Study Material

# Hydrodynamics

## What is hydrodynamics?

The hydrodynamics is the area of fluid mechanics that deals with the study of moving fluids. Its name derives from the Greek “hydro”, which means water , but hydrodynamics is not limited to studying liquids, but also gases.

It is one of the oldest known disciplines, and in its beginnings it almost always focused on hydraulics, which is the study of liquids and in particular water, both at rest and in motion.

It is known that the inhabitants of ancient Mesopotamia practiced the construction of irrigation systems for crops. And likewise, the ancient Egyptians learned to control the waters of the Nile to their advantage.

In the science of fluids, the Roman Empire stood out, due to the degree of sophistication reached by its techniques, thanks to which they built complex systems of aqueducts, baths and irrigation. Some of his works still survive today.

However, for a long time hydrodynamics did not have an adequate mathematical foundation. It was in the 18th century that it received its definitive boost with the works of the Swiss scientist Daniel Bernoulli (1700-1782).

Bernoulli applied the principle of conservation of energy to fluids in motion and derived an expression that governs them. The so-called Bernoulli principle , the foundation of hydrodynamics, is explained in more detail shortly .

## What does hydrodynamics study?

Hydrodynamics studies fluids in motion and their interactions, understanding by fluid not only liquids, but also gases.

Hydraulics is the specific area that deals with liquids and their interactions with different forces, while aerodynamics focuses on the interaction between a gaseous medium and the solid objects that move inside it.

### Ideal fluids

The movement of real fluids can be quite complicated to describe, however, there are initial assumptions that simplify some aspects, achieving a good understanding of various phenomena.

Hydrodynamics starts from the study of ideal fluids. In this way, assume that a fluid is:

• Incompressible, which means that its density is not altered.
• Stationary, so its velocity is the same at a given point and moment.
• Not viscous, that is, it lacks internal friction.
• Irrotational, it does not present eddies or whirlpools.

Once the model for the ideal fluid dynamics has been established, the concept of viscosity is introduced, which is the internal friction between the layers of the fluid. With this, the approximation to a real fluid is better.

Viscosity causes a loss of pressure along the tube through which the fluid travels, and the physical model that describes these effects was discovered by the 19th century French physician, JL Poiseuille (1799-1869), who conducted numerous studies. about the movement of an important viscous fluid: blood.

## Principles of hydrodynamics

The two fundamental principles of hydrodynamics are:

• Conservation of mass
• Conservation of energy

The first principle is expressed through the continuity equation and the second, through the Bernoulli equation .

### Continuity equation

There is a pipe through which a fluid circulates without losses or contributions. This means that the pipe is not leaking and no fluid is added to the amount that is circulating.

A portion of fluid that circulates through the narrow part of the pipe, in light blue, is the same that then passes through the wide part, also in light blue.

Since the mass is conserved, the portion circulating through the section of cross-section A 1 is the same as that circulating through the other section of cross-section A 2 :

Since mass is the product of density ρ and volume V:

ρ ∙ V 1 = ρ ∙ V 2

Where V 1 is the volume in section A 1 and V 2 the volume in section A 2 .

The volume is the cross-sectional area times the length of the span s (see the figure above):

ρ ∙ (A 1 ∙ s 1 ) = ρ ∙ (A 2 ∙ s 2 )

In turn, the length of the section is the product of the fluid velocity and the time interval:

s = v ∙ Δt

Also, since the density of the fluid remains constant (incompressible fluid), it can be canceled, as can time:

1 ∙ v 1 ∙ Δt = A 2 ∙ v 2 ∙ Δt

The continuity equation is finally obtained:

1 ∙ v 1 = A 2 ∙ v 2

The product of the cross-sectional area and the velocity of the fluid is called the flow rate and is usually denoted by Q:

Q = A ∙ v

The units of Q are cubic meters / second in the International System of Units, so the flow is also interpreted as volume per unit of time.

### Bernoulli’s equation

Bernoull’s equation is a consequence of applying the conservation of energy to a fluid. It has the sum of the following terms:

• Pressure P
• Kinetic energy per unit volume: ρv 2 / 2g
• Potential energy per unit volume: ρgh

It is constant, therefore, its value is maintained at all points of the route. Later:

P + ρv 2 / 2g + ρgh = constant

Where v is the velocity of the fluid, g the acceleration of gravity and h the height with respect to the reference level, as it appears in the figure above.

## Hydrodynamic applications

### Torricelli’s theorem

Torricelli’s theorem is derived from Bernoulli’s principle and states that the speed v with which a fluid leaves a small hole is the same as a body has when it falls by gravity from a height h:

### Siphon

The siphon is used to transfer liquids, and consists of a hose or tube bent in an uneven U-shape, with the shortest side immersed in the container where the liquid is, and the longest side in the destination container.

The level of the original container must be above the level of the liquid outlet in the tube, and it must be ensured that the hose is completely filled with liquid, without air bubbles.

As the part of the fluid that is on the longest side is heavier, it causes the liquid to behave like a chain that slides on a pulley, pouring into the arrival container (at a lower height).

### Pitot meter

It consists of a small tube that is usually used in airplanes, to measure its speed with respect to the air. It is also used to measure the speed of water flow in a pipe or that of river currents.

## Examples of hydrodynamics in daily life

The movement of fluids occurs very frequently in daily life, whether in liquids or gases. The following examples demonstrate how important fluid movement is even for the maintenance of life:

### Household plumbing systems

In the houses there is a system of pipes that transports the white water, separated from the black water. Sometimes piping systems are also built for domestic gas, used for cooking and heating.

### The car’s cooling system

When the car’s engine is running, a lot of heat is generated. To remove it, in most models, the engine is cooled with a fluid, which can be water or a coolant with additives to prevent corrosion and optimize cooling.

The liquid is passed through a system of very thin ducts: the radiator, by means of a pump and is cooled with the help of a stream of air driven by a fan. The coolant, which is directed towards the engine, extracts excess heat and transports it to the radiator, in back and forth cycles while the engine is running.