13.17:
Equation of Continuity
The volume of fluid flowing through a point in an area in unit time gives the volume flow rate.
On substituting volume with area times distance, and using the relation between velocity and distance, volume flow rate equals area times velocity of the fluid.
Consider an incompressible fluid with the same density at all points flowing steadily through an irregular cross-sectional pipe. For a steady flow, the velocity and density of the fluid at a point remain constant with time.
The mass of the fluid passing through a point per unit time is termed as mass flow rate, and it equals density times the volume flow rate.
Since the pipe does not have any other source or sink, the mass flowing into the pipe must equal the mass leaving the pipe. This gives the general equation of continuity for fluids.
For incompressible fluids, density cancels out. Hence, the volume flow rate into the pipe equals the volume flow rate out of the pipe.
13.17:
Equation of Continuity
Fluid motion is represented by either velocity vectors or streamlines. The volume of a fluid flowing past a given location through an area during a period of time is called the flow rate Q, or more precisely, the volume flow rate. Flow rate and velocity are related—for instance, a river has a greater flow rate if the velocity of the water in it is greater. However, the flow rate also depends on the size and shape of the river. The relationship between flow rate (Q) and average speed (v) suggests that flow rate is directly proportional to both the average speed of the fluid and the cross-sectional area of a river, pipe or other conduit. The larger the conduit, the greater is its cross-sectional area.
Consider an incompressible fluid flowing through a pipe of decreasing radius. Here, since the fluid is incompressible, the same amount of fluid must flow past any point in the tube in a given time to ensure continuity of flow. The flow is continuous because there are no sources or sinks that add or remove mass, so the mass flowing into the pipe must be equal to the mass flowing out of the pipe. In this case, because the cross-sectional area of the pipe decreases, it is necessary for the velocity to increase. This logic can be extended to say that the flow rate must be the same at all points along the pipe. This is called the equation of continuity, and it is valid for any incompressible fluid (with constant density). Since liquids are essentially incompressible, the equation of continuity is valid for all liquids. However, gases are compressible, so the equation must be applied with caution to gases if they are subjected to compression or expansion.
This text is adapted from Openstax, University Physics Volume 1, Section 14.5: Fluid Dynamics.