Mathematical Equations – Bernoulli's Principle

In fluid dynamics, Bernoulli's principle states that an increase in velocity along a frictionless flow simultaneously causes a decrease in either the pressure or the potential energy of the fluid. Bernoulli's principle is named after the Dutch-Swiss mathematician Daniel Bernoulli. Bernoulli published this principle in his book Hydrodynamica in 1738.

This principle, sometimes referred to as the Bernoulli equation, can be applied to different types of fluid flow rates. In fact, there are different Bernoulli equations for different kinds of fluids. The simplest version of Bernoulli's principle applies to incompressible fluids (eg, most liquid fluids) and compressible fluids (eg, gases) that move at low Mach numbers.

Bernoulli's Principle and the Law of Conservation of Energy

Bernoulli's principle can be derived from the law of conservation of energy. Accordingly, in a constant flow, the sum of all mechanical energies of the fluid moving on a path is equal at every point on that path. This expression means that the sum of the kinetic and potential energy is constant. Therefore, any increase in the velocity of the fluid proportionally increases the dynamic pressure and kinetic energy of the fluid, while decreasing its static pressure and potential energy.

Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small-volume fluid moves horizontally from a high-pressure region to a low-pressure region, at the back; means there is more pressure than at the front. This exerts a net force on the fluid, causing it to accelerate along the streamline.

Bernoulli conducted experiments on fluids, and his equation is valid only for incompressible flows.

In many applications of Bernoulli's equation, the change in the term ρgz along the streamline is so small that it is negligible compared to other terms. For example, the change in height z along streamlines for an aircraft in flight is quite small and the term ρgz can be neglected. Thus, the above equation can also be used in the following simplified form:

Bernoulli's Equation Simplified Form

static pressure + dynamic pressure = total pressure

Every point in a steady flow has its own static pressure p and dynamic pressure q, independent of the fluid velocity at that point. Their sum p + q is defined as the total pressure p0. Bernoulli's principle can thus be summarized as "the total pressure along a streamline is constant".

Irrotational flow can be assumed in any situation where a large mass of fluid passes through a solid body. Examples are airplanes underway and ships moving in open water bodies. On the other hand, it is important to remember that Bernoulli's principle cannot be applied to the boundary layer or to flows in long pipes.

If the flow is non-rotating, the total pressure on each streamline is the same, and Bernoulli's principle can be summarized as "total pressure is constant throughout the flow".

If flow is stopped at a point on a streamline, that point is called the stagnation point and the total pressure at that point is equal to the stagnation pressure.

📩 27/09/2021 18:53