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Fluid dynamics, or fluid dynamics, is a discipline of physics that studies the movement of fluids, that is, liquids and gases, including the interaction between two fluids and that of a fluid with containment or boundary materials. Fluid dynamics is one of the two branches of fluid mechanics, the other being the static or rest study of fluids, that is, fluid statics.
fluid dynamics
Fluid dynamics is a macroscopic model of matter and its interactions. In this context the term “fluid” refers to both liquids and gases; Let’s remember that the difference is that a liquid, or non-compressible fluid, does not change its volume as pressure increases, while a gas, a compressible fluid, decreases its volume as pressure increases. The fundamental hypothesis is that a fluid is a continuous material in the space it occupies, and therefore its microscopic composition, its atoms and molecules or discontinuous components are not considered.
Fluid dynamics is also called fluodynamics; in the case of incompressible fluids, liquids, it is called hydrodynamics, and aerodynamics when compressible fluids are studied, gases. Magnetohydrodynamics studies the dynamics of electrically conductive fluids interacting with electric and magnetic fields. The state of matter called plasma at low temperatures can also be studied with fluid dynamics models.
As in any physical model, fluid dynamics is structured on a series of hypotheses and principles, some of them more general, which correspond to fluid mechanics. One of the first principles that were postulated historically is the one related to buoyancy ; Archimedes’ principle, proposed by the ancient Greek physicist and mathematician in the 3rd century BC. Archimedes’ principle postulates that a body partially or completely submerged in a liquid at rest experiences an upward vertical force equal to the weight of the liquid displaced by the body. As it appears from the postulate, the principle corresponds to the statics of fluids.
When studying a fluid in motion, pressure, velocity, and density are three crucial variables in fluid dynamics. Density is often represented by the symbol ρ , velocity by v , and pressure by p .
Bernoulli’s principle
Bernoulli’s principle is one of the principles of fluid dynamics, postulated by Daniel Bernoulli in 1738. The principle is postulated for an ideal fluid, without viscosity, and says that a fluid circulating in pipes in a closed circuit has an energy that Remains constant. The different forms of energy, kinetic and potential, are balanced to keep the total energy constant . The pressure decreases when the velocity of the fluid increases. Bernoulli’s principle is valid when there is no energy loss in other physical processes, or they are very small and can be neglected, such as heat radiation, viscous forces, or turbulence.
Bernoulli’s principle was expressed mathematically by Leonhard Euler in the so-called Bernoulli equation . The equation expresses the conservation of the sum of the three forms of energy at any point of the fluid in the system; the kinetic energy, the energy of the flow expressed by the pressure and the potential energy.
( ρ .v 2 /2) + p + ρ .gz = k
where ρ is the density of the fluid, v is its speed and p is its pressure; g is the acceleration of gravity and z is the height of the point of the system that is considered with respect to a reference level. The sum of these three forms of energy is equal to a constant k at any point in the system, and therefore this constant can be equalized at two different points a and b, being able to relate the hydrodynamic variables as follows.
( ρ .v a 2 /2) + p a + ρ .gz a = ( ρ .v b 2 /2) + p b + ρ .gz b
Viscosity and the Newtonian fluid
Viscosity is a fundamental parameter of fluids. Viscosity is defined as the resistance of the fluid to its deformation or flow. Two types of viscosity are differentiated: the dynamic viscosity μ , and the kinematic viscosity ν = μ / ρ .
Along with the definition of a viscous fluid, another important concept in fluid dynamics is that of a Newtonian fluid. They are the fluids in which the viscosity can be considered constant at a certain pressure and temperature, and said viscosity does not depend on other variables of the fluid, such as forces or speeds. Newtonian fluids are the easiest to study, with water and oils being the most common examples. This hypothesis allows us to establish a linear relationship between the force to which a fluid is subjected to move between two surfaces, and the fluid flow velocity. The typical case, shown in the following figure, is that of a surface A moving at a speed v over another surface (plane B) separated by a distance y, distance occupied by a Newtonian fluid of viscosity μ .
If the fluid is Newtonian, the force F that opposes the movement is F = μ .A.(v/y) . In this way, if there is a fluid that moves on a surface applying a constant force to it, a linear fluid velocity variation is obtained with the distance to the fixed surface, where the velocity of the fluid is zero.
The flow
Given that fluid dynamics consists of the study of fluids in motion, first of all we must define a fundamental parameter that allows us to approach this analysis. This parameter is the flow , which is the amount of fluid that moves through a certain surface area per unit of time . The concept of flow is used to describe a wide range of situations involving fluids: air blowing through a hole, or liquid moving through a pipe or over a surface.
As already stated, a compressible fluid, typically a gas, is one that decreases in volume with increasing pressure, that is, when compressed. It is possible to reduce the section of an air duct and maintain the same flow by transporting the air at the same speed; For this, the pressure of the system will have to be increased to contain the same mass of air in a smaller volume. When a compressible fluid is in motion there can be spatial variations in its density. By contrast, an incompressible fluid in motion does not change its density at any point in the system.
The flow of a fluid can have various characteristics, depending on the system being studied and its conditions. If the flow does not change with time, it is said to be constant. And if the flow is in a steady state, this implies that the properties of the fluid, such as speed or density at each point, do not vary with time either. It could happen that you have a system in which there is a constant flow but the properties of the fluid vary, in which case the flow would not be steady. On the other hand, the inverse statement is correct: all steady-state flux implies constant flux. A very simple case is water flowing through a pipe driven by a pump. Flow, the amount of water that passes through a section of pipe per unit of time (liters per minute, for example), is constant. Besides,
Conversely, if some property of the fluid varies with time at some point in the system, we have unsteady flow or a transient state of flow. Rain flowing down a gutter during a storm is an example of unsteady flow; The amount of water that passes through a section of the gutter per unit of time varies with the intensity of the rain. Systems in unstable or transitory states are more difficult to study than stationary ones, since variations over time make approaching the situation more complex.
laminar flow and turbulent flow
A first approximation to the idea of laminar flow is to think of the smooth movement of a fluid, like oil flowing slowly on a surface; In contrast, in a turbulent flow, the fluid becomes chaotically mixed within it as the macroscopic volume moves. The following figure schematically shows how the laminar and turbulent flow would be in a fluid that moves in a pipe, where the arrows symbolize the trajectory of small volumes of fluid. According to this definition, a turbulent flow is a state of unstable flow. However, with a turbulent flow you can have a constant flow, because although the fluid mixes within it as it moves, it may be that the total amount of fluid that crosses a surface per unit of time does not vary with time. time.
In both types of flow eddies, vortices and recirculations can be produced. The difference between both flows lies in the chaotic movement of the small volumes of fluid, independently of the macroscopic movement.
The physical parameter that determines whether a flow is laminar or turbulent is the Reynolds number, Re . This parameter was proposed by the Irish engineer and mathematician Osborne Reynolds in 1883. Reynolds’ research work and those developed by the Irish physicist and mathematician George Gabriel Stokes and the French Claude Louis Naiver in the second half of the 19th century allowed the development of the expressions fundamental mathematics of fluid dynamics, the Navier-Stokes equations, valid for Newtonian fluids.
The Reynolds number expresses a relationship between the forces of inertia in a fluid and the forces associated with viscosity. In the case of a liquid flowing through a straight pipe, the Reynolds number has the following expression
Re = ρ .vD/ μ
where ρ is the density of the fluid, μ is its viscosity, v is its velocity in the pipe, and D is the diameter of the pipe.
Although the expression of the Reynolds number depends on the system being studied, it is a dimensionless parameter, without units, and therefore the interpretation of its value is independent of the characteristics of the system. High values of Re correspond to turbulent flow, while low values correspond to laminar flow. The importance in determining this flow characteristic lies in the fact that both the flow properties and the mathematical model with which to study the system are different.
Flow in a pipe and in an open channel
Two systems involving moving fluids that are interesting to compare are flow through a pipe and flow in an open channel. In the first case, the fluid moves contained within the rigid limits of a containment, such as water flowing inside a pipe or air moving inside a conduit. In the case of flow in an open channel, there is a section of the flow that is not in contact with a rigid surface, that is, it is open. This is the case of a river, of rainwater that flows through a gutter or an irrigation channel. In these examples the surface of the water that is in contact with the air is the free surface of the flow.
Flow in a pipe is driven by pressure exerted on the fluid by a pump or other mechanism, or by gravity. But in open channel systems the main force acting is gravity. Drinking water supply systems usually use the force of gravity to distribute water previously stored in tanks elevated above the level of houses. The difference in height generates a pressure on the fluid given by the force of gravity on the free surface of the water stored in the tank.
applications of fluid dynamics
Two thirds of the Earth’s surface is covered by water, and the planet is covered by a layer of gases, the atmosphere. And these fluids are mostly in motion. Therefore, fluid dynamics is closely related to life and nature, in addition to the multiple applications in the technological developments of humanity. Let’s look at four branches of science and technology that are based on applications of fluid dynamics.
Oceanography, meteorology, and the climate sciences . The atmosphere is a mixture of gases in motion that can be analyzed with fluid dynamics models, and is the object of study in atmospheric sciences. Like the study of ocean currents, crucial for understanding and predicting weather patterns , which can also be studied with fluid dynamics models.
Aeronautics . The behavior of airplanes, in all its varieties and in the different aspects in which it is necessary to study them, is the subject of study of compressible fluid dynamics.
Geology and Geophysics . The study of the movement of tectonic plates and volcanic processes is related to the movement of magma, the fluid matter that flows in the depths of the Earth. The application of fluid dynamic models is fundamental in the study of these processes.
Hematology and hemodynamics . The behavior of fluids is essential in all biological processes, both at the cellular level and in the physiology of organisms, in solutions and suspensions, such as blood. Fluid dynamics allows the development of models to study these essential fluids for life.
Sources
Peñaranda Osorio, Caudex Vitelio. Fluid mechanics. ECOE Editions, 2018.
Mott, Robert. Fluid Mechanics . Pearson Education, 6th edition, Mexico, 2006.
