Continuity uses the conservation of matter to describe the relationship between the velocities of a fluid in different sections of a system. The simple observation that the volume flow rate, , must be the same throughout a system provides a relationship between the velocity of the fluid through a pipe and the cross-sectional area.
Continuity works in tandem with Bernoulli's principle in the design and construction of systems of irrigation, plumbing, etc.
The first rate of change of interest for fluid in motion is the mass flow rate: the amount of mass that passes through a checkpoint in one unit time. Or, mathematically,
However, the mass of a fluid is strange to calculate, since there is not necessarily a feesibly measurable amount of, say, water flowing through a pipe. As such, relations to density, are preferred. Combining this equation with the mass flow rate equation above gives
Provided the pipe is prismatic, its volume can be expressed where is the cross-sectional area and is a length perpendicular to that area.
Also of interest is volume flow rate.
The average volume flow of blood through an aortic valve is . If the cross-sectional area of a healthy aortic valve is , what is the average speed of the blood?
The volume flow rate equation is . Since the problem provides the rate and and asks for , use the second definition.
Imagine two pipes of different diameters connected so that all the matter that passes through the first section must pass through the second. This means the mass flow rate of each section must be equal, otherwise some mass would be disappearing between the two sections. Mathematically, this can be expressed as
According to the definition of density, , so
As long as the fluid is incompressible, the density will not change from one section to the next, so cancels out of both sides. Additionally, if the pipes are prismatic (as most conventional pipes are), then the volume can be expressed in terms of the cross-sectional area and length: The equation is now
But since ,
Water flows from a cylindrical pipe of radius 4 into another cylindrical pipe of radius 2 . If the water exhibits a velocity of 20 in the second pipe, what was the velocity in the first?
Solution: Fluid velocity in joined pipes requires the continuity equation:
Since the pipes are cylindrical, each cross-section is a circle with area . Additionally, is given, so