Similarity Solution

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>~=</math>	<math>~4\pi r^2 \rho \, , </math>
<math>~\frac{\partial M_r}{\partial t} </math>	<math>~=</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>~=</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>~\frac{dM_r}{dt}</math>	<math>~=</math>	<math>~0 </math>
<math>~\Rightarrow ~~~ 0</math>	<math>~=</math>	<math>~\frac{\partial M_r}{\partial t} + v_r ~\frac{\partial M_r}{\partial r} </math>
	<math>~=</math>	<math>~\frac{\partial M_r}{\partial t} +4\pi r^2 \rho v_r \, .</math>

See Especially

M. V. Penston (1969, MNRAS, 144, 425): Dynamics of Self-Gravitating Gaseous Sphers - III. Analytic Results in the Free-Fall of Isothermal Cases
Richard B. Larson (1969, MNRAS, 145, 271): Numerical Calculations of the Dynamics of Collapsing Proto-Star
F. H. Shu (1977, ApJ, 214, 488-497): Self-Similar Collapse of Isothermal Spheres and Star Formation
C. Hunter (1977, ApJ, 218, 834-845): The Collapse of Unstable Isothermal Spheres

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Similarity Solution

Establishing Set of Governing Equations

See Especially

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