Streamlines, streaklines, and pathlines

Streamlines, streaklines and pathlines are field lines in a fluid flow.
They differ only when the flow changes with time, that is, when the flow is not steady.
Considering a velocity vector field in three-dimensional space in the framework of continuum mechanics, we have that: 
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 -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 lines 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
where "formula_2" denotes the vector cross product and formula_3 is the parametric representation of "just one" streamline at one moment in time.
If the components of the velocity are written formula_4 and those of the streamline as formula_5 we deduce
which shows that the curves are parallel to the velocity vector. Here formula_7 is a variable which parametrizes the curve formula_8 Streamlines are calculated instantaneously, meaning that at one instance of time they are calculated throughout the fluid from the instantaneous flow velocity field.
A streamtube consists of a 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:
formula_9
The flow velocity in the direction formula_7 of the streamline is denoted by formula_11. formula_12 is the radius of curvature of the streamline. The density of the fluid is denoted by formula_13 and the kinematic viscosity by formula_14. formula_15 is the pressure gradient and formula_16 the velocity gradient along the streamline. For a steady flow, the time derivative of the velocity is zero: formula_17. formula_18 denotes the gravitational acceleration.
Pathlines.
Pathlines are defined by
The suffix formula_20 indicates that we are following the motion of a fluid particle. 
Note that at point formula_21 the curve is parallel to the flow velocity vector formula_22, where the velocity vector is evaluated at the position of the particle formula_21 at that time formula_24.
Streaklines.
Streaklines can be expressed as,
where, formula_26 is the velocity of a particle formula_20 at location formula_28 and time formula_24. The parameter formula_30, parametrizes the streakline formula_31 and formula_32, where formula_24 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, formula_34, further on that streamline the equations governing the flow will send it in a certain direction formula_35. As the equations that govern the flow remain the same when another particle reaches formula_34 it will also go in the direction formula_35. If the flow is not steady then when the next particle reaches position formula_34 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. 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. 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. 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 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 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.