KAHIBARO
Table of Contents
Flow conservation in moving fluids
The continuity equation expresses a simple but powerful idea, mass is conserved as a fluid flows. If fluid is not being created or destroyed inside a region, then the amount of mass entering and leaving must balance properly.
In beginner fluid mechanics, the continuity equation is most often used to relate the speed of a fluid to the cross sectional area of the pipe or channel through which it moves. It explains why water speeds up when a pipe narrows and slows down when a pipe widens.
Volume flow rate and mass flow rate
To build the continuity equation, consider fluid flowing through a pipe. During a short time interval $\Delta t$, a slice of fluid of length $\Delta x$ passes through a cross section. If the fluid speed is $v$, then
$$
\Delta x = v \Delta t
$$
If the cross sectional area is $A$, then the volume that passes through is
$$
\Delta V = A \Delta x = A v \Delta t
$$
So the volume flow rate is
$$
Q = \frac{\Delta V}{\Delta t} = Av
$$
If the fluid density is $\rho$, then the mass of that fluid slice is
$$
\Delta m = \rho \Delta V
$$
and the mass flow rate is
$$
\frac{\Delta m}{\Delta t} = \rho Av
$$
Important definitions:
$$
Q = Av
$$
is the volume flow rate, and
$$
\dot m = \rho Av
$$
is the mass flow rate.
For steady flow, the mass flow rate is the same at every cross section of a single flow tube.
Continuity equation for incompressible flow
For an incompressible fluid, the density stays constant. This is a very good approximation for liquids such as water under ordinary conditions.
If fluid flows through two different sections of a pipe, labeled 1 and 2, conservation of mass gives
$$
\rho A_1 v_1 = \rho A_2 v_2
$$
Since $\rho$ is the same on both sides, it cancels:
$$
A_1 v_1 = A_2 v_2
$$
This is the continuity equation for incompressible steady flow.
It means that when area decreases, speed must increase, and when area increases, speed must decrease.
For steady incompressible flow:
$$
A_1 v_1 = A_2 v_2
$$
Smaller area means larger speed.
Continuity equation for compressible flow
For gases, density can change significantly, so we usually must keep $\rho$ in the equation. Then the continuity relation is
$$
\rho_1 A_1 v_1 = \rho_2 A_2 v_2
$$
This is the more general one dimensional form for steady flow.
For low speed gas flow, density changes may be small, and the simpler incompressible form can sometimes still be used as an approximation.
Physical meaning
Imagine fluid moving through a pipe that narrows. If the same amount of fluid mass must pass every second, then fluid particles in the narrow section must move faster. Otherwise fluid would pile up, which would violate mass conservation in steady flow.
A helpful everyday picture is placing your thumb partly over the end of a garden hose. The opening becomes smaller, and the water comes out faster. That speed increase is a direct consequence of continuity.
Visualizing the equation
The wider section has larger area and smaller speed. The narrower section has smaller area and larger speed.
Comparing sections of a pipe
The continuity equation is especially easy to use in table form.
| Quantity | Section 1 | Section 2 |
|---|---|---|
| Area | $A_1$ | $A_2$ |
| Speed | $v_1$ | $v_2$ |
| Density | $\rho_1$ | $\rho_2$ |
| Mass flow rate | $\rho_1 A_1 v_1$ | $\rho_2 A_2 v_2$ |
For steady flow, the last row must be equal at both sections.
If the fluid is incompressible, then $\rho_1 = \rho_2$, so the product $Av$ must be the same at both sections.
Simple example
Suppose water flows through a pipe that narrows from area $A_1 = 4.0\ \text{cm}^2$ to area $A_2 = 1.0\ \text{cm}^2$. If the speed in the wide section is $v_1 = 2.0\ \text{m/s}$, find the speed in the narrow section.
Using continuity for incompressible flow,
$$
A_1 v_1 = A_2 v_2
$$
so
$$
v_2 = \frac{A_1}{A_2} v_1
$$
Substitute the values:
$$
v_2 = \frac{4.0}{1.0} \times 2.0 = 8.0\ \text{m/s}
$$
The speed becomes four times larger because the area becomes four times smaller.
When the continuity equation applies
The continuity equation relies on conservation of mass. In this chapter, it is usually applied under steady flow conditions, meaning the flow pattern does not change with time at a given point.
It works very naturally for flow in pipes, tubes, and streamlines where fluid passes from one cross section to another. In liquids, the incompressible form is commonly used. In gases, the general mass conservation form is safer unless density changes are known to be negligible.
Common mistakes
A common mistake is to assume that flow rate and speed are the same thing. They are not. Speed is how fast the fluid moves, while flow rate measures how much fluid passes per unit time.
Another common mistake is forgetting that area must be perpendicular to the flow direction. The cross sectional area in $Q = Av$ is not just any surface area, it is the area cut across the stream.
Students also sometimes use $A_1 v_1 = A_2 v_2$ for gases without checking whether density changes matter. For compressible flow, use
$$
\rho_1 A_1 v_1 = \rho_2 A_2 v_2
$$
instead.
Do not confuse speed $v$ with flow rate $Q$.
$$
Q = Av
$$
and only for incompressible steady flow between two sections,
$$
A_1 v_1 = A_2 v_2
$$
Differential idea behind continuity
The full continuity equation in advanced fluid mechanics describes how density and velocity can vary from point to point and in time. For beginners, the key idea is enough, whatever mass enters a flow tube per second must also leave it per second in steady flow.
So the continuity equation is the mathematical statement of mass conservation in fluid motion. It is one of the central tools of fluid dynamics and often works together with other relations to analyze flowing fluids.