|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| [[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]].'')]]
| | RJ Kettlebells is a dedicated company that can provide each and every client with the services of a [http://berojgarjago.com/article/article.php?id=43049 fitness trainer]. Indeed, this gym is built on the idea that sport should be part of life, part of a healthy lifestyle, because only by exercising on a daily basis will anyone be capable of having a long life, not to mention that he or she will feel good in his skin, completely forgetting about complexes. It is true that the specialized market is very rich in options and it is only natural to be so, given the impressive demand. Individuals do have a growing interest in all that is related to sports. However, RJ Kettlebells is the best choice anyone could make and for a few clear reasons. <br><br>First of all, it is worth mentioning that the staff working at RJ Kettlebells is entirely made up of certified instructors and fitness trainers, ready to teach each and every individual how to make the best of all sessions. Also, apart from providing individuals with pieces of advice regarding the exercises that they should be doing, the RJ Kettlebells trainers will offer tips and pieces of advice on diets. Secondly, this establishment also provides interested clients with the Kettlebell classes. This type of program has gained a lot of popularity in the last few years, becoming highly practiced by a large number of individuals. There are plenty of Kettlebell instructors in Arlington, several of them working at RJ Kettlebells. Indeed, this program usually develops in the presence of a personal trainer and by making use of the kettelbells. This training session works all the muscles in the body. In order to obtain the best results and to lose all the extra pounds, clients have to collaborate with the best [http://goarticles.com/article/9344301 personal trainer in Arlington]. On the official RJ Kettlebells online platform, interested clients will be able to read all about the services offered by this gym, including the fitness programs it is offering. RJ Kettlebells will also provide users with a complete online fitness book by means of which they will be able to closely regard their progress. For all those interested in obtaining the above mentioned book, all that is needed is to become a member on the official website. <br><br>RJ Kettlebells is a company based in Arlington, USA. It is an establishment built on passion and dedication, as well as hard work. Interested visitors should know that on the official website, a blog is waiting to be read. Indeed, in this section of the website, you are bound to discover plenty of interesting pieces of information that will completely alter the opinion the general public has regarding sports. To better convince themselves of the professionalism of the services offered by the previously mentioned company, interested clients can read what former or on-going customers have had to say. On the official website, there is a section entitled ‘testimonials’. If clients should require further pieces of information regarding the services offered or the team working at RJ Kettlebells, they are encouraged to visit the online platform and look through the categories you will find there. |
| [[Image:Streamlines and streamtube.svg|thumb|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.]]
| |
| | |
| [[Fluid flow]] is characterized by a [[velocity]] [[vector field]] in [[three-dimensional space]], within the framework of [[continuum mechanics]]. '''Streamlines, streaklines, and pathlines''' are [[field line]]s resulting from this vector field description of the 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>
| |
| | |
| * '''Streamlines''' are a family of [[curve]]s that are instantaneously [[tangent]] to the [[velocity]] vector of the flow. These show the direction a fluid element will travel in at any point in time.
| |
| | |
| * '''Streaklines''' are the [[Locus (mathematics)|locus]] 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 [[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.
| |
| | |
| * '''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. Similarly, streaklines cannot intersect themselves or other streaklines, because two particles cannot be present at the same location at the same instant of time; unless the origin point of one of the streaklines also belongs to the streakline of the other origin 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).
| |
| | |
| Streamlines and timelines provide a snapshot of some flowfield characteristics, whereas streaklines and pathlines depend on the {{Not a typo|full time}}-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]]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 as<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>{d\vec{x}_S\over ds} \times \vec{u}(\vec{x}_S) = 0,</math>
| |
| | |
| with "×" denoting 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> we deduce:<ref name=Granger/>
| |
| | |
| :<math>{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|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]].
| |
| | |
| === Pathlines ===
| |
| [[File:Kaberneeme campfire site.jpg|thumb|A long exposure photo of sparks from a campfire shows the pathlines for the flow of hot air.]]
| |
| Pathlines are defined by
| |
| | |
| :<math>
| |
| \begin{cases}
| |
| \displaystyle \frac{d\vec{x}_P}{dt} = \vec{u}_P(\vec{x}_P,t) \\[1.2ex]
| |
| \vec{x}_P(t_0) = \vec{x}_{P0}
| |
| \end{cases}
| |
| </math>
| |
| | |
| The suffix <math> P </math> indicates that we are following 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>
| |
| \begin{cases}
| |
| \displaystyle \frac{d \vec{x}_{P} }{dt} = \vec{u}_{P} (\vec{x}_{P},t) \\[1.2ex]
| |
| \vec{x}_{P}( t = \tau_{P}) = \vec{x}_{P0}
| |
| \end{cases}
| |
| </math>
| |
| | |
| where, <math> \vec{u}_{P} </math> is the velocity of a particle <math> P </math> at location <math> \vec{x}_{P} </math> and time <math> t </math>. The parameter <math> \tau_{P} </math>, parametrizes the streakline <math> \vec{x}_{P}(t,\tau_{P}) </math> and <math> 0 \le \tau_{P} \le t_0 </math>, where <math> t_0 </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. 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 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. 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. 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.
| |
| | |
| == Applications ==
| |
| | |
| Knowledge of the streamlines can be useful in fluid dynamics. For example, [[Bernoulli's principle]], which describes the relationship between pressure and velocity in an inviscid fluid, is derived for locations along a streamline.
| |
| | |
| 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.
| |
| | |
| [[Engineering|Engineers]] often use [[dye]]s 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]] | accessdate=2009-04-20 }}</ref> The patterns guide their 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 [[steam locomotives]], [[streamliner]]s, [[automobile|cars]] and [[dolphin]]s are often aesthetically pleasing to the eye. The [[Streamline Moderne]] style, an 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.
| |
| | |
| == See also ==
| |
| * [[Drag coefficient]]
| |
| * [[Equipotential surface]]
| |
| * [[Flow visualization]]
| |
| * [[Potential flow around a circular cylinder]]
| |
| * [[Scientific visualization]]
| |
| * [[Stream function]]
| |
| * [[Streamsurface]]
| |
| | |
| == 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 ==
| |
| *[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)]
| |
| | |
| [[Category:Fluid dynamics]]
| |
| [[Category:Aerodynamics]]
| |
| [[Category:Fluid mechanics| ]]
| |
| [[Category:Mechanical engineering]]
| |
| | |
| *[http://www.nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/FLUID-MECHANICS/ui/Course_home-3.htm NPTEL web content]
| |
RJ Kettlebells is a dedicated company that can provide each and every client with the services of a fitness trainer. Indeed, this gym is built on the idea that sport should be part of life, part of a healthy lifestyle, because only by exercising on a daily basis will anyone be capable of having a long life, not to mention that he or she will feel good in his skin, completely forgetting about complexes. It is true that the specialized market is very rich in options and it is only natural to be so, given the impressive demand. Individuals do have a growing interest in all that is related to sports. However, RJ Kettlebells is the best choice anyone could make and for a few clear reasons.
First of all, it is worth mentioning that the staff working at RJ Kettlebells is entirely made up of certified instructors and fitness trainers, ready to teach each and every individual how to make the best of all sessions. Also, apart from providing individuals with pieces of advice regarding the exercises that they should be doing, the RJ Kettlebells trainers will offer tips and pieces of advice on diets. Secondly, this establishment also provides interested clients with the Kettlebell classes. This type of program has gained a lot of popularity in the last few years, becoming highly practiced by a large number of individuals. There are plenty of Kettlebell instructors in Arlington, several of them working at RJ Kettlebells. Indeed, this program usually develops in the presence of a personal trainer and by making use of the kettelbells. This training session works all the muscles in the body. In order to obtain the best results and to lose all the extra pounds, clients have to collaborate with the best personal trainer in Arlington. On the official RJ Kettlebells online platform, interested clients will be able to read all about the services offered by this gym, including the fitness programs it is offering. RJ Kettlebells will also provide users with a complete online fitness book by means of which they will be able to closely regard their progress. For all those interested in obtaining the above mentioned book, all that is needed is to become a member on the official website.
RJ Kettlebells is a company based in Arlington, USA. It is an establishment built on passion and dedication, as well as hard work. Interested visitors should know that on the official website, a blog is waiting to be read. Indeed, in this section of the website, you are bound to discover plenty of interesting pieces of information that will completely alter the opinion the general public has regarding sports. To better convince themselves of the professionalism of the services offered by the previously mentioned company, interested clients can read what former or on-going customers have had to say. On the official website, there is a section entitled ‘testimonials’. If clients should require further pieces of information regarding the services offered or the team working at RJ Kettlebells, they are encouraged to visit the online platform and look through the categories you will find there.