Lagrangian and Eulerian description

Continuity equation:

Current density:[1] \[ \boldsymbol{j}=\rho \boldsymbol{v}\quad(1) \] where the direction of \(\boldsymbol{j}\) is same as \(\boldsymbol{v}\), the value is equal to the product of the fluid density, the velocity of the fluid, and the area of cross section perpendicular to the velocity (the area is typically set to 1).

All these quantities are, in general, functions of the coordinates \(x, y, z\) and of the time \(t\). We emphasize that \(\mathbf{v}(x, y, z, t)\) is the velocity of the fluid at a given point \((x, y, z)\) in space and at a given time \(t\), i.e. it refers to fixed points in space and not to specific particles of the fluid; in the course of time, the latter move about in space. The same remarks apply to \(\rho\) and \(p\).

Continuity equation: \[ \begin{aligned} \partial \rho / \partial t&+\operatorname{div}(\rho \boldsymbol{v})=0\\ &\text{or}\\ \partial \rho / \partial t&+\rho \operatorname{div} \boldsymbol{v}+\boldsymbol{v} \cdot \operatorname{grad} \rho=0 \end{aligned}\quad(2) \]

momentum equation

Euler's equation

assumption: inviscid flow \[ \rho \frac{\mathrm{d} v}{\mathrm{~d} t}=-\operatorname{grad} p\quad(3) \]

The derivative \(\mathrm{d} \boldsymbol{v} / \mathrm{d} t\) which appears here denotes not the rate of change of the fluid velocity at a fixed point in space, but the rate of change of the velocity of a given fluid particle as it moves about in space. This derivative has to be expressed in terms of quantities referring to points fixed in space. To do so, we notice that the change \(\mathrm{d} \boldsymbol{v}\) in the velocity of the given fluid particle during the time \(\mathrm{d} t\) is composed of two parts, namely the change during \(\mathrm{d} t\) in the velocity at a point fixed in space, and the difference between the velocities (at the same instant) at two points \(\mathrm{d} \boldsymbol{r}\) apart, where \(\mathrm{d} \boldsymbol{r}\) is the distance moved by the given fluid particle during the time \(\mathrm{d} t\). The first part is \((\partial \boldsymbol{v} / \partial t) \mathrm{d} t\), where the derivative \(\partial\boldsymbol v / \partial t\) is taken for constant \(x, y, z\), i.e. at the given point in space. The second part is \[ \mathrm{d} x \frac{\partial \boldsymbol {v}}{\partial x}+\mathrm{d} y \frac{\partial \boldsymbol{v}}{\partial y}+\mathrm{d} z \frac{\partial \boldsymbol{v}}{\partial z}=(\mathrm{d}\boldsymbol{r} \cdot \operatorname{grad}) \boldsymbol{v}\quad(4) \]

Thus \[ \mathrm{d} \boldsymbol{v}=(\partial \boldsymbol{v} / \partial t) \mathrm{d} t+(\mathrm{d} \boldsymbol{r} \cdot \mathbf{g r a d}) \boldsymbol{v}\quad(5) \] or, dividing both sides by \(\mathrm{d} t\) \[ \frac{\mathrm{d} \boldsymbol{v}}{\mathrm{d} t}=\frac{\partial \boldsymbol{v}}{\partial t}+(\boldsymbol{v} \cdot \mathbf{g r a d}) \boldsymbol{v}\quad(6) \]

Substituting this in (3), we find \[ \frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v} \cdot \operatorname{grad}) v=-\frac{1}{\rho} \operatorname{grad} p \quad(7) \]

无论何种情况，某个时刻我们所描述流体内的物理量（比如速度，压力等）都是eulerian observer下（例如：测量者只能从流体外部测量速度，意味着，该速度是流体质点相对于观察者的，就算是观察者自己固定在流体某个质点，这是相对于测量者的速度），然而，对于lagrangian form（此时的方程也是场方程，但是是所有质点在某时刻的方程的）不同时刻，不同的质点，我们可以通过以下方程方程进行eulerian -> lagrangian 的转换。 \[ \small \int_\tau\left[\frac{d \rho(\boldsymbol x(t),t)}{d t}+\rho(\boldsymbol x(t),t) \operatorname{div} \boldsymbol v(\boldsymbol x(t),t)\right] \delta \tau=\int_\tau\left[\frac{\partial \rho(\boldsymbol x,t)}{\partial t}+\operatorname{div}(\rho(\boldsymbol x,t)\boldsymbol v (\boldsymbol x,t))\right] \delta \tau=0 \] where \((\boldsymbol x,t)\) is coordinate parameter e.g. \((x,y,z,t)\) while in \((\boldsymbol x(t),t)\) \(t\) is the variable and \(\boldsymbol x(t)\) is the function of \(t\) which represents the relationship between the position of a fluid particle and time \(t\)