# Dictionary Definition

hydrodynamics n : study of fluids in motion [syn:
hydrokinetics]

# User Contributed Dictionary

## English

### Noun

hydrodynamics- the scientific study of fluids in motion

#### Related terms

#### Translations

- Dutch: hydrodynamica
- German: Hydrodynamik
- Russian: гидродинамика
- Polish: hydrodynamika
- Swedish: hydrodynamik

### See also

# Extensive Definition

Fluid dynamics is the sub-discipline of fluid
mechanics dealing with fluid flow: fluids (liquids and gases) in motion. It has several
subdisciplines itself, including aerodynamics (the study of
gases in motion) and hydrodynamics (the study of liquids in
motion). Fluid dynamics has a wide range of applications, including
calculating forces and
moments
on aircraft,
determining the mass flow
rate of petroleum
through pipelines, predicting weather patterns, understanding
nebulae in interstellar space and
reportedly modeling fission weapon detonation. Some of its
principles are even used in
traffic engineering, where traffic is treated as a continuous
fluid.

Fluid dynamics offers a systematic structure that
underlies these practical disciplines and that embraces empirical
and semi-empirical laws, derived from flow
measurement, used to solve practical problems. The solution of
a fluid dynamics problem typically involves calculation of various
properties of the fluid, such as velocity, pressure, density, and temperature, as functions of
space and time.

## Equations of fluid dynamics

The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum (also known as Newton's Second Law of Motion), and conservation of energy (also known as First Law of Thermodynamics). These are based on classical mechanics and are modified in quantum mechanics and general relativity. They are expressed using the Reynolds Transport Theorem.In addition to the above, fluids are assumed to
obey the continuum assumption. Fluids are composed of molecules
that collide with one another and solid objects. However, the
continuum assumption considers fluids to be continuous, rather than
discrete. Consequently, properties such as density, pressure,
temperature, and velocity are taken to be well-defined at
infinitesimally small points, and are assumed to vary continuously
from one point to another. The fact that the fluid is made up of
discrete molecules is ignored.

For fluids which are sufficiently dense to be a
continuum, do not contain ionized species, and have velocities
small in relation to the speed of light, the momentum equations for
Newtonian
fluids are the Navier-Stokes
equations, which is a non-linear set
of differential
equations that describes the flow of a fluid whose stress
depends linearly on velocity gradients and pressure. The
unsimplified equations do not have a general closed-form
solution, so they are only of use in
Computational Fluid Dynamics or when they can be simplified.
The equations can be simplified in a number of ways, all of which
make them easier to solve. Some of them allow appropriate fluid
dynamics problems to be solved in closed form.

In addition to the mass, momentum, and energy
conservation equations, a thermodynamical equation
of state giving the pressure as a function of other thermodynamic
variables for the fluid is required to completely specify the
problem. An example of this would be the perfect gas
equation of state:

- p= \frac

where p is pressure, \rho is density, R_u is the gas
constant, M is the molecular
mass and T is temperature.

### Compressible vs incompressible flow

All fluids are compressible to some
extent, that is changes in pressure or temperature will result in
changes in density. However, in many situations the changes in
pressure and temperature are sufficiently small that the changes in
density are negligible. In this case the flow can be modeled as an
incompressible
flow. Otherwise the more general compressible
flow equations must be used.

Mathematically, incompressibility is expressed by
saying that the density \rho of a fluid parcel does not change as
it moves in the flow field, i.e.,

- \frac = 0 \, ,

For flow of gases, to determine whether to use
compressible or incompressible fluid dynamics, the Mach number
of the flow is to be evaluated. As a rough guide, compressible
effects can be ignored at Mach numbers below approximately 0.3. For
liquids, whether the incompressible assumption is valid depends on
the fluid properties (specifically the critical pressure and
temperature of the fluid) and the flow conditions (how close to the
critical pressure the actual flow pressure becomes). Acoustic problems always
require allowing compressibility, since sound waves
are compression waves involving changes in pressure and density of
the medium through which they propagate.

### Viscous vs inviscid flow

Viscous problems are those in which fluid friction has significant effects on the fluid motion.The Reynolds
number can be used to evaluate whether viscous or inviscid
equations are appropriate to the problem.

Stokes flow
is flow at very low Reynolds numbers, such that inertial forces can
be neglected compared to viscous forces.

On the contrary, high Reynolds numbers indicate
that the inertial forces are more significant than the viscous
(friction) forces. Therefore, we may assume the flow to be an
inviscid
flow, an approximation in which we neglect viscosity at all, compared to
inertial terms.

This idea can work fairly well when the Reynolds
number is high. However, certain problems such as those involving
solid boundaries, may require that the viscosity be included.
Viscosity often cannot be neglected near solid boundaries because
the no-slip
condition can generate a thin region of large strain rate
(known as Boundary
layer) which enhances the effect of even a small amount of
viscosity, and thus
generating vorticity.
Therefore, to calculate net forces on bodies (such as wings) we
should use viscous flow equations. As illustrated by d'Alembert's
paradox, a body in an inviscid fluid will experience no drag
force. The standard equations of inviscid flow are the Euler
equations. Another often used model, especially in
computational fluid dynamics, is to use the Euler
equations away from the body and the boundary
layer equations, which incorporates viscosity, in a region
close to the body.

The Euler
equations can be integrated along a streamline to get Bernoulli's
equation. When the flow is everywhere irrotational
and inviscid, Bernoulli's equation can be used throughout the flow
field. Such flows are called potential
flows.

### Steady vs unsteady flow

When all the time derivatives of a flow field
vanish, the flow is considered to be a steady flow. Otherwise, it
is called unsteady. Whether a particular flow is steady or
unsteady, can depend on the chosen frame of
reference. For instance, laminar flow over a sphere is steady in the frame of
reference that is stationary with respect to the sphere. In a frame
of reference that is stationary than the governing equations of the
same problem without taking advantage of the steadiness of the flow
field.

Although strictly unsteady flows, time-periodic
problems can often be solved by the same techniques as steady
flows. For this reason, they can be considered to be somewhere
between steady and unsteady.

### Laminar vs turbulent flow

Turbulence is flow dominated by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar. It should be noted, however, that the presence of eddies or recirculation does not necessarily indicate turbulent flow--these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via Reynolds decomposition, in which the flow is broken down into the sum of a steady component and a perturbation component.It is believed that turbulent flows obey the
Navier-Stokes
equations.
Direct Numerical Simulation (DNS), based on the incompressible
Navier-Stokes equations, makes it possible to simulate turbulent
flows with moderate Reynolds numbers (restrictions depend on the
power of computer and efficiency of solution algorithm). The
results of DNS agree with the experimental data.

Most flows of interest have Reynolds numbers too
high for DNS to be a viable option (see: Pope), given the state of
computational power for the next few decades. Any flight vehicle
large enough to carry a human (L > 3 m), moving faster than 72
km/h (20 m/s) is well beyond the limit of DNS simulation (Re = 4
million). Transport aircraft wings (such as on an Airbus A300
or Boeing
747) have Reynolds numbers of 40 million (based on the wing
chord). In order to solve these real life flow problems, turbulence
models will be a necessity for the foreseeable future.
Reynolds-Averaged Navier-Stokes equations (RANS) combined with
turbulence
modeling provides a model of the effects of the turbulent flow,
mainly the additional momentum transfer provided by the Reynolds
stresses, although the turbulence also enhances the heat and
mass
transfer. Large
Eddy Simulation (LES) also holds promise as a simulation
methodology, especially in the guise of Detached
Eddy Simulation (DES), which is a combination of turbulence
modeling and large eddy simulation.

### Newtonian vs non-Newtonian fluids

Sir Isaac Newton showed how stress and the rate of strain are very close to linearly related for many familiar fluids, such as water and air. These Newtonian fluids are modeled by a coefficient called viscosity, which depends on the specific fluid.However, some of the other materials, such as
emulsions and slurries and some visco-elastic materials (eg.
blood, some polymers), have more complicated
non-Newtonian
stress-strain behaviours. These materials include sticky liquids
such as latex, honey, and lubricants which are
studied in the sub-discipline of rheology.

### Magnetohydrodynamics

Magnetohydrodynamics is the multi-disciplinary study of the flow of electrically conducting fluids in electromagnetic fields. Examples of such fluids include plasmas, liquid metals, and salt water. The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism.### Other approximations

There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below.- The Boussinesq approximation neglects variations in density except to calculate buoyancy forces. It is often used in free convection problems where density changes are small.
- Lubrication theory exploits the large aspect ratio of the domain to show that certain terms in the equations are small and so can be neglected.
- Slender-body theory is a methodology used in Stokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid.
- The shallow-water equations can be used to describe a layer of relatively inviscid fluid with a free surface, in which surface gradients are small.
- The Boussinesq equations are applicable to surface waves on thicker layers of fluid and with steeper surface slopes.
- Darcy's law is used for flow in porous media, and works with variables averaged over several pore-widths.
- In rotating systems, the quasi-geostrophic approximation assumes an almost perfect balance between pressure gradients and the Coriolis force. It is useful in the study of atmospheric dynamics.

## Terminology in fluid dynamics

The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods.Some of the terminology that is necessary in the
study of fluid dynamics is not found in other similar areas of
study. In particular, some of the terminology used in fluid
dynamics is not used in fluid
statics.

### Terminology in incompressible fluid dynamics

The concepts of total pressure (also known as stagnation pressure) and dynamic pressure arise from Bernoulli's equation and are significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense - they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and can be identified for every point in a fluid flow field.In Aerodynamics, L.J. Clancy writes (page 21):
"To distinguish it from the total and dynamic pressures, the actual
pressure of the fluid, which is associated not with its motion but
with its state, is often referred to as the static pressure, but
where the term pressure alone is used it refers to this static
pressure."

A point in a fluid flow where the flow has come
to rest (i.e. speed is equal to zero adjacent to some solid body
immersed in the fluid flow) is of special significance. It is of
such importance that it is given a special name - a stagnation
point. The pressure
at the stagnation point is of special significance and is given its
own name - stagnation
pressure, which is equal to the total pressure.

### Terminology in compressible fluid dynamics

In a compressible fluid, such as air, the temperature and density are essential when determining the state of the fluid. In addition to the concept of total pressure (also known as stagnation pressure), the concepts of total (or stagnation) temperature and total (or stagnation) density are also essential in any study of compressible fluid flows. To avoid potential ambiguity when referring to temperature and density, many authors use the terms static temperature and static density. Static temperature is identical to temperature; and static density is identical to density; and both can be identified for every point in a fluid flow field.The temperature and density at a stagnation
point are called stagnation temperature and stagnation
density.

Readers might wonder if there are such concepts
as dynamic temperature or dynamic density. There aren't.

A similar approach is also taken with the
thermodynamic properties of compressible fluids. Many authors use
the terms total (or stagnation) enthalpy and total (or
stagnation) entropy. The
terms static enthalpy and static entropy appear to be less common,
but where they are used they mean nothing more than enthalpy and
entropy respectively, and the prefix 'static' is being used to
avoid ambiguity with their 'total' or 'stagnation'
counterparts.

## References

- Acheson, D.J. (1990) "Elementary Fluid Dynamics" (Clarendon Press).
- Batchelor, G.K. (1967) "An Introduction to Fluid Dynamics" (Cambridge University Press).
- Clancy, L.J. (1975) "Aerodynamics" (Pitman Publishing Limited).
- Lamb, H. (1994) "Hydrodynamics" (Cambridge University Press, 6th ed.). Originally published in 1879, the 6th extended edition appeared first in 1932.
- Landau, L.D. and Lifshitz, E.M. (1987) "Fluid Mechanics" (Pergamon Press, 2nd ed.).
- Milne-Thompson, L.M. (1968) "Theoretical Hydrodynamics" (Macmillan, 5th ed.). Originally published in 1938.
- Pope, S.B. (2000) "Turbulent Flows" (Cambridge University Press).
- Shinbrot, Marvin (1973) "Lectures on Fluid Mechanics" (Gordon and Breach)

## Notes

## See also

### Fields of study

### Mathematical equations and concepts

- Bernoulli's equation
- Reynolds transport theorem
- Boussinesq approximation
- Conservation laws
- Euler equations
- Darcy's law
- Dynamic pressure
- Fluid statics
- Helmholtz's theorems
- Kirchhoff equations
- Manning equation
- Navier-Stokes equations
- Pascal's law
- Poiseuille's law
- Pressure
- Static pressure
- Pressure head
- Relativistic Euler equations
- Reynolds decomposition
- Stream function
- Streamlines, streaklines and pathlines

### Types of fluid flow

### Fluid properties

### Fluid phenomena

### Applications

### Miscellaneous

## External links

hydrodynamics in German: Strömungslehre

hydrodynamics in Estonian: Hüdrodünaamika

hydrodynamics in Czech: Proudění

hydrodynamics in Esperanto:
Fluidaĵ-Dinamiko

hydrodynamics in Persian: دینامیک سیالات

hydrodynamics in French: Dynamique des
fluides

hydrodynamics in Korean: 유체동역학

hydrodynamics in Indonesian: Dinamika
fluida

hydrodynamics in Italian: Fluidodinamica

hydrodynamics in Norwegian: Væskedynamikk

hydrodynamics in Norwegian Nynorsk:
Væskedynamikk

hydrodynamics in Oromo: Fluid dynamics

hydrodynamics in Polish: Dynamika płynów

hydrodynamics in Simple English: Fluid
dynamics

hydrodynamics in Finnish:
Virtausdynamiikka

hydrodynamics in Turkish: Akışkanlar
dinamiği

hydrodynamics in Chinese: 流體動力學