Derivatives and trajectories
Methods for Model-Based Development in CE
2025-04-14
All of the following statements implicitly assume differentiability!
The gradient of a scalar field \(\phi: \mathbb R^3 \rightarrow \mathbb R\) is a vector field given by
\[ \nabla \phi (\mathbf x) = \frac{\partial \phi}{\partial x_i} (\mathbf x) \mathbf e_i\]
The gradient of a vector field \(\mathbf v: \mathbb R^3 \rightarrow \mathbb R^3\) is a second order tensor field given by
\[ \nabla \mathbf v (\mathbf x) = \frac{\partial v_i}{\partial x_j} (\mathbf x) \mathbf e_i \otimes \mathbf e_j\]
The divergence of a vector field \(\mathbf v: \mathbb R^3 \rightarrow \mathbb R^3\) is a scalar field given by
\[ \nabla \cdot \mathbf v = tr( \nabla \mathbf v) \qquad (\nabla \cdot \mathbf v) (\mathbf x) = \frac{\partial v_i}{\partial x_i} (\mathbf x) \]
The Laplacians of a scalar field \(\phi: \mathbb R^3 \rightarrow \mathbb R\) and a vector field \(\mathbf v: \mathbb R^3 \rightarrow \mathbb R^3\) are defined as
\[ \triangle \phi = \nabla \cdot ( \nabla \phi) \qquad \triangle \mathbf v = \nabla \cdot ( \nabla \mathbf v)\]
The aforementioned form of the state variables is called the Eulerian formulation.
In contrast to this, we can look at a material particle initially located at \(\mathbf x_0\) and track its trajectory during motion:
\[ \mathcal{\mathbf X} (\mathbf x_0;t), \quad \mathcal{\mathbf X}:\mathbb R^+ \rightarrow \mathbb R^3,\quad \text{with} \quad \mathcal{\mathbf X} (\mathbf x_0;t) = \left(X_1(\mathbf x_0;t),X_2(\mathbf x_0;t),X_3(\mathbf x_0;t) \right) \]
Here, \(\mathbf x_0\) denotes that the trajectory starts at \(\mathbf x_0\) at time zero. It is, hence, no real space coordinate and can rather be seen as a parameter of \(\mathcal{\mathbf X}\).
We will therefore drop \(\mathbf x_0\) for the rest of this section and refer to the trajectory as
\[ \mathcal{\mathbf X} (t) = (X_1(t),X_2(t),X_3(t)). \]
Tracking the evolution of the density along the trajectory of a fluid parcel, hence \(\rho(\mathcal{\mathbf X}(t),t)\), or shorter \(\rho(\mathcal{\mathbf X},t)\), is referred to as the density in Lagrangian formulation:
Let’s look at the total density \(\rho(\mathcal{\mathbf X},t)\) change along the trajectory:
\[ \begin{aligned} \frac{d}{dt} \rho(\mathcal{\mathbf X},t) &= \frac{d}{dt} \rho(X_1(t),X_2(t),X_3(t),t) \\[1em] & = \underbrace{\frac{d}{dt} X_1(t)}_{=v_1(\mathbf x,t)} \cdot \partial_x \rho +\underbrace{\frac{d}{dt} X_2(t)}_{=v_2(\mathbf x,t)} \cdot \partial_y \rho +\underbrace{\frac{d}{dt} X_3(t)}_{=v_3(\mathbf x,t)} \cdot \partial_z \rho +\underbrace{\frac{d}{dt} t}_{=1} \cdot \partial_t \rho &\\[1em] & = \partial_t \rho + v_1 \cdot \partial_{x_1} \rho + v_2 \cdot \partial_{x_2} \rho + v_3 \cdot \partial_{x_3} \rho = \underbrace{\underbrace{\partial_t \rho}_{\text{local/time derivative}} + \underbrace{ \mathbf v \cdot \nabla \rho }_{\text{convective derivative}}}_{\text{material/total derivative denoted by } \frac{D}{Dt}} \end{aligned} \]
The resulting relation between the total derivative and its local and convective counterparts reads:
\[ \begin{equation*} \frac{D}{Dt} \rho = \partial_t \rho + \mathbf v \cdot \nabla \rho \end{equation*} \]
For a vector field \(\mathbf v\), we can do the same, which yields \[ \frac{D}{Dt} \mathbf v = \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v \]
Written componentwise this results in three equations \[ \begin{aligned} \frac{D}{Dt} u &= \partial_t u + \mathbf v \cdot \nabla u \\ \frac{D}{Dt} v &= \partial_t v + \mathbf v \cdot \nabla v \\ \frac{D}{Dt} w &= \partial_t w + \mathbf v \cdot \nabla w \end{aligned} \]
For an arbitrary field \(\phi\), we have to distinguish
\[ \underbrace{\frac{D}{Dt} \phi = 0}_{\phi \text{ constant}} \qquad \text{or} \qquad \underbrace{\partial_t \phi = 0}_{\phi \text{ stationary}} \]
We can consider various aspects of a fluid particle’s motion, each of which requires its own formalization and visualization: pathlines, streamlines and streaklines.
The actual physical motion of a fluid particle is describes by its trajectory or pathline defined as \(\mathcal{\mathbf X} (t) = (X_1(t),X_2(t),X_3(t))\). This corresponds to the Lagrangian formulation of the motion described before. A pathline is what you would see on a photo with a long exposure time when recording a moving object.
Pathline
Given an initial position \(\mathbf{x}_0\) at time \(t=0\), the pathline or trajectory \(\mathbf{X}(\mathbf{x}_0; t)\) can be determined from the following ordinary differential equation:
\[ \begin{eqnarray*} \frac{d}{dt} \mathbf{X}(t) &=& \mathbf{v}(\mathbf{X}(t),t) \\ \mathbf{X}(0) &=& \mathbf{x}_0, \end{eqnarray*} \]
in which \(\mathbf{x}_0\) constitutes a parameter that denotes the dependence of the pathline or trajectory on the particle’s initial position.
The streamline describes a curves, whose tangent vector constitutes the velocity at that location. An analogy would be iron filings in the vacinity of a bar magnet that orient themselves along the magnetic field lines.
Streamline
Per its definition, the streamline \(\frac{d}{ds} \mathbf{\hat{X}}(s)\) has to be tangent to the velocity field \(\mathbf{v}\) at every point in space for a given time. It is defined as
\[ \begin{eqnarray} \frac{d}{ds} \mathbf{\hat X}(s) = \alpha \mathbf{v}( \mathbf{\hat X}(s),t^*). \end{eqnarray} \]
with \(\alpha \in \mathbb R \setminus \{0\}\).
This definition automatically fullfills the required condition: \[ \begin{aligned} \frac{d}{ds} \mathbf{\hat X}(s) \times \frac{d}{ds} \mathbf{\hat X}(s) & = \alpha \mathbf{v}(\mathbf{\hat X}(s),t^*) \times \frac{d}{ds} \mathbf{\hat X}(s) = 0. \end{aligned} \]
A streakline, finally describes the set of all points that have passed through a particular point \(\mathbf{x}_0\) in the past. A typical example is given by a visualization in response to a continuously released tracer, such as ink in a water flow, or smoke in a wind tunnel.
Streakline
Assuming a location \(\mathbf{x}_0\) to be given, the streakline \(\mathbf{\widetilde{X}}(\mathbf{x}_0, \tau_0; \tau)\) is defined as
\[ \begin{eqnarray} \frac{d}{d\tau} \mathbf{\widetilde{X}(\tau)} &=& \mathbf{v}(\mathbf{\widetilde{X}(\tau)}, \tau_0 + \tau) \\ \widetilde{\mathbf{X}}(0) &=& \mathbf{x}_0 \end{eqnarray} \]
Note, that the start time (of the injection) \(\tau_0\) might varies for each particle.
Continuum Mechanical Modeling for Simulation Science - Derivatives and trajectories