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Streamlines and streamtubes

In problems involving electric or magnetic fields we visualize the fields by introducing field lines (often called lines of force), which have the properties that (1) they are everywhere tangent to the field vectors, and (2) the density of the field lines is proportional to the magnitude (strength) of the field. We can use the same method to visualize the velocity field tex2html_wrap_inline887 by introducing streamlines, which have the following properties:

A streamtube is a tubular region of fluid surrounded by streamlines. Since streamlines don't intersect, the same streamlines pass through a streamtube at all points along its length. Let's take two cross-sections of a streamtube, with cross-sectional areas tex2html_wrap_inline931 and tex2html_wrap_inline933 (see Fig. 2.4).   

Figure 2.4: A streamtube in a fluid.

If these areas are made small enough then the fluid velocities across the cross-sections will be constant; let's call the speeds tex2html_wrap_inline935 and tex2html_wrap_inline937 (the directions of the velocities being perpendicular to the cross-sections). The rate at which mass is entering the streamtube is tex2html_wrap_inline939 ; the rate at which it is leaving is tex2html_wrap_inline941 . If the mass inside the streamtube is not changing with time, so that the fluid is either incompressible or in steady-state, we have


If, in addition, the fluid is incompressible, then tex2html_wrap_inline943 , and we have


so that the fluid speed increases when the cross-sectional area of the streamtube decreases. If this seems counter-intuitive, the following analogy might help. You drive down a four lane highway, and due to construction one lane on your side has been closed. The traffic crawls toward the point where the lane has been closed, with cars haphazardly merging into your lane. You finally hit the single lane, and whiz through the construction area. The cross-sectional area has decreased, and your speed has increased.

next up previous
Next: Circulation and vortices Up: The equation of continuity Previous: Incompressible flow

Vittorio Celli
Mon Aug 11 22:46:35 EDT 1997