5
$\begingroup$

In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the equations are:

  1. The mass continuity equation $\frac{\partial \rho}{\partial t}+\nabla (\rho \mathbf{v}) = 0$

  2. The Euler equation (3 components) $\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p$

  3. A statement of the fact, that there is no dissipation of energy $\frac{d s}{d t} = 0$ ($s$ is entropy per unit mass)

My question: how to express the last equation in terms of the actual variables we are using? We have to assume some form of the equation of state for the fluid in order to do it right? Landau in his typical fashion glance over it, assuming the readers' perfect understanding of thermodynamics, which is not my case unfortunately.

PS. The question is a little related to a more general one I posted yesterday: Explicit form of the entropy production in hydrodynamics

$\endgroup$

1 Answer 1

5
$\begingroup$

Generally the fifth equation is pressure conservation: $$ \frac{\partial p}{\partial t}+\mathbf v\cdot\nabla p+\gamma p\nabla\cdot\mathbf v=0 $$ which, depending on your particular subfield, is usually written in terms of the total energy: $$ \frac{\partial E}{\partial t}+\nabla\cdot\left[\left(E+p\right)\mathbf v\right]=0 $$ where $E=\frac12\rho v^2+p\left(\gamma-1\right)^{-1}$. However, if you want to do this in terms of the entropy and you are working with an ideal gas, then we can define a variable $$ S\equiv p\rho^{-\gamma} $$ as a measure for entropy (not entropy itself because it's missing certain constants, e.g. heat capacity at constant volume $C_v$). Then you can use the total derivative to evolve $S$: $$ \frac{DS}{Dt}=\frac{\partial S}{\partial t}+\mathbf v\cdot\nabla S=0 $$


The connection between this measure of entropy, $S$, and the entropy density, $s$, is $$ s=C_v\ln(S)=C_v\ln\left(p\rho^{-\gamma}\right)=C_p\ln\left(p^{1/\gamma}\rho^{-1}\right) $$ where we used $\gamma=C_p/C_v$ between the latter two equalities. The above relation can be found in Section 83 of Landau's Fluid Dynamics text.

$\endgroup$
4
  • $\begingroup$ Would the downvoter like to explain what they believe is wrong with my answer? $\endgroup$
    – Kyle Kanos
    Jun 4, 2014 at 13:39
  • $\begingroup$ OK, thanks! And how to get (or justify somehow) this pressure conservation equation? $\endgroup$
    – Lurco
    Jun 6, 2014 at 19:08
  • $\begingroup$ You can start with thermodynamics & say $\dot{E}=\dot{Q}-\dot{W}$ (total energy = heat transfer - work done), then work your way with ideal gas laws and get to it. $\endgroup$
    – Kyle Kanos
    Jun 6, 2014 at 19:50
  • $\begingroup$ But it's probably easier to start with stresses & strains and get the conservation of energy that way (e.g., the way this site does it). $\endgroup$
    – Kyle Kanos
    Jun 6, 2014 at 19:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.