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Fluid mechanics Lagrange & Euler formalism

  1. Apr 30, 2012 #1
    Lagrange & Euler formalism

    How we get relation

    [tex](\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}[/tex]

    [tex]T[/tex] - temperature
    [tex]t[/tex] - time
  2. jcsd
  3. May 3, 2012 #2
    Lagrange representation
    where coordinates [tex]x_0,y_0,z_0[/tex] are fixed.

    I have a trouble with that. Can you tell me one type of this function?
  4. May 3, 2012 #3


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

    Again, the Euler coordinates [itex]\vec{x}_0[/itex] are labels for a fixed fluid element by noting its position at a fixed (initial) time [itex]t_0[/itex]. 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) [itex]\vec{x}(t,\vec{x}_0)[/itex].

    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 [itex]\vec{x}[/itex] measures the velocity of the fluid element at this position at time, [itex]t[/itex]. This field we denote by [itex]\vec{v}(t,\vec{x})[/itex].

    The relation to the Euler coordinates is then given by

    [tex]\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)].[/tex]

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

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

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


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

    [tex]\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)}.[/tex]

    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.
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