Difference between revisions of "User:Tohline/SSC/Perspective Reconciliation"

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Revision as of 00:59, 12 December 2014

Reconciling Eulerian versus Lagrangian Perspectives

Whitworth's (1981) Isothermal Free-Energy Surface
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Linearizing the Key Relations

Continuity Equation
Lagrangian Perspective Eulerian Perspective

<math>~\frac{d\rho}{dt}</math>

<math>~=</math>

<math>~- \rho \nabla\cdot\vec{v}</math>

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

<math>~=</math>

<math>~- \rho \nabla\cdot\vec{v} - \vec{v}\cdot \nabla\rho</math>

Spherically Symmetric Initial Configurations & Purely Radial Perturbations

<math>~\frac{d\rho}{dt}</math>

<math>~=</math>

<math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr)</math>

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

<math>~=</math>

<math>~- \frac{\rho}{r^2} \frac{\partial}{\partial r} \biggl( r^2 v_r \biggr) - v_r \frac{\partial \rho}{\partial r}</math>

See Also