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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
    [tex]T=T^{L}(x_0,y_0,z_0,t)[/tex]
    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

    vanhees71

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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

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

    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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