Modern Physics

# Hydrodynamics: Laws, Applications and Solved Exercise

The hydrofoil is part of the hydraulic system that focuses on the study of fluid motion and fluid interactions moving its limits. As for its etymology, the origin of the word is in the Latin hydrodynamic term .

Hydrodynamics is named after Daniel Bernoulli. He was one of the first mathematicians to study hydrodynamics, published in 1738 in his hydrodynamic work . Moving fluids are found in the human body, as in the blood flowing through the veins or the air flowing through the lungs.

Fluids are also found in many everyday life and engineering applications; for example, in pipes for supplying water, gas, etc.

For all that, the importance of this branch of physics seems evident; its applications are not in vain in the field of healthcare, engineering and construction.

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

## Approximations

When studying fluids in motion, it is necessary to take a series of approaches that facilitate their analysis.

Thus, it is considered that fluids are incomprehensible and, therefore, their density remains unchanged in the face of pressure changes. Furthermore, the fluid energy losses due to viscosity are assumed to be negligible.

Finally, fluid flows are supposed to occur in the steady state; that is, the speed of all particles passing through the same point is always the same.

## Laws of hydrodynamics

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

### continuity equation

In fact, the continuity equation is the mass conservation equation. It can be summarized as follows:

Given a tube and having two sections S 1 and S 2 , there is a liquid circulating at velocities V 1 and V 2 , respectively.

If in the section connecting the two sections there are no contributions or consumptions, it can be said 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 as follows:

1 ∙ S 1 = v 2 ∙ S 2

### Bernoulli’s principle

This principle states that an ideal fluid (no friction or viscosity) circulating through a closed pipeline will always have a constant energy in its path.

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

2 ƿ 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 due to gravity and z is the height measured in the direction of gravity

### Torricelli 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 contained in a container behaves when it travels through a small orifice, under the effect of gravity.

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

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

r = √2gh

In that equation, V r is the average velocity of the liquid as it leaves the hole, g is the acceleration due to gravity, and h is the distance from the center of the hole to the plane of the liquid’s surface.

## applications

The applications of hydrodynamics 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, study their relief or know the thickness needed for the walls.

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

It has applications in aviation, in the study of conditions that favor the take-off of planes and in the design of ships’ hulls.

## Exercise solved

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

After the pressure at the lower level is known (P = 1.50 atm), determine the pressure at the top level.

The problem can be solved by applying Bernoulli’s principle, so you should:

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

As the speed is constant, it is reduced to:

1 + ∙ g ∙ z 1 = P + ∙ g ∙ z

By replacing and cleaning, you get:

1 = P + ∙ g ∙ z – ƿ ∙ g ∙ z 1

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