Editing Streamlines, streaklines, and pathlines

{{Short description|Field lines in a fluid flow}}
[[Image:Streaklines and pathlines animation (low).gif|right|300px|thumb|The red particle moves in a flowing fluid; its ''pathline'' is traced in red; the tip of the trail of blue ink released from the origin follows the particle, but unlike the static pathline (which records the earlier motion of the dot), ink released after the red dot departs continues to move up with the flow. (This is a ''streakline''.) The dashed lines represent contours of the velocity field (''streamlines''), showing the motion of the whole field at the same time. (''See [[:Image:Streaklines and pathlines animation.gif|high resolution version]].'')]]
[[Image:Streamlines and streamtube.svg|280px|right|thumb|Solid blue lines and broken grey lines represent the streamlines. The red arrows show the direction and magnitude of the flow velocity. These arrows are tangential to the streamline. The group of streamlines enclose the green curves (<math>C_1</math> and <math>C_2</math>) to form a stream surface.]]

'''Streamlines''', '''streaklines''' and '''pathlines''' are [[field line]]s in a [[fluid flow]].
They differ only when the flow changes with time, that is, when the flow is not [[steady flow|steady]].<ref name=Batchelor>{{cite book | author=Batchelor, G. | title=Introduction to Fluid Mechanics | year=2000}}</ref>
<ref name=Kundu>{{cite book | author=Kundu P and Cohen I | title=Fluid Mechanics}}</ref>
Considering a [[velocity]] [[vector field]] in [[three-dimensional space]] in the framework of [[continuum mechanics]]:  
* '''Streamlines''' are a family of [[curve]]s whose [[tangent]] vectors constitute the [[velocity]] vector field of the flow. These show the direction in which a massless [[fluid element]] will travel at any point in time.<ref>{{cite web|url=https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/definition-of-streamlines/|title=Definition of Streamlines|website=www.grc.nasa.gov|access-date=4 October 2023|url-status=live|archive-url=https://web.archive.org/web/20170118122133/https://www.grc.nasa.gov/WWW/k-12/airplane/stream.html|archive-date=18 January 2017}}</ref>
* '''Streaklines''' are the [[Locus (mathematics)|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 (as in [[dye tracing]]) extends along a streakline.
* '''Pathlines''' are the [[trajectory|trajectories]] 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.

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. 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 provide a snapshot of some flowfield characteristics, whereas streaklines and pathlines depend on the {{Not a typo|full time}}-history of the flow. Often, sequences of streamlines or streaklines at different instants, presented either in a single image or with a videostream, may provide insight to the flow and its history.

[[File:Stream function.png|thumb|For an incompressible-flow velocity vector field in 2D (red, top), its streamlines (dashed) can be computed as the contours of the [[stream function]] (bottom).]]

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]]s define the streamlines is known as the '''[[stream function]]'''.

== Mathematical description ==

=== Streamlines ===
[[Image:Magnet0873.png|thumb|The direction of [[magnetic field]] lines are streamlines represented by the alignment of [[iron filings]] sprinkled on paper placed above a bar magnet]]
[[Image:Streamlines around a NACA 0012.svg|thumb|300px|right|[[Potential flow|Potential-flow]] streamlines achieving the [[Kutta condition]] around a [[NACA airfoil]] with upper and lower streamtubes identified.]]

Streamlines are defined by<ref name=Granger>{{Cite book | title=Fluid Mechanics | first=R.A. | last=Granger | year=1995 | isbn=0-486-68356-7 | publisher=Dover Publications }}, pp. 422–425.</ref>
<math display="block">{d\vec{x}_S\over ds} \times \vec{u}(\vec{x}_S) = \vec{0},</math>
where "<math>\times</math>" denotes the [[vector (geometry)|vector]] [[cross product]] and <math>\vec{x}_S(s)</math> is the [[parametric equation|parametric representation]] of ''just one'' streamline at one moment in time.

If the components of the velocity are written <math>\vec{u} = (u,v,w),</math> and those of the streamline as <math>\vec{x}_S=(x_S,y_S,z_S),</math> then<ref name=Granger/>
<math display="block">{dx_S\over u} = {dy_S\over v} = {dz_S\over w},</math>
which shows that the curves are parallel to the velocity vector. Here <math>s</math> is a [[Variable (mathematics)|variable]] which [[Parametrization (geometry)|parametrizes]] the curve <math>s\mapsto \vec{x}_S(s).</math>  Streamlines are calculated instantaneously, meaning that at one instance of time they are calculated throughout the fluid from the instantaneous [[flow velocity]] [[field (physics)|field]].

{{anchor|Streamtube}}A '''streamtube''' consists of a [[Bundle (mathematics)|bundle]] of streamlines, much like communication cable.

The equation of motion of a fluid on a streamline for a flow in a vertical plane is:<ref>{{Cite web|title=Equation of motion of a fluid on a streamline|url=https://www.tec-science.com/mechanics/gases-and-liquids/equation-of-motion-of-a-fluid-on-a-streamline/|last=tec-science|date=2020-04-22|website=tec-science|language=en-US|access-date=2020-05-07}}</ref>
<math display="block">\frac{\partial c}{\partial t} + c\frac{\partial c}{\partial s}=\nu \frac{\partial^2 c}{\partial r^2} - \frac{1}{\rho}\frac{\partial p}{\partial s}-g\frac{\partial z}{\partial s}</math>

The flow velocity in the direction <math>s</math> of the streamline is denoted by <math>c</math>. <math>r</math> is the radius of curvature of the streamline. The density of the fluid is denoted by <math>\rho</math> and the kinematic viscosity by <math>\nu</math>. <math>\frac{\partial p}{\partial s}</math> is the pressure gradient and <math>\frac{\partial c}{\partial s}</math> the velocity gradient along the streamline. For a steady flow, the time derivative of the velocity is zero: <math>\frac{\partial c}{\partial t}=0</math>. <math>g</math> denotes the gravitational acceleration.

=== Pathlines ===
[[File:Kaberneeme campfire site.jpg|thumb|A [[long-exposure photo]] of [[spark (fire)|spark]] from a [[campfire]] shows the pathlines for the flow of hot air.]]
Pathlines are defined by
<math display="block">
\begin{cases}
    \dfrac{d\vec{x}_P}{dt}(t) = \vec{u}_P(\vec{x}_P(t),t) \\[1.2ex]
    \vec{x}_P(t_0) = \vec{x}_{P0}
\end{cases}
</math>

The subscript <math> P </math> indicates a following of the motion of a fluid particle. 
<!-- The following statement does not seem to make sense to me. Please correct it and remove the comment.

For pathlines there is a <math>t</math> (time) dependence.  This is because they are how the fluid moves one particle so if the flow changes in time this is reflected affects the path. 
--> 
Note that at point <math> \vec{x}_P </math> the curve is parallel to the flow velocity vector <math> \vec{u} </math>, where the velocity vector is evaluated at the position of the particle <math> \vec{x}_P </math> at that time <math> t </math>.

=== Streaklines ===
[[File:Aeroakustik-Windkanal-Messhalle.JPG|thumb|Example of a streakline used to visualize the flow around a car inside a wind tunnel.]]
Streaklines can be expressed as,
<math display="block">
\begin{cases} 
       \displaystyle \frac{d \vec{x}_{str} }{dt} = \vec{u}_{P} (\vec{x}_{str},t) \\[1.2ex]
       \vec{x}_{str}( t = \tau_{P}) =  \vec{x}_{P0}
\end{cases} 
</math>
where, <math> \vec{u}_{P}(\vec{x},t) </math> is the velocity of a particle <math> P </math> at location <math> \vec{x} </math> and time <math> t </math>. The parameter <math> \tau_{P} </math>, parametrizes the streakline <math> \vec{x}_{str}(t,\tau_{P}) </math> and  <math> t_0 \le \tau_{P} \le t </math>, where <math> t </math> 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, <math>a_0</math>, further on that streamline the equations governing the flow will send it in a certain direction <math>\vec{x}</math>.  As the equations that govern the flow remain the same when another particle reaches  <math>a_0</math> it will also go in the direction  <math>\vec{x}</math>. If the flow is not steady then when the next particle reaches position <math>a_0</math> 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. 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 frame of reference|inertial reference frame]] are different from those observed in another inertial reference frame. For instance, the streamlines in the [[Earth's atmosphere|air]] around an [[aircraft]] [[wing]] are defined differently for the passengers in the aircraft than for an [[Observation|observer]] 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<!-- [[Streamlined]] redirects here --> ==

Knowledge of the streamlines can be useful in fluid dynamics. The curvature of a streamline is related to the [[pressure]] 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]] can be used in water, or [[smoke]] 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.<ref>{{cite web | url=http://modular.mit.edu:8080/ramgen/ifluids/Flow_Visualization.rm | title=Flow visualisation | publisher=National Committee for Fluid Mechanics Films (NCFMF) | format=[[RealMedia]] | access-date=2009-04-20 | url-status=dead | archive-url=https://web.archive.org/web/20060103012451/http://modular.mit.edu:8080/ramgen/ifluids/Flow_Visualization.rm | archive-date=2006-01-03 }}</ref> 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]]s, [[streamliner]]s, [[automobile|cars]] and [[dolphin]]s are often aesthetically pleasing to the eye. The [[Streamline Moderne]] style, a 1930s and 1940s offshoot of [[Art Deco]], brought flowing lines to architecture and design of the era. The canonical example of a streamlined shape is a chicken [[Oval (geometry)|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.{{Citation needed|date=September 2021}}

== See also ==
{{Portal|Physics}}
* [[Drag coefficient]]
* [[Elementary flow]]
* [[Equipotential surface]]
* [[Flow visualization]]
* [[Flow velocity]]
* [[Scientific visualization]]
* [[Seeding (fluid dynamics)]]
* [[Stream function]]
* [[Streamsurface]]
* [[Streamlet (scientific visualization)]]

== Notes and references ==

=== Notes ===
{{Reflist}}

===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 ==
*[https://web.archive.org/web/20051126010928/http://www.centennialofflight.gov/essay/Dictionary/streamlining/DI44.htm Streamline illustration]
* [http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec02/tutorial/tutorial-spsl.html Tutorial - Illustration of Streamlines, Streaklines and Pathlines of a Velocity Field(with applet)]
* [http://prj.dimanov.com/ Joukowsky Transform Interactive WebApp]

[[Category:Continuum mechanics]]
[[Category:Numerical function drawing]]