User:Tohline/AxisymmetricConfigurations/SolutionStrategies
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Axisymmetric Configurations (Structure — Part II)
Equilibrium, axisymmetric structures are obtained by searching for time-independent, steady-state solutions to the identified set of simplified governing equations. We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:
<math> \frac{\partial f}{\partial t} + \biggl[ \dot\varpi \frac{\partial f}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial f}{\partial z} \biggr] </math>
Equation of Continuity
<math>\cancel{\frac{\partial\rho}{\partial t}} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr]
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math>
The Two Relevant Components of the
Euler Equation
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<math> \cancel{\frac{\partial \dot\varpi}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] </math> |
= |
<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3} </math> |
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<math> \cancel{\frac{\partial \dot{z}}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] </math> |
= |
<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math> |
Adiabatic Form of the
First Law of Thermodynamics
<math>
\biggl\{\cancel{\frac{\partial \epsilon}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \epsilon}{\partial\varpi} \biggr] +
\biggl[ \dot{z} \frac{\partial \epsilon}{\partial z} \biggr]\biggr\} +
\biggl\{\cancel{\frac{\partial }{\partial t}\biggl(\frac{1}{\rho}\biggr)} + \biggl[ \dot\varpi \frac{\partial f}{\partial\varpi} \biggr] +
\biggl[ \dot{z} \frac{\partial f}{\partial z} \biggr] \biggr\} = 0
</math>
Poisson Equation
<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho .
</math>
To find steady-state solutions, we must first rewrite each time derivative in Eulerian rather than Lagrangian form. Specifically, for any scalar function, <math>f</math>,
<math> \frac{df}{dt} \rightarrow \frac{\partial f}{\partial t} + (\vec{v}\cdot \nabla)f . </math>
terms of For a time-independent structure, the <math>4^\mathrm{th}</math> (first law of thermodynamics) equation is trivially satisfied.
Let's begin by rewriting the identified set of equations in terms of Eulerian rather than Lagrangian time derivatives:
The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_r v_r = 0</math>. After setting the radial velocity, <math>v_r</math>, and all time-derivatives to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>3^\mathrm{rd}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> (Euler) and <math>4^\mathrm{th}</math> give, respectively,
Hydrostatic Balance
<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,
and,
Poisson Equation
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> .
(We recognize the first of these expressions as being the statement of hydrostatic balance appropriate for spherically symmetric configurations.)
We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions — <math>~P</math>(r), <math>~\rho</math>(r), and <math>~\Phi</math>(r). As has been outlined in our discussion of supplemental relations for time-independent problems, in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between <math>~P</math> and <math>~\rho</math>. (See below.)
Solution Strategies
When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where <math>~\rho</math> is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,
<math>dm_r = \rho dr \oint dS = r^2 \rho dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\varphi = 4\pi r^2 \rho dr</math> ,
where we have pulled from the Wikipedia discussion of integration and differentiation in spherical coordinates to define the spherical surface element <math>dS</math>. Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,
<math>M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
We can also state that,
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<math>~\frac{dM_r}{dr} = 4\pi r^2 \rho</math> |
This differential relation is often identified as a statement of mass conservation that replaces the equation of continuity for spherically symmetric, static equilibrium structures.
Technique 3
As in Technique #2, we replace <math>dP/\rho</math> by d<math>~H</math> in the hydrostatic balance relation, but this time we realize that the resulting expression can be written in the form,
<math>\frac{d}{dr}(H+\Phi) = 0</math> .
This means that, throughout our configuration, the functions <math>~H</math>(<math>~\rho</math>) and <math>~\Phi</math>(<math>~\rho</math>) must sum to a constant value, call it <math>C_\mathrm{B}</math>. That is to say, the statement of hydrostatic balance reduces to the algebraic expression,
<math>H + \Phi = C_\mathrm{B}</math> .
This relation must be solved in conjunction with the Poisson equation,
<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> ,
giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, <math>~H</math>, <math>~\rho</math>, and <math>~\Phi</math>
See Also
- Part I of Axisymmetric Configurations: Simplified Governing Equations
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© 2014 - 2021 by Joel E. Tohline |