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.