U.S. Centennial of Flight Commission home page


Particle pathline and streamlines

Particle pathline and streamlines. Figure (a) shows a particle pathline in this example of a transmitting ocean buoy at six-hour intervals over a period of several days. Figure (b) shows the surface ocean currents at a particular fixed time. The lines comprising this flow field are called streamlines.


Streamlines at time 0 and time 1

Figure (a) shows the fluid flow (or air) about a house on a windy day at one instant in time. Figure (b) shows the flow an instant of time later. One can see that this flow is unsteady. There are many areas where the flow pattern is different.


Steady flow – particle pathline = streamline

This figure shows unsteady and steady flow around a nicely "streamlined'' body in a wind tunnel. At time t0 the tunnel is not running and no air is flowing. At time t1 the tunnel is started and air begins flowing about the body.


Three types of flows

This figure shows an object like a small paddle wheel immersed in a moving fluid. If the wheel translates (moves) without rotating, the motion is irrotational.


Fluids and Fluid Flow

There are basically three states of matter: solid, liquid, and gas. The substance H2O is commonly called "ice" in its solid state, "water" in its liquid state, and "water vapor" in its gaseous state. When side forces, called shearing forces, are applied to a solid piece of ice, very large forces are needed to deform or break it. The solid has a very high internal friction, or resistance to shearing. The word for internal friction is viscosity and for a solid, its value is generally very large. Liquids have a lower viscosity than solids, and gases have a still lower viscosity, in other words, they have less internal friction.

Liquids and gases are both considered fluids since they behave differently from solids. Imagine two layers of water or air. If shear forces are applied to these layers, there will be a substantial and sustained relative motion of the layers with the air layers sliding faster over one another than the water layers. However, the fact that a shear force must be applied to deform both of these fluids indicates that they also possess internal friction.

Water, under normal temperatures, is about 50 times more viscous than air. Ice is 5 x 1016 times more viscous than air. One concludes that, in general, solids have extremely high viscosities whereas fluids have low viscosities. In the category of fluids, liquids generally possess higher viscosities than gases. Air, of primary interest in aerodynamics, has a relatively small viscosity, and in some theories, it is described as a perfect fluid—one that has zero viscosity or is "inviscid." But it will be shown that even this small viscosity of air (or internal friction) has important effects on an airplane in terms of lift and drag.

All fluids are compressible (that is, their density increases under increasing pressure) to some extent, but liquids are much less compressible than gases and are generally considered incompressible. Even gases may be treated as incompressible provided the airflow speeds involved are not great. For subsonic airflow over an airplane below about 150 meters per second (492 feet per second or about 336 miles per hour), air may be treated as incompressible, i.e., the density remains the same throughout the flow. At higher speeds, the effects of compressibility must be taken into account.

The Flow

A fluid flow (both liquid and air) may be described in two different ways: the Lagrangian approach (named after the famous French mathematician Joseph Louis Lagrange), and the Eulerian approach (named after Leonhard Euler, a famous Swiss mathematician). In the Lagrangian approach, one particle is chosen and is followed as it moves through space with time. The line traced out by that one particle is called a particle pathline. An example is a transmitting ocean buoy that observes a set path over regular intervals over a period of time. The path observed is the particle pathline.

A Eulerian approach is used to obtain a clearer idea of the airflow at one particular instant. One can look at a "photograph" of the flow of, for instance, surface ocean currents at a particular fixed time. The entire flow field is easily visualized. The lines comprising this flow field are called streamlines.

Thus, a pathline refers to the trace of a single particle in time and space whereas a streamline presents the line of motion of many particles at a fixed time. The question of whether particle pathlines and streamlines are ever the same is considered next.

Of basic importance in understanding fluid movements about an object is the concept of a "steady flow." On a windy day a person calls the wind steady if, from where he stands, it blows constantly from the same direction at a constant speed. If, however, the speed or direction changes, the wind is "gusty" or unsteady. In a similar manner the flow of a fluid (both liquid and air) about an object is steady if its velocity (speed and direction) at each point in the flow remains constant—this does not necessarily require that the velocity be the same at all points in the fluid.

This means that for unsteady flows, particle pathlines (the Langranian point of view) and streamlines (the Eulerian approach) are not equivalent. For a steady flow, however, a particle pathline and streamline are equivalent, and the Lagrangian point of view is the same as the Eulerian approach for flow visualization.

As well as steady or unsteady, fluid flow can be rotational or irrotational. If the elements of fluid at each point in the flow have no net angular (spin) velocity about the points, the fluid flow is said to be irrotational. One can imagine a small paddle wheel immersed in a moving fluid. If the wheel translates (or moves) without rotating, the motion is irrotational. If the wheel rotates in a flow, the flow is rotational.

According to a theorem of Hermann von Helmholtz, a German physicist who contributed much to theoretical aerodynamics, assuming zero viscosity, if a fluid flow is initially irrotational, it remains irrotational. In real life, viscosity effects are limited to a small region near the surface of the airfoil and in its wake. Most of the flow may still be treated as irrotational.

A simplifying argument often used to aid in understanding basic ideas about fluid flow is that of a one-dimensional fluid flow. Flows may be considered one-dimensional where the flow parameters (for example: density, velocity, temperature, pressure) vary as a function of one spatial variable (for example, length) and variations in the other two spatial dimensions (i.e., y and z) are negligible by comparison.

To make study of fluids somewhat easier, simplifying assumptions about fluids are made. The first is that fluid is considered to be inviscid (no viscosity); the second is that it is incompressible. Further, the flow is considered steady and one-dimensional. Fluids with these characteristics are said to be ideal fluids or perfect fluids. Once solutions of problems relating to the lift and drag of ideal fluids, or the inviscid flow, have been made, a solution of the viscous flow in the thin boundary layer allows the effects of skin friction drag to be calculated.

—Adapted from Talay, Theodore A. Introduction to the Aerodynamics of Flight. SP-367, Scientific and Technical Information Office, National Aeronautics and Space Administration, Washington, D.C. 1975

For Further Reading:

Anderson, Jr., John D. A History of Aerodynamics. Cambridge, England: Cambridge University Press, 1997.

Katz, Joseph and Plotkin, Allen. Low-Speed Aerodynamics, 2nd edition. Cambridge, England: Cambridge University Press, 2001.

Wegener, Peter P. What Makes Airplanes Fly? New York: Springer-Verlag, 1991.

“Flow Conditions.” Allstar Project. http://www.allstar.fiu.edu/aero/Hydr15.htm

Educational Organization

Standard Designation  (where applicable

Content of Standard

International Technology Education Association

Standard 2

Students will develop an understanding of the core concepts of technology.

American Association for the Advancement of Science


Understand principles of motion and forces

National Science Education Standards


Students should develop an understanding of conservation of energy and increase in disorder.

National Council of Teachers of Mathematics


Understand numbers, ways of representing numbers, relationships among numbers, and number systems.


Flow profiles

If a wheel rotates in a flow, as illustrated in this figure, the flow is rotational.


This figure shows airflow about an airfoil section. The flow far ahead of the airfoil section is uniform and of constant velocity. It is irrotational. It remains irrotational if zero viscosity is assumed.


Figure (a) shows a bundle of streamlines of a simple flow. Each streamline can be thought of as a stream tube since fluid flows along it as if in a tube. In the case of steady flow, the stream tube is permanent. Taken together, the bundle of stream tubes comprises an even larger stream tube. Fluid flows through it as, for example, water flows through a pipe or channel. The velocity varies across the tube, in general, according to the individual streamline velocity variation, as shown in figure (b). An "average" uniform value of velocity at the cross section can represent the actual varying value, as indicated in figure (c).