Changes between Version 64 and Version 65 of SelfGravityDevel

Timestamp:

10/22/19 12:26:37 (6 years ago)

Author:

Jonathan

Comment:

—

SelfGravityDevel

@@ -8 +8 @@
 [[latex($\frac{d \rho \mathbf{v}}{dt} = \mathbf{f}_g = -\rho \nabla \phi$)]]
 
-and energy equation
+and the energy equation now includes the gravitational self-energy on the left
 
 [[latex($\frac{d E}{dt} = \mathbf{v} \cdot \mathbf{f}_g = -\rho \mathbf{v} \cdot \nabla \phi$)]]
@@ -24 +24 @@
 AstroBEAR uses hypre's linear solver to solve Poisson's equation for the potential.  However, there are two methods for including the source terms for momentum and energy.  The non-conservative approach simply includes the terms during a source update, while the conservative approach recasts the RHS of the momentum and energy equations as total divergences - so they can be differenced conservatively.
 
-
-
-
-
+Substituting the Poisson equation into the momentum equation we have
+
+[[latex($\frac{d \rho \mathbf{v}}{dt} = (\frac{\nabla^2\phi}{4 \pi G}+\rho_0) \nabla \phi = \nabla \cdot \left [ \frac{1}{4 \pi G} \left ( \nabla \phi \nabla \phi - \frac{1}{2} \left ( \nabla \phi \cdot \nabla \phi + 4 \pi G \rho_0 \phi \right) \right ) \right ]$)]]
+
+
+
+
+
+For the energy equation, we can only find a conservative approach - if we consider the evolution of the total combined energy (hydro + gravitational).  
+
+[[latex($\frac{d \left(E + \frac{1}{2} \left (\rho - \rho_0 \right) \phi \right)}{dt} = -\nabla \cdot \mathbf{F_g} $)]]
+
+
+Combining this with the our previous energy equation we arrive at
+
+[[latex($\nabla \cdot \mathbf{F_g} = -\frac{1}{2} \frac{d \left (\rho \phi \right)}{dt} + \frac{1}{2} \rho_0 \frac{d \phi}{dt}+ \rho \mathbf{v} \cdot \nabla \phi$)]]
+
+Expanding the time derivative and using the continuity equation, we arrive at
+
+
+[[latex($\nabla \cdot \mathbf{F_g} = \frac{1}{2} \phi \nabla \cdot \left ( \rho \mathbf{v} \right) - \frac{1}{2} \left (\rho - \rho_0 \right) \dot{\phi}  + \rho \mathbf{v} \cdot \nabla \phi$)]]
+
+
+Now from time differencing the poisson equation, we arrive at
+
+[[latex($\nabla^2 \dot{\phi} = 4 \pi G \dot{\rho} = - 4 \pi G \nabla \cdot (\rho \mathbf{v})$)]]
+
+and multiplying both sides by [[latex($\frac{\phi}{8 \pi G}$)]] gives
+
+[[latex($\frac{1}{2} \nabla \cdot (\rho \mathbf{v}) \phi = -\frac{1}{8 \pi G} \phi \nabla^2 \dot{\phi}$)]]
+
+
+Adding the LHS and subtracting the RHS and substituting 
+
+[[latex($\rho - \rho_0 = \frac{1}{4 \pi G} \nabla^2 \phi$)]]
+
+into the divergence equation gives us
+
+[[latex($\nabla \cdot \mathbf{F_g} = \nabla \cdot   \left (\phi \rho \mathbf{v} \right) + \frac{1}{8 \pi G} \left (\phi \nabla^2 \dot{\phi} - \dot{\phi}\nabla^2\phi \right) $)]]
+
+Now we can use the relationship 
+
+[[latex($\nabla \cdot \left [ \phi \nabla \dot{\phi} - \dot{\phi} \nabla \phi \right ]= \phi \nabla^2 \dot{\phi} - \dot{\phi} \nabla^2 \phi$)]]
+
+to arrive at
+
+[[latex($\mathbf{F_g} = \phi \rho \mathbf{v} + \frac{1}{8 \pi G} \left (\phi \nabla \dot{\phi} - \dot{\phi}\nabla \phi \right) $)]]
+
+
+[[CollapsibleStart(Notes)]]
 The electric field generated by a point charge[[latex($Q$)]]is
 
@@ -231 +277 @@
   * Solve for gas potential after accretion?
   * 
+
+[[CollapsibleEnd]]