Law of continuity

The law of continuity states (in integral form) that the mass flow of a fluid (liquid or gas) in a pipe is independent of where it is measured. The differential form is the continuity equation . It applies in both frictionless and frictionless cases for stationary (time-independent) and unsteady flows of incompressible fluids, but not for unsteady flows of compressible fluids.

For incompressible (non-compressible) fluids, continuity also applies to the volume flow .

Incompressible fluids

Change in cross-section of a pipe

According to the law of continuity for incompressible fluids, the same volume emerges from a pipe section as it does at the other end. In the figure on the right, the two volumes are marked in gray, whereby plug flow has been assumed for the sake of simplicity . The volume entering is the volume exiting . Because in the narrower part of the tube, the displacement is greater by the same factor than by which the cross section is greater than . The same applies to the flow velocities (averaged over the cross section) . ${\displaystyle V_{1}=A_{1}\Delta x_{1}}$ ${\displaystyle V_{2}=A_{2}\Delta x_{2}}$ ${\displaystyle V_{1}=V_{2}}$ ${\displaystyle \Delta x_{2}}$ ${\displaystyle \Delta x_{1}}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle v=\Delta x/\Delta t}$

Giovanni Battista Venturi found this connection , see Bernoulli's equation .

Compressible fluids

For compressible (not volume-stable) fluids or fluids that can change their density, the following applies to the mass flow:

${\displaystyle Q=v\cdot A\cdot \rho ={\text{konstant}}}$

${\displaystyle \rho }$ = Density of the fluid

or.

${\displaystyle v_{1}\cdot A_{1}\cdot \rho _{1}=v_{2}\cdot A_{2}\cdot \rho _{2}}$

${\displaystyle \rho _{1,2}}$ = Density in pipe 1 or 2

Thus, the following applies: the mass that goes in on one side must come out on the other side.

The density of the fluid can change, for example, when the temperature of the fluid changes between the beginning and the end of the pipe. When the density decreases, a larger volume must come out in the same time.

Trivia

In contrast to the first appearance, even cars in traffic jams behave according to the law of continuity when the lane is narrowed. The distance between the cars must be considered as the density. With a large cross-section, the density is low, the speed is high and the traffic flows freely. In the traffic jam before the narrowing, the density is high and the speed is low. In the constriction, the cross-section is small, the speed and the density are medium, and the vehicle throughput is the same in all cases, provided that no car leaves the road or comes onto it.

Web links

• The law of continuity
• The volume flow ( Memento from August 11, 2014 in the Internet Archive )
• Yves Mayer: Effects of winglets on the aerodynamics of an aircraft . Matura thesis, p. 11 (PDF file; 4663 kB).

Individual evidence

1. ^ Matt Anderson: Continuity Equation Moving fluids and traffic. In: youtube.com. June 24, 2014, accessed January 26, 2018 . 
2. ^ Fluid Dynamics Explains Some Traffic Jams. In: insidescience.org. American Institute of Physics, November 8, 2013, accessed January 26, 2018 .