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Homework Help: Euler's equations in differential forms

  1. Oct 30, 2016 #1
    1. The problem statement, all variables and given/known data

    Euler's equations can be written using vector calculus as

    ##\displaystyle{\frac{\partial v_{i}}{\partial t}+v^{j}\left(\frac{\partial v_{i}}{\partial x^{j}}\right) = -\left(\frac{1}{\rho}\right)\frac{\partial p}{\partial x^{i}}+f_{i}}.##



    Euler's equations can also be written using differential forms as

    ##\displaystyle{\mathcal{L}_{{\bf{v}}+\partial / \partial t}(\nu) = \textbf{d} \left\{\frac{1}{2}||{\bf{v}}||^{2}+\phi-\int\frac{dp}{\rho}\right\}}##



    Under the assumption that ##f_{i} = \text{grad}_{i}\phi##, that ##p=p(\rho)## and that ##\nu## is the ##1##-form with components ##v_i##, how can you prove that the two formulations above are equivalent?

    2. Relevant equations


    3. The attempt at a solution

    ##\displaystyle{\frac{\partial v_{i}}{\partial t}+v^{j}\left(\frac{\partial v_{i}}{\partial x^{j}}\right) = -\left(\frac{1}{\rho}\right)\frac{\partial p}{\partial x^{i}}+f_{i}}##

    ##\displaystyle{\frac{\partial v_{i}}{\partial t}+\frac{\partial}{\partial x^{j}}\left(v^{j}v_{i}\right)-v_{i}\frac{\partial v^{j}}{\partial x^{j}} = \text{grad}_{i}\ \left(-\int\frac{dp}{\rho}\right)+\text{grad}_{i}\ \phi}##

    ##\displaystyle{\frac{\partial v_{i}}{\partial t}+\frac{\partial}{\partial x^{j}}\left(v^{j}v_{i}\right) = v_{i}\frac{\partial v^{j}}{\partial x^{j}}+\text{grad}_{i}\ \left(-\int\frac{dp}{\rho}\right)+\text{grad}_{i}\ \phi}##

    ##\text{I have to fill up this missing line}##

    ##\displaystyle{\frac{\partial}{\partial x^{j}}\left(v^{j}v_{i}\right) + \frac{\partial v_{i}}{\partial t} = \text{grad}_{i}\left(\frac{1}{2}||{\bf{v}}||^{2}\right)+\text{grad}_{i}\ \left(-\int\frac{dp}{\rho}\right)+\text{grad}_{i}\ \phi}##

    ##\displaystyle{\frac{\partial\nu}{\partial t}+\mathcal{L}_{\bf{v}}(\nu) = \textbf{d}\left\{\frac{1}{2}||{\bf{v}}||^{2}+\phi-\int\frac{dp}{\rho}\right\}}##

    ##\displaystyle{\mathcal{L}_{{\bf{v}}+\partial / \partial t}(\nu) = \textbf{d} \left\{\frac{1}{2}||{\bf{v}}||^{2}+\phi-\int\frac{dp}{\rho}\right\}}##
     
  2. jcsd
  3. Nov 5, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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