In fluid dynamics, streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. Pathlines are the trajectory that an infinitesimally small point would make if it followed the flow of the fluid in which it was embedded. A streakline is the locus of points of the all fluid particles that have passed continuously though a particular spatial point in the past. In steady flow (which is time-independent), the streamlines, pathlines, and streaklines coincide. A scalar function whose contours define the streamlines is known as the streamfunction.

Mathematical description

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Streamlines are defined as

$\{d\backslash mathbf\{x\}\backslash over\; ds\}\; =\; \backslash lambda\; \backslash mathbf\{u\},\; \backslash lambda\; \backslash in\; \backslash mathbb\{R\}$

If $\backslash mathbf\{u\}\; =\; (u,v,w)$ then at $t=t\_0\; ,\; \{dx\backslash over\; u\}\; =\; \{dy\backslash over\; v\}\; =\; \{dz\backslash over\; w\}$ shows that the curves are parallel to the velocity vector. Pathlines by

$\{d\backslash mathbf\{x\}\backslash over\; dt\}\; =\; \backslash mathbf\{u\}$ where $\backslash mathbf\{x\}(t\_0)\; =\; \backslash mathbf\{x\_0\}$ is the initial condition.

Streaklines have no single formula.

Frame Dependence

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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 a 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.

By definition, streamlines defined at a single instant in a flow do not intersect if the flow is incompressible. They cannot begin or end inside the fluid.

A region bounded by streamlines is called a stream tube. Because the streamlines are tangent to the flow velocity, fluid that is inside a stream tube must remain forever within that same stream tube.

Applications

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Knowledge of the streamlines can be useful in fluid dynamics. For example, Bernoulli's principle, which expresses conservation of mechanical energy, is only valid along a streamline. Also, the curvature of a streamline is an indication of the pressure change perpendicular to the streamline. The instantaneous centre of curvature of a streamline is in the direction of increasing pressure, and the magnitude of the pressure gradient can be calculated from the curvature of the streamline.

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 smoothes an operation. For instance, it is common to hear references to streamlining a business practice, or operation.

See also

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• Drag coefficient
• Equipotential surface

References

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External links

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• Streamline illustration
• Flow Description at University of Sydney
• Flow Visualization at the College of Engineering at Purdue University

fluid dynamics | Aerodynamics | Fluid mechanics | Mechanical engineering

Stromlinie | Linea di flusso