Fluid Parcels and the Continuum Hypothesis

When working out the equations of motion (and the physics) of fluids, we tend to inherit an 18th-century perspective on the nature of matter: a fluid is a continuum, not a collection of molecules. This makes terminology rather confusing to modern students, since atomic theory is taught in primary school.

Textbook definitions explain a fluid parcel is an infinitesimal amount of fluid, presumably in a simply-connected (and, for algebraic topologists, connected) region. Older texts may use the term "fluid particle" as a synonym for fluid parcel. Another synonym found in the literature (classic and modern) is "fluid element".

The idea is that a fluid may be described as a continuous collection of fluid parcels, just as a domain may be described as a continuous collection of "infinitesimal volumes".

Puzzle 1. Why not describe a fluid parcel as a volume element or more rigorously as a volume form or density?

Well, that may all be well-and-good for a mathematical formalization describing a fluid parcel, but (a) what does it mean for the equations? And (b) when does a physical system behave like a fluid?

Continuum Hypothesis

What this means for the equations is that we can conceptually "keep zooming in" on a fluid parcel, and each fluid parcel "acts like" a physical body. So we can meaningfully describe a fluid parcel's velocity, density, pressure, temperature, etc.

Proposition 2 (Continuum Hypothesis). A fluid parcel may be described like a point-particle, and has the physical characteristics of the bulk (e.g., velocity, pressure, density, etc.).

Remark 3. I have seen the literature argue the "amount of matter" in a fluid parcel is constant (which means its volume can vary, otherwise density is constant). I'm not sure if this is standard or not? If so, I suppose we could conceptually think of the number of molecules in a parcel is fixed...but this may be "chalk and cheese".

Knudsen Number

When can we apply fluid mechanics to describe a physical body? Fortunately, we have a (dimensionless) parameter measuring how badly fluid mechanics approximates a physical body: the Knudsen number.

Definition 4. The Knudsen Number for a body is \begin{equation} \mathrm{Kn} := \lambda/L \end{equation} where L is the length scale of the body, and $\lambda$ is the mean free path of its constituent molecules.

The rule/heuristic is: we may apply fluid mechanics if \begin{equation} \mathrm{Kn}\ll 1. \end{equation} If $\mathrm{Kn}\lt0.01$, then it's perfectly fine to use fluid mechanics. For the values between $0.01\leq\mathrm{Kn}\lt0.10$, then it's debatable-but-doable...the cut-off here is rather fuzzy. Larger values demand using statistical mechanics instead of fluid mechanics.

Example 5. Air at 1013 hPa has a mean free path of approximately 68nm (see Jennings' The mean free path in air Journal of Aerosol Science 19, no.2 (1988) pp.159–166). For geophysical models, the length scale would be on the order of 8.5 kilometers (the scale atmosphere height is about 8.5km). Thus the Knudsen number for air would be about $\mathrm{Kn}\approx (68\times 10^{-9})/(8.5\times 10^{3}) = 8\times 10^{-12}\sim 10^{-11}$. Hence fluid mechanics would be a good approximation at this scale. For a more careful analysis of the mean-free path of air, see some calculations by David Pace.

Example 6. Water can be estimated to have a mean free path of $\lambda\approx 2.5\times10^{-10}\,\mathrm{m}$ (c.f., handout). For a cup of water, which holds 1 cup (approximately 236588 cubic millimeters), its scale is the cuberoot of this number: $L\approx 62\times10^{-3}\,\mathrm{m}$. Hence the Knudsen number for a cup of water is approximately $\mathrm{Kn}\approx (2.5\times 10^{-10})/(62\times 10^{-3})\approx 4\times 10^{-9}$. Fluid mechanics can describe a cup of water very well, as we expect!

Remark 7. There is a more generally setting, treating matter as continuous, which is the field of "continuum mechanics". The condition for determining appropriateness of continuum mechanics to a problem seems to be the Hill–Mandel principle, where if $F$ is the deformation gradient and $P$ is the stress tensor in equilibrium, then $\langle P : F\rangle = \langle P \rangle : \langle F \rangle$, where $A : B = \mathrm{tr}(AB^{T}) = \sum_{i,j}(A)_{ij}(B)_{ij}$ and brackets indicate microscopic averaging. For a review with emphasis on finite elements, see arXiv:1509.06621.

There are three physical length scales we can work with:

1. $L_{\text{molec}}$ Molecular scale (which is approximated by the mean free path)
2. $L_{\text{fluid}}$ Fluid parcel scale (which could be approximated by the size of a drop or droplet, which pharmacists have standardized and ordained to be 1/20 of a milliliter — approximately a sphere with radius π millimeters)
3. $L_{\text{macro}}$ Macro-scale of the phenomena (the size of the container of fluid, or some standard scale).

This is another way to heuristically reason about applicability, and the condition here becomes $L_{\text{molec}}\ll L_{\text{fluid}}$...but we usually have this for most engineering purposes (unless someone is trying to use individual water molecules to cool their quantum computer, or something) and geophysical models.

References

1. Simon J.A. Malham, Introductory fluid mechanics. (PDF) Lecture notes dated September 15, 2014.
2. Pieter Wesseling, Principles of Computational Fluid Dynamics. Springer, 2000.
   This is my only book which discusses Knudsen number.
3. Carlo Marchioro, Mario Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids. Springer, 1994.
   Chapter 1, §1, discusses "number of molecules in a parcel", for example.

Continuity Equation

We had introduced the Euler flow equations, and saw it gave us (in $n$ dimensions) $n$ equations in $n+2$ unknowns. This requires introducing another 2 equations for us to solve the equations of motion. Today we will do half the required job, we will introduce one additional equation: the conservation of mass, or continuity equation.

Euler's Derivation

A relatively clever derivation begins by examining a fluid parcel of "small volume" $V$ and density $\rho$, so small such that $\int_{V}\rho\,\D^{n}x\approx\rho V$. Since the particles in the fluid parcel remain in the parcel over time, we have its total mass be constant: \begin{equation} \frac{\mathrm{D}(\rho V)}{\mathrm{D} t} = 0. \end{equation} Since this fluid parcel is nonempty, $\rho V\gt0$, so taking the logarithm of the mass and then taking its time derivative gives us: \begin{equation} \frac{1}{\rho}\frac{\mathrm{D}\rho}{\mathrm{D} t} + \frac{1}{V}\frac{\mathrm{D} V}{\mathrm{D} t} = 0. \end{equation} Supposing we have the fluid parcel be described as a rectangle with edges $\delta x$, $\delta y$, $\delta z$, then we see \begin{equation} V = \delta x\,\delta y\,\delta z \end{equation} hence Taylor expanding the volume as a function of time gives us \begin{equation} V + \frac{\mathrm{D} V}{\mathrm{D} t}\delta t = \left[1 + \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right)\delta t\right]\delta x\,\delta y\,\delta z. \end{equation} Collecting these together give us the equation \begin{equation} \frac{\mathrm{D}\rho}{\mathrm{D} t} + \rho\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) = 0. \end{equation} This is the continuity equation.

Remark 1. We should observe that the divergence of the velocity vector field, as it appears in Eq (4), describes the dilation of the volume element. It is the "expansion" of the fluid at that point.

Puzzle 2. This derivation seems to use the Lagrangian description; in general, however, we would need to include a factor of the Jacobian to describe the volume of the fluid parcel. Work out this derivation. Cheaters may consult Childress.

Modern Textbook Derivation

Consider now a fixed cube in the fluid, whose side lengths are again $\delta x$, $\delta y$, $\delta z$. What amount of mass flows into the cube across the $\delta y\delta z$ face?

We find, assuming the flow moves away from the origin, \begin{equation} \begin{pmatrix}\mbox{mass which}\\ \mbox{flows in}\end{pmatrix} = \left(\rho - \frac{1}{2}\frac{\partial(\rho u)}{\partial x}\delta x\right)\delta y\,\delta z. \end{equation} Similarly, the amount flowing out \begin{equation} \begin{pmatrix}\mbox{mass which}\\ \mbox{flows out}\end{pmatrix} = \left(\rho + \frac{1}{2}\frac{\partial(\rho u)}{\partial x}\delta x\right)\delta y\,\delta z. \end{equation} We then have the net mass gained be $-\delta x\,\delta y\,\delta z\partial_{x}(\rho u)$. Doing this for the other faces, we get \begin{equation} \begin{pmatrix}\mbox{net mass}\\ \mbox{flows}\end{pmatrix} = \nabla\cdot(\rho\vec{u})\,\delta x\,\delta y\,\delta z. \end{equation} On the other hand, the mass flux varies over time as \begin{equation} \begin{pmatrix}\mbox{mass}\\ \mbox{flux}\end{pmatrix} = \frac{\partial}{\partial t}(\rho\,\delta x\,\delta y\,\delta z). \end{equation} Setting equals to equals, and simplifying, we find \begin{equation} \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\vec{u})=0 \end{equation} which is another, equivalent, form of the continuity equation.

Exercise 3. Prove Eq (10) is equivalent to Eq (5).

References

I have profited most from Lamb's presentation, though Childress takes greater care when working through the derivation in the Lagrangian description.

1. Leonhard Euler, "Principes généraux du mouvement des fluides". Mémoires de l'académie des sciences de Berlin 11 (1757) 274–315; English translation arXiv:0802.2383
2. Horace Lamb, Hydrodynamics. Dover, sixth ed., 1932; §7 (archive)
3. Stephen Childress, An Introduction to Theoretical Fluid Mechanics. AMS Press, 2009.

Derivation of Euler Equations for Fluid Flow

Overview: the Euler [flow] equations are the equations of motion for an idealized fluid. We just derive them and conclude with a few remarks.

Derivation

The derivations of the Euler equation boil down to the same game plan: derive $F=ma$ for a fluid parcel. (A "fluid parcel" is an infinitesimal volume of fluid, more a heuristic than a rigorous notion.)

The forces acting on a fluid parcel, in a simplified idealized setting, are just the fluid's pressure. We may find it by integrating over the surface of the fluid parcel: \begin{equation} \vec{F} = -\oint_{\partial V}p(\vec{x},t)\vec{n}\,\D A \end{equation} where $\vec{n}$ is the outward-pointing normal vector on the bounding surface $\partial V$ of the fluid parcel. (The negative sign is due to the normal vector being outward-pointing.) The divergence theorem yields \begin{equation} \vec{F} = -\int_{V}(\nabla p(\vec{x},t))\,\D V. \end{equation} This is half the equation of motion.

The other half is the "mass times acceleration" term. We generically describe a fluid by its velocity vector field $\vec{u}(\vec{x}, t)$. Now we can sum over its acceleration vector for each point in the fluid parcel, weighted by its density $\rho$, to give us \begin{equation} m\vec{a} = \int_{V} \rho(\vec{x}, t)\frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t}\,\D V. \end{equation} Here we use Stokes' notation \begin{equation} \frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t} = \partial_{t}\vec{u}(\vec{x},t) + (\vec{u}(\vec{x},t)\cdot\nabla)\vec{u}(\vec{x},t) \end{equation} which is the material time derivative.

Now we set equals to equals by Newton's second Law, we get \begin{equation} \int_{V} \rho(\vec{x}, t)\frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t}\,\D V=-\int_{V}(\nabla p(\vec{x},t))\,\D V \end{equation} for arbitrary regions $V$. This gives us Euler's equations of motion \begin{equation} \rho(\vec{x}, t)\frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t} =-\nabla p(\vec{x},t). \end{equation} Usually books divide through by density $\rho$, and add an external force term (usually Earth's gravitational field $\vec{g}$) to the right-hand side.

Theorem 1. Euler's flow equations of motion for fluid is \begin{equation} \frac{\mathrm{D}\vec{u}(\vec{x},t)}{\mathrm{D}t} = \frac{-1}{\rho(\vec{x}, t)}\nabla p(\vec{x},t). \end{equation}

A few remarks are in order:

1. This is an idealization insofar as friction between fluid parcels is negligible (which is reflected by a very small viscosity).
2. Exact solutions to the Euler flow may be found in Marchioro and Pulvirenti's Mathematical Theory of Incompressible Nonviscous Fluids. One large class of solutions include potential flow of incompressible fluids, where $\vec{u}=\nabla\varphi$ the velocity is the gradient of a scalar potential. Incompressibility amounts to $\nabla\cdot\vec{u}=0$ in this case. (Note to future me: insert link to incompressibility condition when I get around to writing it)
3. Intuitively, compressible fluids are gases, incompressible fluids are liquids.
4. In $n$ dimensions, Euler equations has $n$ unknowns from the velocity vector, 1 from the pressure, and 1 from the density. But there is one equation per dimension, i.e., there are $n$ equations for $n+2$ unknowns. Consequently, Euler equations are under-determined, and we need 2 more equations to determine all unknowns. We usually add an equation for the conservation of mass (the continuity equation) and either work with incompressible fluid or compressible fluids with additional equations from thermodynamics.

References

The most elegant (and difficult) presentation of Euler's equations I have found is in Landau and Lifshitz. It's the subject of the first chapter of their Fluid Mechanics, which is just amazing (but nearly impenetrable if you don't already have a good familiarity with fluids).

1. Leonhard Euler, "Principes généraux du mouvement des fluides". Mémoires de l'académie des sciences de Berlin 11 (1757) 274–315; English translation arXiv:0802.2383
2. Landau and Lifshitz, Fluid Mechanics.
3. Demetrios Christodoulou, "The Euler Equations of Compressible Fluid Flow". Bulletin of the American Mathematical Society 44, no.4 (2007) 581–602
4. John K. Hunter, An Introduction to the Incompressible Euler Equations. Notes dated Sep 25, 2006.
5. M. Zingale, Notes on the Euler equations. Dated April 16, 2013.
6. Chorin and Marsden, A mathematical introduction to fluid mechanics. Springer, third ed., 1993.
7. C. Marchioro, M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids. Springer, 1994. Discusses exact solutions to Euler flow.

Describing Fluids

There are two ways to describe fluids: the Lagrangian frame, and the Eulerian frame.

Consider some fluid, which has its initial position in some region, say $\mathcal{B}_{0}\subset\RR^{3}$ (for "body" at time 0). We take $\mathcal{B}_{0}$ to be an open subset of 3-space, though we could consider $\mathcal{B}_{0}\subset\RR^{n}$ open, for some fixed $n$.

As the fluid moves, its constituent particles also move, occupying a (potentially different) region $\mathcal{B}_{t}$ at time $t$. We could introduce a Time Evolution Operator \begin{equation} \mathcal{M}_{t}\colon\mathcal{B}_{0}\mapsto\mathcal{B}_{t} \end{equation} which takes some time $t$ and transforms the initial body to the body at the specified time.

Suppose we take $\vec{a}$ to be a point in the initial body $\vec{a}\in\mathcal{B}_{0}$. Take the position of that point to be a function \begin{equation} \vec{x} = \mathcal{X}(\vec{a}, t) \end{equation} which is such that $\vec{a} = \mathcal{X}(\vec{a}, 0)$. This is the Lagrangian Coordinate of the fluid identified by initial position $\vec{a}$.

Remark 1. We could actually choose any arbitrary parametrization of the fluid; i.e., we don't need $\vec{a}$ to be the initial position of the fluid particle. We could have chosen $\vec{a}$ to be the position at some other particular moment, or we could use some other scheme (provided the label was unique).

But in practice, it's usually the case that $\vec{a}$ is the initial position of a fluid particle. Take care when reading the literature, because someone may have a brilliant new idea which requires a particularly idiosyncratic choice for $\vec{a}$.

The Lagrangian picture resembles that in analytical mechanics: we care about the history of each particle. (It's also coincidentally the starting point for the Hamiltonian/"Canonical" formalism in fluid dynamics, just as it's the starting point for the Hamiltonian formalism in analytical mechanics.) Although mathematically and physically intuitive, elegant, and appealing...it has the disadvantage that, when we do measurements in the lab, it's at a specific point in time (not over the history of the particle).

We could take this opposite view, starting with a fixed point $\vec{x}$ inside the fluid $\vec{x}\in\mathcal{B}_{t}$. Then we take the quantities of interest as functions $f(\vec{x},t)$. This is the Eulerian description of the fluid. Convention compels us to then examine the velocity vector $\vec{u}(\vec{x}, t)$ at each point in the fluid...although there's nothing to stop us from extending $\vec{u}$ to $\RR^{n}$ in general, provided we demand $\vec{u}=0$ outside the fluid.

Relating the two pictures. The velocity vector field relates to the Lagrangian description by taking the time derivative of the Lagrange coordinates: \begin{equation} \left.\frac{\partial\mathcal{X}}{\partial t}\right|_{\vec{a}} = \vec{u}(\mathcal{X}(\vec{a}, t), t). \end{equation} This gives us a system of differential equations we'd need to solve.

Remark 2: Caution with notation. Here I must stress the warning which Abelson and friends give in The Structure and Interpretation of Classical Mechanics about ambiguity with notation. Some authors (especially from physics) write the partial derivative with respect to, say, time to mean: \begin{equation} \frac{\partial}{\partial t}f(\vec{x}(t), t) = \left.\frac{\partial f(\vec{q}, t)}{\partial t}\right|_{\vec{q}=\vec{x}(t)}. \end{equation} Undergraduates and mathematicians are understandably confounded by this choice of notation (what ever happened to the "chain rule"?). Well, that's handled by the total derivative with respect to time: \begin{equation} \frac{\D}{\D t}f(\vec{x}(t), t) = \left.\frac{\partial f(\vec{q}, t)}{\partial t} +\sum_{j}\frac{\partial f(\vec{q},t)}{\partial q_{j}}\frac{\D x_{j}(t)}{\D t}\right|_{\vec{q}=\vec{x}(t)}. \end{equation} This notation has been "grandfathered in", and there's nothing we can do to change it. I'm so, so sorry.

Also note, if we wanted to be "completely general", we would need to consider derivatives with respect to the Lagrange parameter $\vec{a}$ — the curious reader may consult with pleasure Lamb's Hydrodynamics (§§13–14).

Example 1 (Acheson). Consider a 2-dimensional fluid with $\vec{u}=(u(x,y,t), v(x,y,t))=(-\Omega y, \Omega x)$ where $\Omega\gt0$ is a constant angular speed. We have two differential equations \begin{equation*} \partial_{t}x = -\Omega y(t),\quad x(0)=a \end{equation*} and \begin{equation*} \partial_{t}y = \Omega x(t),\quad y(0)=b. \end{equation*} Then we see (by differentiating with respect to time again, in both equations) that \begin{equation*} \partial_{t}^{2}x = -\Omega^{2}x,\quad\mbox{and}\quad\partial_{t}^{2}y=-\Omega^{2}y \end{equation*} which have general solutions of the form \begin{equation} (x(t),y(t)) = (a\cos(\Omega t)-b\sin(\Omega t), b\cos(\Omega t) + a\sin(\Omega t)). \end{equation} (End of Example 1)

Example 2 (Acheson). Consider now the Rankine Vortex in two dimensions. In polar coordinates, it's defined as \begin{equation} \dot{\theta} = \begin{cases} \Omega & r\leq R\\ \Omega(R/r)^{2} & r\geq R \end{cases} \end{equation} and $\dot{r}=0$, where $\Omega\gt0$ is some fixed angular speed, $R\gt0$ is the width of the vortex. We see then that $r=r_{0}$ is constant, and \begin{equation} \theta(t) = \theta_{0} + \begin{cases} \Omega t & r\leq R\\ \Omega t (R/r)^{2} & r\geq R \end{cases} \end{equation} where $\theta_{0}$ is the initial angle at time zero. We can change coordinates to see \begin{equation} x(t) = r\cos(\theta(t)),\quad y(t)=r\sin(\theta(t)) \end{equation} which lets us determine $r$ and $\theta_{0}$ in terms of $x(0)=a$ and $y(0)=b$. So $r^{2} = a^{2} + b^{2}$ and $\theta_{0} = \operatorname{atan2}(b,a)$ (recall the definition of atan2). Also note there is a singularity when $a=b=0$, since $\theta_{0}$ is not well-defined. (End of Example 2)

Note, I lifted these example flows from D.J. Acheson's excellent Elementary Fluid Dynamics; but Acheson doesn't really discuss the Lagrangian treatment of fluids (at least, not in the first few chapters I've read).

References

For the most part, I follow Childress's An Introduction to Theoretical Fluid Mechanics, though I have independently arrived at a similar line of reasoning. It is a joy to see it elsewhere, and Dr Childress probably thought of this before I was even born.

1. Stephen Childress, An Introduction to Theoretical Fluid Mechanics. AMS Press, 2009.
2. Landau and Lifshitz, Fluid Mechanics.
3. Chorin and Marsden, A mathematical introduction to fluid mechanics. Springer, third ed., 1993.
4. D.J. Acheson, Elementary Fluid Dynamics. Oxford University Press, 2009 reprint, chapter 1.

Configuring Mathjax

Mathjax is mildly straightforward, thanks to its instructions.

The `tags: 'ams'` is responsible for automatic equation numbering.

I am also trying to use $ for inline math (TeX delimiters which I prefer).

I've also added a few custom macros, some borrowed from Springer's style classes, others just shorthand. They're implemented using mathjax's configmacros, which may turn out to be a terrible idea...

If (somehow) I botch things later on, and need to backtrack to start over with the mathjax configuration, then this is what it looks like:

<script>
MathJax = {
  loader: {
    load: ['[tex]/ams']
  },
  tex: { 
    macros: {
      RR: "\\mathbb{R}",
      D: "\\mathrm{d}",
      E: "\\mathrm{e}",
      I: "\\mathrm{i}"
    },
    tags: 'ams',
    inlineMath: [['$', '$'], ['\\(', '\\)']],
    packages: {'[+]': ['ams']},
    autoload: {color: []}
  }
};
</script>
<script type="text/javascript" id="MathJax-script" async
  src="https://cdn.jsdelivr.net/npm/mathjax@3.0.0/es5/tex-chtml.js">
</script>