Path line and Stream lineDistinguish between stream line and path line1. A stream line is an imaginary line drawn in a flow field such that a tangent drawn at any point on this line represents the direction of the velocity vector. Path line is the line traced by a single fluid particle as it moves over a period of time.

2. Stream line shows the direction of velocity of a number of fluid particles at the same instant of time. Path line shows the direction of velocity of the same fluid particle at successive instants of time.

Can the path line and a streamline cross each other at right angles?A fluid particle always moves tangent to the streamline. In steady flow, the path lines and streamlines are identical. In unsteady flow, a fluid particle follows one stream line at one instant and another at the next instant and so on, so that the path line have no resemblance to any given instantaneous streamline.

Streamlines, streaklines, and pathlinesFrom Wikipedia, the free encyclopedia

This article may require copy editing for grammar, style, cohesion, tone, or spelling. You can assist by editing it. (February 2012)

The red particle moves in a flowing fluid; its pathlineis 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 high resolution version.)

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 (  and  ) 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 arefield lines 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.[1] [2]

Streamlines are a family of curves that are instantaneously tangent to the velocityvector of the flow. These show the direction a fluid element will travel in at any point in time.

Streaklines are the 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 trajectories that individual fluid particles follow. These can be thought of as a "recording" of the path a fluid element in the flow takes 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 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 linesdefine 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.

Contents

  [hide] 

1   Mathematical description

o 1.1   Streamlines

o 1.2   Pathlines

o 1.3   Streaklines

2   Steady flows

3   Frame dependence

4   Applications

5   See also

6   Notes and references

o 6.1   Notes

o 6.2   References

7   External links

[edit]Mathematical description

[edit]Streamlines

Streamlines are defined as[3]

with "×" denoting the vector cross product and   is the parametric representation of just one streamline at one moment in time.

If the components of the velocity are written   and those of the streamline as   we deduce:[3]

which shows that the curves are parallel to the velocity vector. Here   is a variable which parametrizes the curve   Streamlines are calculated instantaneously,

meaning that at one instance of time they are calculated throughout the fluid from the instantaneous flow velocity field.

[edit]Pathlines

Pathlines are defined by

The suffix   indicates that we are following the motion of a fluid particle. Note that at point   the curve is parallel to the flow velocity vector  , where the velocity vector is

evaluated at the position of the particle   at that time  .

[edit]Streaklines

Streaklines can be expressed as,

where,   is the velocity of a particle   at location   and time  . The parameter  , parametrizes the streakline   and  , where   is a

time of interest.

[edit]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,  , further on that streamline the equations governing the flow will send it in a certain direction  . As the equations that govern the flow

remain the same when another particle reaches   it will also go in the direction  . If the flow is not steady then when the next particle reaches position   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.

[edit]Frame dependence

Streamlines are frame-dependent. That is, the streamlines observed in one inertial reference frame are different from those observed in another inertial reference frame. For

instance, the streamlines in the air around an aircraft wing are defined differently for the passengers in the aircraft than for an 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.

[edit]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.

Engineers often use dyes 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.[4] 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, streamliners, cars and dolphins are often aesthetically pleasing to the

eye. The Streamline Modernestyle, 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 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.

[edit]

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10 - FLUID MECHANICS Page 5 Fluid Dynamics:10.4 Characteristics of fluid flow( 1 ) Steady flow: In a steady flow of fluid, the velocity of the fluid at each point remains constant with time. Every particle of the fluid passing through a given point will have the same velocity. Let the particles of fluid, P, Q and R have velocities →vP , →vQ and →vR respectively which may all be different. But these velocities do not change with time and all particles of the fluid in its flow passing through these points will have these velocities at all times. Such a condition is achieved only at low speeds, e.g., a gently flowing stream. ( 2 ) Unsteady flow: In an unsteady flow of fluid, the velocity of the fluid at a given point keeps on changing with time as in motion of water during ebb and tide. ( 3 ) Turbulent flow: In turbulent flow, the velocity of fluid changes erratically from point to point and from time to time as in waterfalls, breaking of the sea waves. ( 4 ) Irrotational flow: In irrotational flow, no element of fluid has net angular velocity. A small paddle wheel placed

in such a flow will move without rotating. ( 5 ) Rotational flow: In rotational flow, an element of fluid at each point has net angular velocity about that point. A paddle wheel kept in such a flow has turbulent motion while rotating. Rotational flow includes vortex motion such as whirlpools, the air thrown out of exhaust fans, etc. ( 6 ) Incompressible flow: In incompressible flow, the density of fluid remains constant with time everywhere. Generally, liquids and sometimes even a highly compressed gas flow incompressibly. Flow of air relative to the wings of an aeroplane flying below sonic velocity is incompressible. ( 7 ) Compressible flow: In compressible flow, density of fluid changes with position and time. ( 8 ) Non-viscous flow: In non-viscous flow, fluid with small co-efficient of viscosity flows readily. Normally, the flow of water is non-viscous. ( 9 ) Viscous flow: In viscous flow, fluid having large co-efficient of viscosity cannot flow readily. Castor oil, tar have viscous flow. 10.5 Streamlines, Tube of flow and Equation of continuityThe path of motion of a fluid particle is called a line of flow. In a steady flow, velocity of each particle arriving at a point on this path remains constant with time. Hence, every particle reaching this point moves in the same direction with the same speed. However, when this particle moving on the flow line reaches a different point, its velocity may be different. 10 - FLUID MECHANICS Page 6 But this different velocity also remains constant with respect to time. The path so formed is called a streamline and such a flow is called a streamline flow. In unsteady flow, flow lines can be defined, but they are not streamlines as the velocity at a point on the flow line may not remain constant with time. Streamlines do not intersect each other, because if they do then two tangents can be drawn at the point of intersection and the particle may move in the direction of any tangent which is not possible. Tube of flow:The tubular region made up of a bundle of streamlines passing through the boundary of any surface is called a tube of flow.

The tube of flow is surrounded by a wall made of streamlines. As the streamlines do not intersect, a particle of fluid cannot cross this wall. Hence the tube behaves somewhat like a pipe of the same shape. Equation of continuity:In a tube of flow shown in the figure, the velocity of a particle can be different at different points, but is parallel to the tube wall. In a non-viscous flow, all particles in a given crosssection have the same velocity. Let the velocity of the fluid at cross-section P, of area A1, and at crosssection Q, of area A2, be v1 and v2 respectively. Let ρ1 and ρ2 represent density of the fluid at P and Q respectively. Then, as the fluid can not pass through the wall and can neither be created or destroyed, the mass flow rate ( also called mass flux ) at P and Q will be equal and is given by dtdm = ρ1 A1 v1 = ρ2 A2 v2This equation is known as the law of conservation of mass in fluid dynamics. For liquids, which are almost incompressible, ρ1 = ρ2.

A1 v1 = A2 v2 … … ( 1 ) which implies Av = constant … … ( 2 ) or, v ∴ ∝A1Equations ( 1 ) and ( 2 ) are known as the equations of continuity in liquid flow. The product of area of cross-section, A and the velocity of the fluid, v at this cross section, i.e., Av, is known as the volume flow rate or the volume flux. Thus, velocity of liquid is larger in narrower cross-section and vice versa. In the narrower cross-section of the tube, the streamlines are closer thus increasing the liquid velocity. Thus, widely spaced streamlines indicate regions of low speed and closely spaced streamlines indicate regions of high speed.