Relating Lagrange and Euler temperature derivatives in fluid mechanics

[LagrangeEuler]

Lagrange & Euler formalism

How we get relation

$$(\frac{\partial T^{(L)}}{\partial t})_{r_0}=(\frac{\partial T^{(E)}}{\partial t})_{r}+\frac{\partial T^{(E)}}{\partial x}(\frac{\partial x}{\partial t})_{r_0}+\frac{\partial T^{(E)}}{\partial y}(\frac{\partial y}{\partial t})_{r_0}+\frac{\partial T^{(E)}}{\partial z}(\frac{\partial z}{\partial t})_{r_0}$$

$$T$$ - temperature
$$t$$ - time

 

[LagrangeEuler]

Lagrange representation
$$T=T^{L}(x_0,y_0,z_0,t)$$
where coordinates $$x_0,y_0,z_0$$ are fixed.

I have a trouble with that. Can you tell me one type of this function?

 

[vanhees71]

I think, I've explained this already some days ago in this forum.

Again, the Euler coordinates $\vec{x}_0$ are labels for a fixed fluid element by noting its position at a fixed (initial) time $t_0$. Then you follow this fluid element along its trajectory in time. This is given by the position-vector field (as a function of the Euler coordinates) $\vec{x}(t,\vec{x}_0)$.

In the Lagrange description you switch to a field description, i.e., you characterize the fluid by the velocity field, where an observer at point $\vec{x}$ measures the velocity of the fluid element at this position at time, $t$. This field we denote by $\vec{v}(t,\vec{x})$.

The relation to the Euler coordinates is then given by

$$\vec{v}_0(t,\vec{x}_0)=\frac{\partial \vec{x}(t,\vec{x}_0)}{\partial t}=\vec{v}[t,\vec{x}(t,\vec{x}_0)].$$

It gives the velocity of the fluid element, which has been at position $\vec{x}_0$ at time $t$.

The same procedure holds for any other physical quantity which has a specific meaning for a given fluid element. An example is the temperature $T$. In the Euler description, the temperature of the fluid element which has been at $\vec{x}_0$ at time $t_0$ at time $t$ is given by the function $T_0(t,\vec{x}_0)$.

In the Lagrange description, you consider the temperature at the position $\vec{x}$ of the fluid element that is there at time, $t$. This function is called $T(t,\vec{x})$. Again, the relation between the two points of view are given by

$$T_0(t,\vec{x}_0)=T[t,\vec{x}(t,\vec{x}_0)].$$

Now if you want the time derivative of the temperature of the specific fluid element that has been at position $\vec{x}_0$ at time $t_0$, you have to use the chain rule of multi-variable calculus,

$$\partial_t T_0(t,\vec{x}_0)=\left (\frac{\partial T(t,\vec{x})}{\partial t} \right)_{\vec{x}=\vec{x}(t,\vec{x}_0)}+ \frac{\partial \vec{x}(t,\vec{x}_0)}{\partial t} \cdot [\vec{\nabla} T(t,\vec{x})]_{\vec{x}=\vec{x}(t,\vec{x}_0)}=\left (\frac{\partial T(t,\vec{x})}{\partial t} \right)_{\vec{x}=\vec{x}(t,\vec{x}_0)}+\vec{v}[t,\vec{x}(t,\vec{x}_0)] \cdot [\vec{\nabla}\vec{\nabla} T(t,\vec{x})]_{\vec{x}=\vec{x}(t,\vec{x}_0)}.$$

I hope with this very detailed writing of all the arguments, the two (equivalent) descriptions of fluid mechanics and how to switch from one to the other has become more clear.