Lagrangian Description of Fluids

I'd like to just summarize the Lagrangian description of fluids quickly, because some texts have horrible typos and errors.

We begin with describing the position of fluid parcels using a function $\vec{x} = \vec{x}(\vec{x}_{0}, t)$ parametrized by position $\vec{x}_{0}$ at time $t=t_{0}$. Time evolution is described by the function \begin{equation} \vec{x}(\vec{x}_{0}, t) = \phi_{t}(\vec{x}_{0}) \end{equation} such that (locally) $\phi_{t}$ is invertible; i.e., we can write \begin{equation} \vec{x}_{0}(\vec{x}, t) = {\phi_{t}}^{-1}(\vec{x}(\vec{x}_{0},t)). \end{equation} Now we may introduce the Fluid Velocity in the Lagrangian description as \begin{equation} \vec{q} = \frac{\partial\vec{x}(\vec{x}_{0},t)}{\partial t}. \end{equation}

The Eulerian picture may be obtained by writing \begin{equation} \vec{u}(\vec{x}, t) = \vec{q}(\vec{x}_{0}(\vec{x},t), t) = \left.\frac{\partial\vec{x}(\vec{x}_{0},t)}{\partial t}\right|_{\vec{x}_{0}={\phi_{t}}^{-1}(\vec{x})} \end{equation}

Example 1. Consider a flow on the unit disc, with trajectories described by \begin{equation} \begin{pmatrix} x(t)\\ y(t) \end{pmatrix} = \begin{pmatrix}\cos(\omega t) & -\sin(\omega t)\\ \sin(\omega t) & \cos(\omega t) \end{pmatrix} \begin{pmatrix} x_{0}\\ y_{0} \end{pmatrix}. \end{equation} Observe that $x(0)=x_{0}$ and $y(0)=y_{0}$. We find we can invert this equation to determine the initial positions from the trajectory: \begin{equation} \begin{pmatrix} x_{0}\\ y_{0} \end{pmatrix} = \begin{pmatrix}\cos(\omega t) & \sin(\omega t)\\ -\sin(\omega t) & \cos(\omega t) \end{pmatrix} \begin{pmatrix} x(t)\\ y(t) \end{pmatrix} \end{equation} which permits us to write the Lagrangian parameters (initial positions) in terms of the current position of the fluid parcel. We find the time derivative of the trajectories \begin{equation} \vec{q} = \frac{\D}{\D t} \begin{pmatrix} x(t)\\ y(t) \end{pmatrix} = \omega\begin{pmatrix}-\sin(\omega t) & -\cos(\omega t)\\ \cos(\omega t) & -\sin(\omega t) \end{pmatrix} \begin{pmatrix} x_{0}\\ y_{0} \end{pmatrix}. \end{equation} Plugging in the equations relating $x_{0}$ and $y_{0}$ in terms of $x(t)$ and $y(t)$ gives us (after some simple matrix multiplication): \begin{equation} \begin{pmatrix}u(x,y,t)\\ v(x,y,t)\end{pmatrix} =\left.\frac{\D}{\D t} \begin{pmatrix} x(t)\\ y(t) \end{pmatrix}\right|_{\vec{x}_{0}=\vec{x}_{0}(\vec{x},t)} = \begin{pmatrix} -\omega y\\ \omega x \end{pmatrix}. \end{equation} This is a steady flow (since time doesn't explicit appear in the equation for $u(x,y,t)$ or $v(x,y,t)$), and just a rotation. If we want to generalize this, we could replace $\omega t$ with a dimensionless, strictly increasing function of time $f(t)$, which would usually not be steady (e.g., if $f(t) = \omega t + \alpha t^{2}$ for positive constants $\omega$, $\alpha$, then $u=-(\omega + 2\alpha t)y$ and $v=(\omega + 2\alpha t)x$).

Exercise 1. Find the density $\rho(x,y,t)$ for the flow from example 1 by solving the continuity equation, or some other way. If you do it some other way, prove your solution satisfies the equation of continuity.

Exercise 2. Is the flow from example 1 compressible or incompressible?

Example 2 (Childress, Example 2.2). Consider the one-dimensional flow with $u(x,t) = 2xt/(1 + t^{2})$. Suppose we want to find the Lagrangian description of the flow, then we can rewrite this as \begin{equation} \frac{\D x(t)}{\D t} = \frac{2x(t)\cdot t}{1 + t^{2}} \end{equation} with initial condition $x(0)=a$. This has the obvious solution \begin{equation} x(t) = a(1 + t^{2}). \end{equation} This illustrates how to translate a solution in the Euler description to find the trajectories in the Lagrangian description.

Exercise 3. Solve the Euler flow equations and continuity equation for both the pressure and density. "Someone on the internet" asserts $\rho(x,t)=x$ is a valid solution: is it? [Hint: use method of characteristics to determine density.]

References

1. C.C. Mei, Methods of Describing Fluid Motion. MIT course notes for 1.63 "Advanced Fluid Mechanics", 2001.
2. Lei Li, Math 575-Lecture 1. Duke University
3. Stephen Childress, An Introduction to Theoretical Fluid Mechanics. AMS Press, 2009.