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Streamlines, streaklines and pathlines are
field line A field line is a graphical visual aid for visualizing vector fields. It consists of an imaginary directed line which is tangent to the field vector at each point along its length. A diagram showing a representative set of neighboring field ...
s in a
fluid flow 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) an ...
. They differ only when the flow changes with time, that is, when the flow is not steady. Considering a
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 ...
vector field in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
in the framework of
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 ...
, we have that: * Streamlines are a family of
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s whose
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
vectors constitute 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 ...
vector field of the flow. These show the direction in which a massless fluid element will travel at any point in time. * Streaklines are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline. * Pathlines are the
trajectories A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time. * Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move. By definition, different streamlines at the same instant in a flow do not intersect, because a fluid particle cannot have two different velocities at the same point. However, pathlines are allowed to intersect themselves or other pathlines (except the starting and end points of the different pathlines, which need to be distinct). Streaklines can also intersect themselves and other streaklines. Streamlines and timelines provide a snapshot of some flowfield characteristics, whereas streaklines and pathlines depend on the -history of the flow. However, often sequences of timelines (and streaklines) at different instants—being presented either in a single image or with a video stream—may be used to provide insight in the flow and its history. If a line, curve or closed curve is used as start point for a continuous set of streamlines, the result is a stream surface. In the case of a closed curve in a steady flow, fluid that is inside a stream surface must remain forever within that same stream surface, because the streamlines are tangent to the flow velocity. A scalar function whose
contour line A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional gr ...
s define the streamlines is known as the stream function. Dye line may refer either to a streakline: dye released gradually from a fixed location during time; or it may refer to a timeline: a line of dye applied instantaneously at a certain moment in time, and observed at a later instant.


Mathematical description


Streamlines

Streamlines are defined by, pp. 422–425. : \times \vec(\vec_S) = 0, where "\times" denotes the
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
and \vec_S(s) is the
parametric representation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric o ...
of ''just one'' streamline at one moment in time. If the components of the velocity are written \vec = (u,v,w), and those of the streamline as \vec_S=(x_S,y_S,z_S), we deduce : = = , which shows that the curves are parallel to the velocity vector. Here s is a variable which parametrizes the curve s\mapsto \vec_S(s). Streamlines are calculated instantaneously, meaning that at one instance of time they are calculated throughout the fluid from the instantaneous
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
field. A streamtube consists of a
bundle Bundle or Bundling may refer to: * Bundling (packaging), the process of using straps to bundle up items Biology * Bundle of His, a collection of heart muscle cells specialized for electrical conduction * Bundle of Kent, an extra conduction path ...
of streamlines, much like communication cable. The equation of motion of a fluid on a streamline for a flow in a vertical plane is: \frac + c\frac=\nu \frac - \frac\frac-g\frac The flow velocity in the direction s of the streamline is denoted by c. r is the radius of curvature of the streamline. The density of the fluid is denoted by \rho and the kinematic viscosity by \nu. \frac is the pressure gradient and \frac the velocity gradient along the streamline. For a steady flow, the time derivative of the velocity is zero: \frac=0. g denotes the gravitational acceleration.


Pathlines

Pathlines are defined by : \begin \displaystyle \frac(t) = \vec_P(\vec_P(t),t) \\ .2ex \vec_P(t_0) = \vec_ \end The suffix P indicates that we are following the motion of a fluid particle. Note that at point \vec_P the curve is parallel to the flow velocity vector \vec , where the velocity vector is evaluated at the position of the particle \vec_P at that time t .


Streaklines

Streaklines can be expressed as, : \begin \displaystyle \frac = \vec_ (\vec_,t) \\ .2ex \vec_( t = \tau_) = \vec_ \end where, \vec_(\vec,t) is the velocity of a particle P at location \vec and time t . The parameter \tau_ , parametrizes the streakline \vec_(t,\tau_) and t_0 \le \tau_ \le t , where t is a time of interest.


Steady flows

In steady flow (when the velocity vector-field does not change with time), the streamlines, pathlines, and streaklines coincide. This is because when a particle on a streamline reaches a point, a_0, further on that streamline the equations governing the flow will send it in a certain direction \vec. As the equations that govern the flow remain the same when another particle reaches a_0 it will also go in the direction \vec. If the flow is not steady then when the next particle reaches position a_0 the flow would have changed and the particle will go in a different direction. This is useful, because it is usually very difficult to look at streamlines in an experiment. However, if the flow is steady, one can use streaklines to describe the streamline pattern.


Frame dependence

Streamlines are frame-dependent. That is, the streamlines observed in one
inertial reference frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
are different from those observed in another inertial reference frame. For instance, the streamlines in the air around an
aircraft An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...
wing A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is e ...
are defined differently for the passengers in the aircraft than for an
observer An observer is one who engages in observation or in watching an experiment. Observer may also refer to: Computer science and information theory * In information theory, any system which receives information from an object * State observer in co ...
on the ground. In the aircraft example, the observer on the ground will observe unsteady flow, and the observers in the aircraft will observe steady flow, with constant streamlines. When possible, fluid dynamicists try to find a reference frame in which the flow is steady, so that they can use experimental methods of creating streaklines to identify the streamlines.


Application

Knowledge of the streamlines can be useful in fluid dynamics. The curvature of a streamline is related to the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
gradient acting perpendicular to the streamline. The center of curvature of the streamline lies in the direction of decreasing radial pressure. The magnitude of the radial pressure gradient can be calculated directly from the density of the fluid, the curvature of the streamline and the local velocity.
Dye A dye is a colored substance that chemically bonds to the substrate to which it is being applied. This distinguishes dyes from pigments which do not chemically bind to the material they color. Dye is generally applied in an aqueous solution and ...
can be used in water, or
smoke Smoke is a suspension of airborne particulates and gases emitted when a material undergoes combustion or pyrolysis, together with the quantity of air that is entrained or otherwise mixed into the mass. It is commonly an unwanted by-produc ...
in air, in order to see streaklines, from which pathlines can be calculated. Streaklines are identical to streamlines for steady flow. Further, dye can be used to create timelines. The patterns guide design modifications, aiming to reduce the drag. This task is known as ''streamlining'', and the resulting design is referred to as being ''streamlined''. Streamlined objects and organisms, like
airfoil An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbin ...
s,
streamliner A streamliner is a vehicle incorporating streamlining in a shape providing reduced air resistance. The term is applied to high-speed railway trainsets of the 1930s to 1950s, and to their successor " bullet trains". Less commonly, the term i ...
s,
cars A car or automobile is a motor vehicle with wheels. Most definitions of ''cars'' say that they run primarily on roads, Car seat, seat one to eight people, have four wheels, and mainly transport private transport#Personal transport, people in ...
and
dolphin A dolphin is an aquatic mammal within the infraorder Cetacea. Dolphin species belong to the families Delphinidae (the oceanic dolphins), Platanistidae (the Indian river dolphins), Iniidae (the New World river dolphins), Pontoporiidae (the b ...
s are often aesthetically pleasing to the eye. The
Streamline Moderne Streamline Moderne is an international style of Art Deco architecture and design that emerged in the 1930s. Inspired by aerodynamic design, it emphasized curving forms, long horizontal lines, and sometimes nautical elements. In industrial desig ...
style, a 1930s and 1940s offshoot of
Art Deco Art Deco, short for the French ''Arts Décoratifs'', and sometimes just called Deco, is a style of visual arts, architecture, and product design, that first appeared in France in the 1910s (just before World War I), and flourished in the Unit ...
, brought flowing lines to architecture and design of the era. The canonical example of a streamlined shape is a chicken egg with the blunt end facing forwards. This shows clearly that the curvature of the front surface can be much steeper than the back of the object. Most drag is caused by eddies in the fluid behind the moving object, and the objective should be to allow the fluid to slow down after passing around the object, and regain pressure, without forming eddies. The same terms have since become common vernacular to describe any process that smooths an operation. For instance, it is common to hear references to streamlining a business practice, or operation.


See also

*
Drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag e ...
* Elementary flow * Equipotential surface * Flow visualization *
Flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
* Scientific visualization *
Seeding (fluid dynamics) Seeding a material is a concept used in fluid dynamics to describe the act of introducing specific particulates or other foreign substances into a stream of fluid being evaluated. An altered fluid will be described as having a seeded flow. De ...
* Stream function * Streamsurface * Streamlet (scientific visualization)


Notes and references


Notes


References

*{{cite book , first = T.E. , last = Faber , year = 1995 , title = Fluid Dynamics for Physicists , publisher = Cambridge University Press , isbn = 0-521-42969-2


External links


Streamline illustration


* ttp://prj.dimanov.com/ Joukowsky Transform Interactive WebApp Continuum mechanics Numerical function drawing