HYDRODYNAMICS: LAWS, APPLICATIONS AND SOLVED EXERCISE - PHYSICAL - 2025

The hydrodynamics is part of the hydraulics that focuses on the study of the movement of fluids and interactions of fluids moving its limits. As for its etymology, the origin of the word is in the Latin term hydrodynamics.

The name of hydrodynamics is due to Daniel Bernoulli. He was one of the first mathematicians to carry out hydrodynamic studies, which he published in 1738 in his work Hydrodynamica. Fluids in motion are found in the human body, such as in the blood that circulates through the veins, or the air that flows through the lungs.

Fluids are also found in a multitude of applications both in everyday life and in engineering; for example, in water supply pipes, gas pipes, etc.

For all this, the importance of this branch of physics seems evident; not for nothing its applications are found in the fields of health, engineering and construction.

On the other hand, it is important to clarify that hydrodynamics as a science part of a series of approaches when dealing with the study of fluids.

Approaches

When studying fluids in motion, it is necessary to carry out a series of approximations that facilitate their analysis.

In this way, it is considered that fluids are incomprehensible and that, therefore, their density remains unchanged under pressure changes. Furthermore, the viscosity fluid energy losses are assumed to be negligible.

Finally, it is assumed that fluid flows occur in a steady state; that is, the speed of all the particles that pass through the same point is always the same.

Laws of hydrodynamics

The main mathematical laws that govern the motion of fluids, as well as the most important quantities to consider, are summarized in the following sections:

Continuity equation

Actually, the continuity equation is the equation for the conservation of mass. It can be summarized like this:

Given a pipe and given two sections S ₁ and S ₂, we have a liquid circulating at speeds V ₁ and V ₂, respectively.

If the section connecting the two sections does not produce inputs or consumption, then it can be stated that the amount of liquid that passes through the first section in a unit of time (which is called mass flow) is the same that passes through the second section.

The mathematical expression of this law is the following:

v ₁ ∙ S ₁ = v ₂ ∙ S ₂

Bernoulli's principle

This principle establishes that an ideal fluid (without friction or viscosity) that is in circulation regime through a closed conduit will always have a constant energy in its path.

Bernoulli's equation, which is nothing other than the mathematical expression of his theorem, is expressed as follows:

v ² ∙ ƿ / 2 + P + ƿ ∙ g ∙ z = constant

In this expression v represents the velocity of the fluid through the section considered, ƿ is the density of the fluid, P is the pressure of the fluid, g is the value of the acceleration of gravity and z is the height measured in the direction of the gravity.

Torricelli's Law

Torricelli's theorem, Torricelli's law or Torricelli's principle consists of an adaptation of Bernoulli's principle to a specific case.

In particular, it studies the way in which a liquid enclosed in a container behaves when it moves through a small hole, under the force of gravity.

The principle can be stated in the following way: the speed of displacement of a liquid in a vessel that has an orifice is the one that any body would have in free fall in a vacuum, from the level at which the liquid is to the point where which is the center of gravity of the hole.

Mathematically, in its simplest version it is summarized as follows:

V _r = √2gh

In this equation V _r is the average velocity of the liquid when it leaves the hole, g is the acceleration of gravity and h is the distance from the center of the hole to the plane of the surface of the liquid.

Applications

Hydrodynamic applications are found both in everyday life and in fields as diverse as engineering, construction and medicine.

In this way, hydrodynamics is applied in the design of dams; for example, to study the relief of the same or to know the necessary thickness for the walls.

Similarly, it is used in the construction of canals and aqueducts, or in the design of the water supply systems of a home.

It has applications in aviation, in the study of the conditions that favor the take-off of airplanes and in the design of ship hulls.

Exercise resolved

A pipe through which a liquid with a density of 1.30 ∙ 10 ³ Kg / m ³ circulates runs horizontally with an initial height z ₀ = 0 m. To overcome an obstacle, the pipe rises to a height of z ₁ = 1.00 m. The cross section of the pipe remains constant.

Knowing the pressure at the lower level (P ₀ = 1.50 atm), determine the pressure at the upper level.

You can solve the problem by applying Bernoulli's principle, so you have to:

v ₁ ² ∙ ƿ / 2 + P ₁ + ƿ ∙ g ∙ z ₁ = v ₀ ² ∙ ƿ / 2 + P ₀ + ƿ ∙ g ∙ z ₀

Since the velocity is constant, it reduces to:

P ₁ + ƿ ∙ g ∙ z ₁ = P ₀ + ƿ ∙ g ∙ z ₀

By substituting and clearing, you get:

P ₁ = P ₀ + ƿ ∙ g ∙ z ₀ - ƿ ∙ g ∙ z ₁

P ₁ = 1.50 ∙ 1.01 ∙ 10 ⁵ + 1.30 ∙ 10 ³ ∙ 9.8 ∙ 0- 1.30 ∙ 10 ³ ∙ 9.8 ∙ 1 = 138 760 Pa

References

1. Hydrodynamics. (nd). On Wikipedia. Retrieved on May 19, 2018, from es.wikipedia.org.
2. Torricelli's theorem. (nd). On Wikipedia. Retrieved on May 19, 2018, from es.wikipedia.org.
3. Batchelor, GK (1967). An Introduction to Fluid Dynamics. Cambridge University Press.
4. Lamb, H. (1993). Hydrodynamics (6th ed.). Cambridge University Press.
5. Mott, Robert (1996). Applied Fluid Mechanics (4th ed.). Mexico: Pearson Education.