simplified version of Bernoulli's equation for reservoirs with little fluid pressure. Its accompanying theorem is a little more interesting but before we go into it, let's mathematically discuss the equation itself for this reservoir.

This equation can be made through many simplifying assumptions and was soon discovered to be a very special case of Bernoulli's equation.

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For Torricelli's theorem, Evangelista Torricelli made some other very interesting observations for the trajectory of the fluid from the hole. All the potential trajectories of the fluid will fall "lower" than a limiting envelope. These paths all follow parabolic trajectories with the water's surface being the observed directrix.

The power of a fluid is just like mechanical power: it's the change in the fluid's energy per unit of time, measured in Watts. This is mathematically defined as follows: