The Reynolds Transport Theorem is a fundamental concept in fluid mechanics and transport phenomena. It essentially extends the Leibniz integral theorem to scenarios where we are interested in analyzing changes within a control volume that may be moving and deforming. This theorem provides a crucial link between Lagrangian and Eulerian descriptions of fluid motion, allowing us to bridge the gap between tracking individual fluid particles and observing changes within a fixed or moving control volume.
Delving into the Reynolds Transport Theorem
Imagine a region in space, $R$, containing a field – consider it a fluid for simplicity. Within this region, we define an arbitrary volume, $Omega(t)$, which can change its shape, volume, and position over time as it moves through $R$. We want to study a scalar quantity, $X$, associated with this volume, defined as:
$$ X = int_{Omega(t)} rho ,dV $$
Here, $rho$ represents the density of the quantity $X$ within the volume $Omega(t)$. The notation $Omega(t)$ highlights the time-dependent nature of our region of interest. Our goal is to determine the rate of change of $X$ with respect to time, $frac{dX}{dt}$.
Directly differentiating under the integral sign is not possible because the integration domain, $Omega(t)$, is itself a function of time. To overcome this, we employ the three-dimensional Leibniz integral theorem. Applying this theorem, coupled with the Divergence Theorem, we arrive at the Reynolds Transport Theorem:
$$ begin{align} frac{dX}{dt} &= frac{d}{dt}left(int{Omega(t)} rho ,dV right) &stackrel{text{Leibniz}}{=} int{Omega(t)} left( frac{partialrho}{partial t} + nabla cdot (rho mathbf{v}) right) ,dV &stackrel{text{Divergence}}{=} int{Omega(t)} frac{partialrho}{partial t} ,dV + int{partialOmega(t)} left( rho mathbf{v} cdot mathbf{n} right) , dA end{align} $$
In this formulation, $mathbf{v}$ represents the velocity field of the surface of our moving volume $Omega(t)$, denoted as $partial Omega(t)$. When $Omega(t)$ is considered to always encompass the same “material” as it moves – meaning it deforms with the flow itself – then $mathbf{v}$ is termed the “flow velocity,” and the Leibniz integral theorem in this context is specifically known as the Reynolds Transport Theorem.
Connecting to the Balance Equation
To further clarify the implications, let’s derive a “Balance Equation.” We assume that the rate of change of $X$ can be expressed in terms of sources or sinks within the density field:
$$ frac{dX}{dt} = int_{Omega(t)} gamma,dV $$
Where $gamma$ represents the source or sink density of $X$. If $X$ is a conserved quantity (like mass or energy), then $frac{dX}{dt} = 0$, implying $gamma = 0$. Substituting this into the Reynolds Transport Theorem equation:
$$ int{Omega(t)} gamma,dV = int{Omega(t)} left( frac{partialrho}{partial t} + nabla cdot (rho mathbf{v}) right) ,dV implies 0 = int_{Omega(t)} left( frac{partialrho}{partial t} + nabla cdot (rho mathbf{v}) – gamma right) ,dV $$
For this integral to be zero for any arbitrary volume $Omega(t)$, the integrand itself must be zero:
$$ 0 = frac{partialrho}{partial t} + nabla cdot (rho mathbf{v}) – gamma $$
This resulting equation is a local field equation, valid at every point within the region $R$, including all points in $Omega(t)$.
Control Volume Perspective
Now, let’s introduce another concept: the control volume, $K(t)$, also within region $R$. The control volume is a deformable region, but its surface, $partial K(t)$ (the control surface), has a velocity field $mathbf{w}$, which may be different from the flow velocity $mathbf{v}$. Integrating our local field equation over the control volume $K(t)$ and applying the Divergence Theorem:
$$ begin{align} 0 &= int{K(t)} left( frac{partialrho}{partial t} + nabla cdot (rho mathbf{v}) – gamma right) ,dV &stackrel{text{Divergence}}{=} int{K(t)} frac{partialrho}{partial t} ,dV + int{partial K(t)} left( rho mathbf{v} cdot mathbf{n} right) , dA – int{K(t)} gamma ,dV end{align} $$
Separately, applying the Leibniz theorem and Divergence theorem directly to the control volume $K(t)$ yields the rate of change of $X$ within the control volume, $X_{CV}$:
$$ begin{align} frac{dX{CV}}{dt} &= frac{d}{dt}left(int{K(t)} rho ,dV right) &stackrel{text{Leibniz}}{=} int{K(t)} left( frac{partialrho}{partial t} + nabla cdot (rho mathbf{w}) right) ,dV &stackrel{text{Divergence}}{=} int{K(t)} frac{partialrho}{partial t} ,dV + int_{partial K(t)} left( rho mathbf{w} cdot mathbf{n} right) , dA end{align} $$
Combining these results, we arrive at a macroscopic balance equation for the control volume:
$$ frac{dX{CV}}{dt} = – int{partial K(t)} left( rho left(mathbf{v} – mathbf{w}right) cdot mathbf{n} right) , dA + int_{K(t)} gamma ,dV $$
This equation, often referred to as the Reynolds Transport Theorem in control volume form, describes the rate of change of quantity $X$ within the control volume $K(t)$. This change is due to two factors: the net flux of $X$ across the control surface $partial K(t)$ (represented by the surface integral) and the generation or consumption of $X$ within the control volume itself (represented by the volume integral). Note that $mathbf{v} – mathbf{w}$ represents the relative velocity of the flow with respect to the control surface.
Limiting Cases of the Control Volume
Let’s consider some specific scenarios for the control volume velocity $mathbf{w}$:
- Stationary Control Volume ($mathbf{w} = 0$): The control volume is fixed in our frame of reference. In this case, the Reynolds Transport Theorem simplifies to:
$$ frac{dX{CV}}{dt} = – int{partial K(t)} left( rho mathbf{v} cdot mathbf{n} right) , dA + int_{K(t)} gamma ,dV $$
This form is frequently used when analyzing systems with fixed boundaries.
- Material Control Volume ($mathbf{w} = mathbf{v}$): The control volume moves with the flow velocity. In this scenario, no fluid flows across the control surface, and the Reynolds Transport Theorem reduces to:
$$ frac{dX{CV}}{dt} = int{K(t)} gamma ,dV $$
This case is often referred to as a material volume or system volume, where we are essentially tracking a specific mass of fluid as it moves and deforms.
The Reynolds Transport Theorem is a powerful tool for analyzing a wide range of problems in fluid mechanics and other transport phenomena. It provides a systematic way to move between integral and differential formulations and to analyze systems from both Lagrangian (material volume) and Eulerian (control volume) perspectives.
Further Reading:
- Transport Phenomena by Bird, Stewart, and Lightfoot.
- Advanced Transport Phenomena by Slattery.
- Continuum mechanics textbooks.
- Leibniz Integral Theorem (The 3D Version is discussed in Ref. 1)
