In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
the flow velocity in
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, also macroscopic velocity in
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, or drift velocity in
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, is a
vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar.
It is also called velocity field; when evaluated along a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
, it is called a velocity profile (as in, e.g.,
law of the wall).
Definition
The flow velocity ''u'' of a fluid is a vector field
:
which gives the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of an ''
element of fluid'' at a position
and time
The flow speed ''q'' is the length of the flow velocity vector
:
and is a scalar field.
Uses
The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow:
Steady flow
The flow of a fluid is said to be ''steady'' if
does not vary with time. That is if
:
Incompressible flow
If a fluid is incompressible the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of
is zero:
:
That is, if
is a
solenoidal vector field.
Irrotational flow
A flow is ''irrotational'' if the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
of
is zero:
:
That is, if
is an
irrotational vector field.
A flow in a
simply-connected domain which is irrotational can be described as a
potential flow, through the use of a
velocity potential with
If the flow is both irrotational and incompressible, the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
of the velocity potential must be zero:
Vorticity
The ''vorticity'',
, of a flow can be defined in terms of its flow velocity by
:
If the vorticity is zero, the flow is irrotational.
The velocity potential
If an irrotational flow occupies a
simply-connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
fluid region then there exists a
scalar field
In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantity ...
such that
:
The scalar field
is called the
velocity potential for the flow. (See
Irrotational vector field.)
Bulk velocity
In many engineering applications the local flow velocity
vector field is not known in every point and the only accessible velocity is the bulk velocity (or average flow velocity)
which is the ratio between the
volume flow rate and the cross sectional area
, given by
:
.
See also
*
Velocity gradient
*
Velocity potential
*
Drift velocity
In physics, a drift velocity is the average velocity attained by charged particles, such as electrons, in a material due to an electric field. In general, an electron in a conductor will propagate randomly at the Fermi velocity, resulting in an a ...
*
Group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
*
Particle velocity
*
Vorticity
*
Enstrophy
*
Strain rate
*
Stream function
*
Pressure gradient
*
Wind velocity
In meteorology, wind speed, or wind flow speed, is a fundamental atmospheric quantity caused by air moving from high to low pressure, usually due to changes in temperature. Wind speed is now commonly measured with an anemometer.
Wind spee ...
References
{{Authority control
Fluid dynamics
Continuum mechanics
Vector calculus
Velocity
Spatial gradient
Vector physical quantities