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Whitworth's (1981) Isothermal Free-Energy Surface
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Establishing Set of Governing Equations

Accompanying chapter discussion

The lefthand side of the two equations containing time derivatives will take a different form if, alternatively, the choice is made to view the evolution from an Eulerian frame of reference. In this case the set of governing equations becomes,

Eulerian Frame

<math>~\frac{\partial M_r}{\partial r} </math>


<math>~4\pi r^2 \rho \, , </math>

<math>~\frac{\partial M_r}{\partial t} </math>


<math>~- 4\pi r^2 \rho v_r \, ,</math>

<math>~\frac{\partial v_r}{\partial t} + v_r \frac{\partial v_r}{\partial r}</math>


<math>~- c_s^2 \biggl( \frac{\partial \ln \rho}{\partial r}\biggr) - \frac{GM_r}{r^2} \, .</math>

Notice that, in place of the standard continuity equation, we will use the following equivalent statement of mass conservation:



<math>~0 </math>

<math>~\Rightarrow ~~~ 0</math>


<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math>



<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math>

See Especially

Whitworth's (1981) Isothermal Free-Energy Surface

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