Continuity Equation Derivation

1. Frameworks for Describing Fluid Flow

Before we dive into derivations, we must first establish how we describe fluid motion. There are two primary viewpoints, which you will see used throughout your study of physics.

To summarize:

• Lagrangian Viewpoint: You follow an individual fluid particle as it moves through space and time. This is like tracking a single car on a highway. The governing equations describe the properties (like velocity, acceleration, density) of that specific particle.
• Eulerian Viewpoint: You stay at a fixed point in space and observe the fluid as it flows past you. This is like standing on an overpass and watching the traffic go by. The governing equations describe the properties of the flow at fixed spatial coordinates $(x,y,z)$.

Most of fluid mechanics, including the derivations we will do today, uses the Eulerian framework because it is often impractical to track every single fluid particle. However, the concept of a particle's acceleration is fundamentally Lagrangian. To bridge these two viewpoints, we need a special mathematical tool.

The Substantial Derivative

The substantial derivative, denoted $D/Dt$, represents the rate of change of a property (like temperature or density) for a fluid particle that is moving with the flow. It connects the Lagrangian concept of a particle's change to the Eulerian field description. It is composed of a local part (change at a fixed point) and a convective part (change due to moving to a new location in the field).

The key result is the definition of the substantial derivative operator:
$\frac{D}{Dt}\equiv \frac{\partial }{\partial t}+\vec{V}\cdot \nabla$
where $\frac{\partial }{\partial t}$ is the local (unsteady) derivative, and $\vec{V}\cdot \nabla$ is the convective derivative. Understanding this operator is essential for the full derivation of the Navier-Stokes equations later on.

2. Derivation 1: The Differential Control Volume

Our first approach is to apply the principle of mass conservation to an infinitesimally small, stationary cubic volume of fluid. We will account for the mass flowing in and out of each face and equate the net outflow to the rate at which mass decreases inside the volume.

This method relies on using Taylor series to express the change in flow properties across this tiny volume. The following video provides a clear, step-by-step walkthrough of this derivation.

After summing the contributions from all six faces and equating the net outflow to the decrease of mass inside the volume, we arrive at the continuity equation in Cartesian coordinates:
$\frac{\partial \rho }{\partial t}+\frac{\partial (\rho u)}{\partial x}+\frac{\partial (\rho v)}{\partial y}+\frac{\partial (\rho w)}{\partial z}=0$

Using the divergence operator $\nabla \cdot$, which in Cartesian coordinates is $\nabla =(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z})$, we can write this much more compactly. If $\vec{V}=(u,v,w)$, then the equation becomes:
$\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \vec{V})=0$
This is the conservation form of the continuity equation. It is one of the most important equations in fluid dynamics.

3. Derivation 2: The Integral Control Volume (Gauss's Theorem)

Our second approach is more general and showcases the power of vector calculus. We will start with a finite control volume $V$ with surface $A$, and state the principle of mass conservation in an integral form.

The principle is:
The time rate of change of mass inside the volume is equal to the negative of the net rate at which mass flows out across the surface.

Let's trace the key steps of the derivation:

1. Integral Statement: The principle of mass conservation is written as:
   $\frac{d}{dt}{\iiint }_V\rho dV=-\iint (\rho \vec{V})\cdot \hat{n}dA$
   The left side is the rate of mass increase in volume $V$. The right side is the net mass flux into the volume.

2. Applying Gauss's Divergence Theorem: The divergence theorem states $\iint \vec{F}\cdot \hat{n}dA={\iiint }_V(\nabla \cdot \vec{F})dV$. Applying this to the right side of our equation (with $\vec{F}=\rho \vec{V}$) transforms the surface integral into a volume integral:
   $\frac{d}{dt}{\iiint }_V\rho dV=-{\iiint }_V\nabla \cdot (\rho \vec{V})dV$

3. Final Step: Since the control volume $V$ is fixed in space, we can bring the time derivative inside the integral. Combining all terms under one integral gives:
   ${\iiint }_V\left(\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \vec{V})\right)dV=0$
   Because this equation must hold for any arbitrary control volume $V$, the only way for the integral to always be zero is if the integrand itself is zero everywhere. This gives us the differential form:
   $\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \vec{V})=0$
   We have arrived at the exact same equation as our first derivation, but through a more powerful and generalizable method.

4. Conservation vs. Non-Conservation Forms

The equation $\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \vec{V})=0$ is in conservation form. We can use a vector identity to rewrite it in a different, but equivalent, non-conservation form that involves the substantial derivative we learned earlier.

As the text shows, by expanding $\nabla \cdot (\rho \vec{V})=\rho (\nabla \cdot \vec{V})+\vec{V}\cdot \nabla \rho$, the continuity equation becomes:
$\frac{\partial \rho }{\partial t}+\vec{V}\cdot \nabla \rho +\rho (\nabla \cdot \vec{V})=0$
Recognizing that the first two terms are the substantial derivative of density, $D\rho /Dt$, we get the non-conservation form:
$\frac{D\rho }{Dt}+\rho (\nabla \cdot \vec{V})=0$

5. Incompressible Flow: A Major Simplification

Many liquids (like water) and gases at low speeds can be modeled as incompressible, meaning their density $\rho$ is constant everywhere. This leads to a massive simplification of the continuity equation.

If $\rho$ is constant, then its derivative in time and space is zero. Therefore, the substantial derivative $D\rho /Dt=0$. The continuity equation $\frac{D\rho }{Dt}+\rho (\nabla \cdot \vec{V})=0$ reduces to:
$\rho (\nabla \cdot \vec{V})=0$
Since $\rho \ne 0$, we must have:
$\nabla \cdot \vec{V}=0$
This is the incompressible continuity equation. It is a simple but profound statement: the velocity field of an incompressible flow must be divergence-free. This means that fluid cannot accumulate at or spread out from any point in the flow.

To see what this looks like in different coordinate systems and how it's used in practice, let's return to Dr. Cimbala's video.

The incompressible continuity equation in Cartesian coordinates is:
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$

This is a powerful constraint that any valid incompressible velocity field must satisfy.

Conclusion

In this lesson, we have rigorously derived the continuity equation, the mathematical statement of mass conservation in a fluid. This is the first of the fundamental governing equations of fluid dynamics.

Key Takeaways:

• The principle of mass conservation can be expressed for a fluid flow as the continuity equation.
• The general, compressible form is $\frac{\partial \rho }{\partial t}+\nabla \cdot (\rho \vec{V})=0$.
• This equation can be derived by analyzing either a differential element (using Taylor series) or an integral volume (using the Divergence Theorem).
• For an incompressible flow ($\rho =\text{constant}$), the equation simplifies dramatically to $\nabla \cdot \vec{V}=0$. This means the velocity field must be divergence-free.

Preview of the Next Lesson:
With mass conservation established, we will next turn to the conservation of energy. In our next lesson, we will derive and apply Bernoulli's equation for ideal, incompressible, steady fluid flow. This famous equation relates pressure, velocity, and height in a moving fluid and is a cornerstone of aerodynamics and hydraulics.