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# Streamlines, streaklines, and pathlines

Fluid flow is described in general by a vector field in three (for steady flows) or four (for non-steady flows including time) dimensions. Pathlines, streamlines, and streaklines are field lines of different vector field descriptions of the flow. For steady flow (see below), the three are the same.

• Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. This means that if a point is picked then at that point the flow moves in a certain direction. Moving a small distance along this direction and then finding out where the flow now points would draw out a streamline.
• Streaklines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past. This can be found experimentally by releasing dye into the fluid in a time period at a fixed point and then at a later time finding out where the dye was.
• Pathlines are the trajectory that a fluid particle would make as it moves around with the flow.

By definition, streamlines defined at a single instant in a flow do not intersect. This is so 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 instance of time. 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).

In simple terms, streamlines and streaklines are like a snapshot of the flowfield whereas pathlines are time-history of the flow.

A region bounded by streamlines is called a streamtube. Because the streamlines are tangent to the flow velocity, fluid that is inside a stream tube must remain forever within that same stream tube. A scalar function whose contours define the streamlines is known as the stream function.

## Mathematical description

### Streamlines

Streamlines are defined as

${d\vec{x}\over ds} = \vec{u}(\vec{x}).$

If the components of the velocity are written $\vec{u} = (u,v,w)$, we deduce ${dx\over u} = {dy\over v} = {dz\over w}$, which shows that the curves are parallel to the velocity vector. Here s is a variable which parametrizes the curve $s\mapsto \vec{x}(s)$. For streamlines there is no t (time) dependence. This is because they are calculated instantaneously, meaning that at one instance of time they are calculated throughout the fluid.

### Pathlines

Pathlines are defined by

$\begin{cases} \frac{d\vec{x}_P}{dt} = \vec{u}_P(\vec{x}_P,t) \\ \vec{x}_P(t_0) = \vec{x}_{P0} \end{cases}$

The suffix P indicates that we are following the motion of a fluid particle. Note that at point $\vec{x}_P$ the curve is parallel to the flow velocity vector $\vec{u}$, where the velocity vector is evaluated at the position of the particle $\vec{x}_P$ at that time t.

### Streaklines

Streaklines can be expressed as,

$\begin{cases} \frac{d \vec{x}_{P} }{dt} = \vec{u}_{P} (\vec{x}_{P},t) \\ \vec{x}_{P}( t = \tau_{P}) = \vec{x}_{P0} \end{cases}$

where, $\vec{u}_{P}$ is the velocity of a particle P at location $\vec{x}_{P}$ and time t. The parameter τP, parametrizes the streakline $\vec{x}_{P}(t,\tau_{P})$ and $0 \le \tau_{P} \le t_0$, where t0 is a time of interest.

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, a0, further on that streamline the equations governing the flow will send it in a certain direction $\vec{x}$. As the equations that govern the flow remain the same when another particle reaches a0 it will also go in the direction $\vec{x}$. If the flow is not steady then when the next particle reaches position a0 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 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.

## 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 radius of curvature of the streamline is 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, and then use the patterns to 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 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 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.