Hydrodynamics

The first attempts to study the resistance of the medium to body movement were made by Leonardo da Vinci and Galileo Galilei . It is generally accepted that Galileo conducted experiments on dropping balls of various densities from the Leaning Tower of Pisa, this experience is described in the educational literature and therefore is known to everyone from school (there is no reliable information confirming the conduct of this experiment by Galileo today). In 1628, Benedetto Castelli published a small work in which he very well for his time explained several phenomena when fluid flows in rivers and canals. However, the work contained an error, since it assumed the rate of fluid leakage from the vessel in proportion to the distance of the hole to the water surface. Torricelli noticed that the water flowing from the fountain rises to a height of the order of the water level of the supply reservoir. Based on this, he proved theorem on the proportionality of the rate of leakage to the square root from the distance from the hole to the surface of the liquid. The theorem was experimentally verified on water flowing from various nozzles. Edmé Mariotto, in a work published after his death, first explained the discrepancy between theory and experiment by taking into account the effects of friction. In the work of Isaac Newton's “philosophie naturalis principia mathematica”, the concepts of viscosity and friction were used to explain the decrease in speed of running water. Also, in the work of Newton, Mariotto's ideas about the flow of water as a set of rubbing threads developed. This theory is already comparable with the modern theory of motion transfer in liquids.

After Newton's publication of his works, scientists around the world began to use its laws to explain various physical phenomena. After 60 years, Leonard Euler received an analog of Newton’s second law for liquids. In 1738, Daniel Bernoulli published a work explaining the theory of fluid motion. He used two assumptions: the surface of the fluid flowing from the vessel always remains horizontal and the fact that the lowering speed of the water layers is inversely proportional to their width. In the absence of demonstrations of these principles, the theory of trust has not received.

Colin Maclaurin and John Bernoulli wanted to create a more general theory, depending only on the fundamental laws of Newton. The scientific community found their methods not rigorous enough. The theory of Daniel Bernoulli met resistance from Jean Leron Dalamber , who developed his theory. He applied the principle obtained by Jacob Bernoulli , who reduced the laws of motion of bodies to the law of their equilibrium. D'Alembert applied this principle to describe the movement of fluids. He used the same hypotheses as Daniel Bernoulli, although his calculus was built in a different manner. He considered at every moment the motion of the liquid layer composed of motion at the last moment in time and the motion that he lost. The laws of equilibrium between loss and loss of motion have given an equation representing the equation of fluid motion. It remained to express the equations of motion of a fluid particle in any given direction. These equations were found by D'Alembert from two principles: a rectangular channel, allocated in a mass of liquid in equilibrium, is itself in equilibrium, and a part of a fluid moving from one place to another retains the same volume if it is incompressible and changes the volume taking into account laws of elasticity, otherwise. This method was adopted and perfected by Leonard Euler. The solution of the problem of the motion of fluids was carried out using the method of partial derivatives of Euler. This calculus was first applied to the movement of water by D'Alembert. The method allowed us to present the theory of fluid motion in a formulation that is not limited by any special assumptions.

Ideal environment

From the point of view of mechanics , a fluid is a substance in which there are no tangential stresses in equilibrium. If the fluid motion does not contain sharp velocity gradients, then the tangential stresses and the friction caused by them can be neglected when describing the flow. If, in addition, the temperature gradients are small, then the thermal conductivity can be neglected, which is the approximation of an ideal fluid . In an ideal fluid, therefore, only normal stresses, which are described by pressure, are considered . In an isotropic fluid, pressure is the same in all directions and is described by a scalar function.

Hydrodynamics of laminar flows

The hydrodynamics of laminar flows studies the behavior of regular solutions to the equations of hydrodynamics, in which the first derivatives of velocity with respect to time and space are finite. In some cases with special geometry, the equations of hydrodynamics can be solved exactly . Some of the most important tasks of this section of hydrodynamics:

• stationary flow of an ideal incompressible fluid under various boundary conditions
• stationary flow of a viscous fluid , Navier-Stokes equations
• waves on the surface of an ideal incompressible fluid and other non-stationary phenomena
• laminar flow around finite bodies
• flows in various immiscible liquids, tangential discontinuities and their stability
• jets , drops and other flows of finite dimensions

Turbulence

Turbulence is the name of such a state of a continuous medium, gas, liquid, their mixtures, when chaotic fluctuations of instantaneous values ​​of pressure , velocity , temperature , density relative to some average values ​​are observed in them, due to the nucleation, interaction, and disappearance of vortex movements of different scales in them, and also linear and nonlinear waves, solitons, jets. Their nonlinear vortex interaction and propagation in space and time occur. Turbulence occurs when the Reynolds number exceeds the critical.

Turbulence can also occur when the continuity of the medium is disturbed, for example, during cavitation (boiling). When capsizing and breaking the waves of the surf, a multiphase mixture of water, air, foam. Instantaneous environmental parameters become chaotic.

There are three zones of turbulence, depending on the transition Reynolds numbers: the zone of smooth-wall friction, the transition zone (mixed friction) and the zone of hydraulically rough pipes (the zone of quadratic friction). All main oil and gas pipelines are operated in the area of ​​hydraulically roughened pipes.

The turbulent flow, apparently, can be described by a system of nonlinear differential equations. It includes the Navier - Stokes equations , continuity and energy.

Turbulence modeling is one of the most difficult and unsolved problems in hydrodynamics and theoretical physics. Turbulence always occurs when certain critical parameters are exceeded: the speed and size of the streamlined body or a decrease in viscosity . It can also occur under strongly non-uniform boundary and initial conditions at the boundary of a streamlined body. Or, it can disappear with a strong acceleration of the flow on the surface, with a strong stratification of the medium. Since turbulence is characterized by random behavior of instantaneous values ​​of velocity and pressure, temperature at a given point in a liquid or gas, this means that under the same conditions a detailed picture of the distribution of these quantities in a liquid will be different and will almost never be repeated. Therefore, the instantaneous velocity distribution at various points of the turbulent flow is usually not of interest, and the averaged values ​​are important. The problem of describing hydrodynamic turbulence lies, in particular, in the fact that, so far, it is not possible to predict, based only on the equations of hydrodynamics, when exactly should the turbulent regime begin and what exactly should happen without experimental data. On supercomputers, only certain types of flows can be modeled. As a result, one has to be content with only a phenomenological, approximate description. Until the end of the 20th century, two results describing the turbulent motion of a fluid were considered unshakable - the von Karman-Prandtl “universal” law on the distribution of the average local fluid flow velocity (water, air) in smooth pipes at high Reynolds numbers and the Kolmogorov-Obukhov theory on the local structure turbulence.

A significant breakthrough in the theory of turbulence at very high Reynolds numbers is associated with the works of Andrei Nikolaevich Kolmogorov of 1941 and 1962, which established that for a certain range of Reynolds numbers the local statistical structure of turbulence is universal, depends on several internal parameters and does not depend on external conditions.

Supersonic fluid dynamics

This section studies the behavior of flows at their velocities near or exceeding the speed of sound in a medium. A distinctive feature of this mode is that shock waves occur during it. In certain cases, for example, during detonation , the structure and properties of the shock wave are complicated. Also interesting is the case when the speeds of the currents are so high that they become close to the speed of light . Such flows are observed in many astrophysical objects, and relativistic hydrodynamics studies their behavior.

Heat and Mass Transfer

Often the flow of liquids is accompanied by an uneven distribution of temperature (cooling of bodies in a fluid, flow of hot fluid through pipes). Moreover, the properties of the liquid ( density , viscosity , thermal conductivity ) may themselves depend on the local temperature. In this case, the heat distribution problem and the fluid motion problem become related. The additional complexity of such tasks is that often the simplest solutions become unstable ...

Geophysical Hydrodynamics

It is devoted to the study of the phenomena and physical mechanisms of natural large-scale turbulent flows on a rotating planet (atmospheric dynamics, current dynamics in the seas and oceans, circulation in the liquid core, the origin and variability of the planetary magnetic field).

Magnetic Hydrodynamics

Describes the behavior of electrically conductive media (liquid metals , electrolytes , plasma ) in a magnetic field .

The theoretical basis of magnetic hydrodynamics is the equations of hydrodynamics taking into account electric currents and magnetic fields in the medium and Maxwell's equations . In environments with high conductivity (hot plasma ) and (or) large sizes ( astrophysical objects ), magnetic pressure and magnetic tension are added to the usual gas-dynamic pressure, which leads to the appearance of Alfvén waves .

Using magnetohydrodynamics, many phenomena of space physics are described: planetary and stellar magnetic fields, the origin of the magnetic fields of galaxies , the solar cycle , chromospheric flares in the sun , sunspots .

Applied Hydrodynamics

This includes various specific scientific and technical tasks. Among other tasks, we mention

• task flow around aircraft and water
• Hydrophysics and atmospheric physics
• combustion hydrodynamics
• microhydrodynamics

Rheology

Rheology is a branch of hydrodynamics that studies the behavior of nonlinear fluids, that is, fluids for which the dependence of the flow velocity on the applied force is nonlinear. Examples of nonlinear fluids are pastes, gels, vitreous bodies, pseudoplastics, viscoelastics. Rheology is actively used in materials science , in geophysics .

In hydrodynamics, there are hundreds of unsolved problems, including the problem of fluid leakage from a bath through a pipe ^[1] .

• Hydrophysics
• Aerodynamics
• Theory of hydrodynamic stability
• Theory of differential equations
• Mathematical physics

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• Ivanov B. N. The world of physical hydrodynamics: From problems of turbulence to space physics. Vol. 2, Moscow: URSS, 2010 .-- 240 s.
• Falkovich, G (2011), Fluid Mechanics (A short course for physicists) , Cambridge Univ Press, ISBN 978-1-107-00575-4 , < http://www.cambridge.org/us/academic/subjects/physics / nonlinear-science-and-fluid-dynamics / fluid-mechanics-short-course-physicists >   * Falkovich G. (2014), Modern Hydrodynamics , RHD , < http://www.weizmann.ac.il/complex/falkovich/fluid-mechanics >  
• Truesdell, Clifford Ambrose. Rational fluid mechanics, 1687-1765. Editor's introduction to Euleri Opera omnia II 12 // Leonardi Euleri. Opera Omnia. - Lausanne: Auctoritate et Impensis, Societas Scientiarum Naturalium Helveticae, 1954. - T. 12. - C. I — CXXV. - (II).

• A site dedicated to fluid dynamics with videos, questions, etc.
• Fluid Mechanics // Prof. MS Cramer's (Virginia Tech )
• Navier-Stokes Equations: Foundations of Fluid Mechanics // Prof. MS Cramer's (Virginia Tech )
• Magnetohydrodynamic site
• History of Hydrodynamics