Continuity Equation for Incompressible Flow

Hello! Welcome to your next lesson in our Fluid Mechanics module.

In the previous lesson, we established a crucial first step in analyzing fluid motion: distinguishing between laminar and turbulent flow using the Reynolds number. This allowed us to characterize the nature of the flow. Now, we will begin applying fundamental physical laws to predict the flow's behavior quantitatively.

Today's lesson focuses on one of the most fundamental principles in all of physics and engineering: the conservation of mass. In fluid mechanics, this principle is expressed through the continuity equation. Your learning outcome is to apply the continuity equation (conservation of mass) to incompressible flows. This powerful and intuitive tool allows you to relate fluid velocity to the geometry of the system, a common task in engineering design, from sizing fuel lines in an aircraft to designing wind tunnels.

1. The Core Idea: What Goes In Must Come Out

At its heart, the continuity equation is a mathematical statement of a simple idea: for a steady flow in a contained system (like a pipe), the mass of fluid entering a section per unit time must equal the mass leaving that section per unit time. Mass can't just appear or disappear.

To apply this principle, we first need to define two key terms: volume flow rate and mass flow rate.

2. Defining Flow Rates

Volume Flow Rate ($Q$)

The volume flow rate, often denoted by $Q$, is the volume of fluid that passes through a given cross-section per unit of time. If a fluid is moving with an average velocity $v$ through a pipe with a cross-sectional area $A$, the volume flow rate is:

$Q=A\cdot v$

The typical SI units for $Q$ are cubic meters per second (m³/s).

Mass Flow Rate ($\dot{m}$)

The mass flow rate, denoted by $\dot{m}$ (pronounced "m-dot"), is the mass of fluid passing through a cross-section per unit of time. It's related to the volume flow rate by the fluid's density $\rho$:

$\dot{m}=\rho \cdot Q=\rho \cdot A\cdot v$

The typical SI units for $\dot{m}$ are kilograms per second (kg/s).

The following video provides a clear introduction to volume flow rate and mass flow rate and derives the continuity equation we will be using.

3. The Continuity Equation for Incompressible Flow

As explained in the video, the principle of conservation of mass for a steady flow between two points (1 and 2) in a pipe means:
${\dot{m}}_1={\dot{m}}_2$
Substituting the formula for mass flow rate, we get the general form of the continuity equation:
${\rho }_1{A}_1{v}_1={\rho }_2{A}_2{v}_2$

This lesson focuses on incompressible flows. A fluid is considered incompressible if its density $\rho$ is constant throughout the flow. Most liquids, like water, are treated as incompressible. Gases, like air, can also be treated as incompressible if they are moving at relatively low speeds (as a rule of thumb, below about 30% of the speed of sound).

For an incompressible flow, ${\rho }_1={\rho }_2$, so the density term cancels out, leaving us with the simplified continuity equation:

$A_1{v}_1=A_2{v}_2$

This also means the volume flow rate is constant: $Q_1=Q_2$. This is the key formula for this lesson. It tells us that for a given flow rate, if the pipe area decreases, the velocity must increase, and vice versa.

Let's watch another example to solidify this. This one uses the pipe radius, which is a common variation.

4. Continuity for Branching Flows

The principle of mass conservation also applies to more complex systems, like pipes that split into multiple branches. For an incompressible fluid, the total volume flow rate entering a junction must equal the sum of the volume flow rates leaving the junction.

$Q_{in}=\sum Q_{out}$

For a pipe that splits from one inlet (1) into two outlets (2 and 3), this becomes:
$A_1{v}_1=A_2{v}_2+A_3{v}_3$

The following reading, from an aerospace-focused university textbook, explains and illustrates this concept with several clear examples.

Now, let's see this applied in a video problem that involves a branching pipe system.

Conclusion

In this lesson, you've learned to apply the principle of mass conservation to incompressible fluid flows. This has given us the continuity equation, a fundamental tool for any fluid mechanics analysis.

Key Takeaways:

• Mass Flow Rate: $\dot{m}=\rho Av$ is the mass of fluid passing a point per unit time.
• Volume Flow Rate: $Q=Av$ is the volume of fluid passing a point per unit time.
• Conservation of Mass: For steady flow, the mass entering a system must equal the mass leaving it.
• Continuity Equation (Incompressible): For a fluid with constant density, this simplifies to $A_1{v}_1=A_2{v}_2$. This means velocity is inversely proportional to the cross-sectional area.
• Branching Flows: For a junction, the total flow rate entering must equal the sum of the flow rates exiting: $Q_{in}=\sum Q_{out}$.

Next Steps:
We have now covered the first fundamental law (conservation of mass). The next logical step is to introduce the conservation of energy. For fluid dynamics, this leads to the famous Bernoulli's equation, which will allow us to relate pressure, velocity, and elevation in a moving fluid. You've already had a sneak peek of it in the last video, and we will explore it in detail in our next lesson.