Experiment of The Month
Velocity Field in a Pipe
When real fluids flow in a pipe, flow is limited by a kind of liquid friction called viscosity. As a first model for viscosity, we can imagine a trough of viscous fluid, like sugar syrup, as sketched below. We place a microscope slide, edge down, in the fluid, and pull it to the right. The viscous force, like friction, acts to oppose
We use a simple rule to think about how the fluid behaves when we pull the slide through it: Syrup touching a surface sticks to it. That is, the fluid which touches the walls is at rest with respect to the walls, and the fluid which touches the microscope slide moves along with the slide. In the regions between the slide and the wall, the fluid velocity decreases as we look nearer to the wall.
Detailed calculations show that the fluid velocity is proportional to the distance from the wall. A sketch of the fluid velocity field is shown below. The length of each arrow indicates the size of the fluid velocity in the neighborhood of the arrow.
As a step towards flow in a pipe, we imagine replacing the microscope slide with fluid. Of course, it's
The resultant force due to pressure depends on the pressure difference, as indicated by the equation in the figure. If this pressure difference were applied only to this sample of fluid, the results would be the same as with the microscope slide. In practice, though moving water with pressure is a little different. We generally apply high pressure at the
The pressure is applied to all of the fluid on the left, and not just our sample. This means that each bit of fluid feels a pressure force pushing it to the right. It is still true that the fluid which touches the wall sticks to the wall, but now there are two forces pushing downstream: the same viscous force as before, and also the pressure force. Near the center of the tube, a sample of water does not fall behind the middle sample as rapidly, because the pressure force helps it along downstream.
Detailed analysis shows that the velocity field should have a parabolic shape, rather than the linear shape for the microscope slide. This derivation is done in Physics 334, Macroscopic Physics and Thermodynamics, for our sophomore physics majors. The figures below show a demonstration of this shape. Chris Kunkle filled a 3 cm diameter tube
|Yellow dish soap above green soap in
||At the beginning of fluid flow, the
interface bows down in the center
|A more fully developed flow pattern|
In each succeeding snapshot,
The assertion that the fluid has zero velocity at the walls is seen to be nearly verified by the thin blue layer that remains on the higher parts of the wall.
|Parabola sketched on interface||Parabola sketched on
A video record of the experiment is available here.
The assertion that the velocity of each bit is constant was tested by estimating the frame-by-frame change in position of the lowest part of the parabola. A graph of the measurements is at the right. The linear shape indicates