# Bernoulli’s principle

Daniel Bernoulli is, together with Leonhard Euler and Jean le Rond d’Alembert, one of the most prominent names in fluid dynamics, and the equation that holds his name is arguably the most famous in this field.

He was born in 1700 in Groningen, Netherlands, into a family of distinguished mathematicians. The family moved to Basel, Switzerland, in 1705. After studying philosophy and logic at the University of Basel, he studied medicine in Heidelburg and Strasbourg and earned a Ph.D. in anatomy and botany in 1721. In 1724 he published Exercitationes Mathematicae, which was his earliest mathematical work. In 1725 he moved to St. Petersburg, Russia, to join the academy where he wrote most of his famous book Hydrodynamica, finally published in 1738. Daniel returned to Basel in 1733, from where he contributed to mathematics and physics until his death on March 17, 1782.

Bernoulli studied how fluids flow. In particular, he was interested in the relationship between the speed at which blood flows and its pressure. His work became the basis for the development of a method of measuring blood pressure by sticking point-ended glass tubes directly into the arteries. This method was used for 170 years, but it was eventually superseded by another less painful. However, Bernoulli’s method can be found today in many applications, for example, in modern aircraft to measure a plane’s airspeed.

What is today known as Bernoulli’s equation does not appear in his book Hydrodynamica. In fact, during Bernoulli’s most productive years, partial differential equations had not yet been introduced into mathematics and physics; hence, he could not approach the derivation of Bernoulli’s equation. The introduction of partial differential equations to mathematical physics was due to d’Alembert in 1747. Although Bernoulli deduced that pressure decreases when the flow speed increases, and increases when the flow speed decreases, it was Euler who derived Bernoulli’s equation in its usual form in 1752:

p + ½ ρ V² = constant

Where:

• p is the pressure of the flow;
• ρ is the density of the fluid;
• V is the velocity of the flow;

The first term of the equation is sometimes also referred to as static pressure, and the second one as dynamic pressure:

p + ½ ρ V² = static pressure + dynamic pressure = constant

The physical significance of Bernoulli’s equation is what is known as Bernoulli’s principle. It states that when the velocity of a fluid increases, the pressure decreases, and when the velocity decreases, the pressure increases.

It is important to note that Bernoulli’s equation was derived by applying Newton’s second law(*), also known as the momentum equation, to inviscid (no viscosity, i.e., no friction), incompressible flows with no body forces (such as gravity). That means it applies to inviscid, incompressible flows only (it is not valid for compressible flows). Besides, the equation also assumes the flows are steady.

Looking closer at Bernoulli’s equation, we find that even though it was derived from the momentum equation, its dimensions are energy per unit volume. The equation states that for an inviscid, incompressible flow that does not undergo significant changes in height during its motion (i.e., the variation of potential energy due to the gravity force is neglected), the sum of the pressure and kinetic energy in a point remains constant:

p + ½ ρ V² = pressure energy + kinetic energy = constant

This fact means that Bernoulli’s equation is also the instantiation of the principle of the conservation of energy for inviscid, incompressible flows. Indeed, Bernoulli’s equation can also be derived from the energy equation that is a statement of the fundamental principle of conservation of energy.

Considering that the three fundamental equations of fluid dynamics are the momentum equation, continuity, and energy, what does it mean that Bernoulli’s equation can be interpreted as either Newton’s second law or an energy equation? It means that the energy equation is redundant for the analysis of inviscid, incompressible flows.

What about rotational and irrotational flows? Bernoulli’s equation is valid along streamlines for both rotational and irrotational flows. For rotational flows, the value of the constant will change from one streamline to another, while in the case of irrotational flows, the equation will also hold between any two points in the flow, not just on the same streamline. Rotational flow: p + ½ ρ V² = constant along a streamline. Irrotational flow: p + ½ ρ V² = constant throughout the flow.

## The pressure coefficient

The pressure coefficient, Cp, is commonly used to illustrate whether the static pressure increases, stays the same, or decreases at a certain point of a flow with respect to the pressure of the undisturbed flow ahead. It is defined as the ratio between the static pressure in a particular point minus the static pressure at the undisturbed flow and the dynamic pressure of the undisturbed flow:

Cp = (p – p) / ( ½ ρ V² )

Where:

• p is the pressure at the point of analysis;
• p is the pressure of the undisturbed flow;
• ρ is the density of the fluid;
• V is the velocity of the undisturbed flow;

Based on the equation above, we can state that the pressure coefficient at a point becomes negative if the static pressure at that point is lower than the static pressure at the undisturbed flow and becomes positive if the static pressure at the point is bigger than the static pressure at the undisturbed flow. When the static pressure is the same as the static pressure of the undisturbed flow, then the pressure coefficient is zero.

Using Bernoulli’s equation, we can say that for points belonging to the same streamline (for rational and irrotational, inviscid, incompressible flows) but also for points belonging to different ones (only in case of irrotational, inviscid, incompressible flows):

p + ½ ρ V² = p + ½ ρ V² = constant

Thus, the pressure coefficient expression above can also be written as follows:

Cp = 1 – ( V / V

Where:

• V is the velocity at the point of analysis;
• V is the velocity of the undisturbed flow;

Therefore, we can also state that the pressure coefficient at a point becomes negative if the flow accelerates at that point with respect to the undisturbed flow. Likewise, the pressure coefficient becomes positive if the flow slows down with respect to the undisturbed flow. The pressure coefficient is zero when the speed at the point of analysis and the speed of the undisturbed flow are equal.

(*) Newton’s second law states that the rate of change of momentum of a body over time is directly proportional to the force applied. Momentum is equal to the mass times the velocity ( m v ), and the rate of change of velocity over time is the acceleration. Thus, for objects and systems with constant mass, Newton’s second law can be re-stated in terms of acceleration: the acceleration of an object as produced by a force is directly proportional to the magnitude of the force and inversely proportional to the mass of the object. In mathematical notation: a = F / m.

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