F 1 = 20.4 N
13.12 - Streamline flow
Streamline flow: A fluid flow in which the
fluid’s velocity remains constant at any
particular point.
Steady, or streamline, flow is one of the characteristics of ideal fluid flow. Streamline
flow is particularly easy to demonstrate with the flow of a gas, although since gasses
are compressible they are not ideal fluids.
At the right, you see one streamline traced by smoke in the diagram, flowing around an
automobile. As long as the car does not rotate, the streamline stays the same over time.
Any particle of the fluid will follow some streamline, visible or not, as it flows past the
car. As the car is rotated in the video, you get a chance to see the paths followed by
different streamlines of air flowing around various parts of its body.
Engineers use wind tunnels to photograph streamlines. Powerful fans blow air past an
object like a car or an airplane, and dyes or smoke are injected into the airflow at
several points and carried downstream so that the streamlines are made visible.
Engineers analyze the streamlines to investigate the air resistance of a particular car design, or the amount of lift (upward force) generated by
an airplane wing.
The velocity of streamline fluid flow can vary from point to point. Air moves past an airplane wing or auto body with different velocities at
different points (for example, it moves faster over the tops of these objects than beneath them). The tangent to a streamline at a point coincides
with the direction of the velocity vectors of the fluid particles passing by the point.
In streamline flow, the fluid has a constant velocity at all times at a given point. The velocity of any given air particle in the visible streamline
on the right may change as the flow carries it downstream past the stationary automobile. In particular, a change in the direction of the
streamline reflects a change in velocity. However, all the particles in the streamline pass the same point with the same velocity.
How can you conclude that the velocity remains constant at each point? Consider what would happen if the speed of the fluid flow at a point
were to vary over time. If the speed increased, the affected particles would collide with particles ahead of them; if it decreased, particles from
behind would collide. The resulting collisions would cause an erratic flow, changing the streamline, as would changes in the direction of the
particles’ motions. The constancy of the streamlines over time indicates that the velocity at each point does not change.
Streamline flow
At different points, velocities can differ
At any point, velocity constant over timeImage courtesy of Lexus13.13 - Fluid continuity
Fluid equation of continuity: The volume flow
rate of an ideal fluid flowing through a closed
system is the same at every point.
Turn on a hose and watch the water flow out, and then cover half the hose end with
your thumb. The water flows faster through the narrower opening. You have just
demonstrated the fluid equation of continuity: How much volume flows per unit time í
the volume rate of flow í stays constant in a closed system. The increased speed of the
flow at the opening balances the decreased cross sectional area there. Of course after
the water leaves the hose with its new speed, it is no longer in the closed system, and
you cannot apply the equation of continuity to the resulting spray of droplets. They
spread out without slowing down.
The fluid equation of continuity can be observed in rivers, whose courses approximate
closed systems. River water flows more quickly through narrow or shallow channels, called rapids, and more slowly where riverbeds are wider
and deeper. This relationship is alluded to by the proverb, “Still waters run deep.”
The constancy of fluid flow rate is summarized in the continuity equation that appears as the first line in Equation 1. In this equation, stated for
two arbitrary points P 1 and P 2 , the rate of flow is measured as the mass flow rate. The mass flow rate equals the product of the speed of the
fluid, its density, and the cross sectional area it flows through. It is measured in kilograms per second (kg/s). In a closed system (no leaks, no
inflows) the mass flow rate is the same past all points.
Since we assume that an ideal fluid is incompressible, having a density that is constant, we can cancel the density factor from both sides of the
first continuity equation. This enables us to say that the speed of the fluid times its cross sectional area is everywhere the same. This is stated
by the second equation, which expresses continuity in terms of the volume flow rate, represented by R. The volume flow rate is measured in
cubic meters per second (m^3 /s). If the cross sectional area decreases (as when the pipe illustrated in Equation 1 narrows), the speed of the
fluid flow increases, and R remains the same.
Fluid continuity
Fluid flows past each point at same rate