AxisymmetricConfigurations/Equilibria

Axisymmetric Configurations (Steady-State Structures)

Constructing Steady-State Axisymmetric Configurations

Equilibrium, axisymmetric structures are obtained by searching for time-independent, steady-state solutions to the identified set of simplified governing equations.  
 
 
 
 
 

Cylindrical Coordinate Base

We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:

Equation of Continuity

${\xcancel{\frac{\partial \rho }{\partial t}}}^0+\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\rho \varpi \dot{\varpi }\right]+\frac{\partial }{\partial z}\left[\rho \dot{z}\right]=0$

The Two Relevant Components of the
Euler Equation

${\xcancel{\frac{\partial \dot{\varpi }}{\partial t}}}^0+\left[\dot{\varpi }\frac{\partial \dot{\varpi }}{\partial \varpi }\right]+\left[\dot{z}\frac{\partial \dot{\varpi }}{\partial z}\right]$	$=$	$-[\frac{1}{\rho }\frac{\partial P}{\partial \varpi }+\frac{\partial \mathrm{\Phi }}{\partial \varpi }]+\frac{j^2}{{\varpi }^3}$
${\xcancel{\frac{\partial \dot{z}}{\partial t}}}^0+\left[\dot{\varpi }\frac{\partial \dot{z}}{\partial \varpi }\right]+\left[\dot{z}\frac{\partial \dot{z}}{\partial z}\right]$	$=$	$-[\frac{1}{\rho }\frac{\partial P}{\partial z}+\frac{\partial \mathrm{\Phi }}{\partial z}]$

Adiabatic Form of the
First Law of Thermodynamics
$\{\cancel{\frac{\partial ϵ}{\partial t}}+[\dot{\varpi }\frac{\partial ϵ}{\partial \varpi }]+[\dot{z}\frac{\partial ϵ}{\partial z}\left]\right\}+P\{\cancel{\frac{\partial }{\partial t}\left(\frac{1}{\rho }\right)}+[\dot{\varpi }\frac{\partial }{\partial \varpi }\left(\frac{1}{\rho }\right)]+[\dot{z}\frac{\partial }{\partial z}\left(\frac{1}{\rho }\right)\left]\right\}=0$

Poisson Equation

$\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \frac{\partial \mathrm{\Phi }}{\partial \varpi }\right]+\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}=4\pi G\rho .$

The steady-state flow field that will be adopted to satisfy both an axisymmetric geometry and the time-independent constraint is, $\overrightarrow{v}={\hat{e}}_{\phi }(\varpi \dot{\phi })$. That is, $\dot{\varpi }=\dot{z}=0$ but, in general, $\dot{\phi }$ is not zero and can be an arbitrary function of $\varpi$ and $z$, that is, $\dot{\phi }=\dot{\phi }(\varpi ,z)$. We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity $\dot{\phi }(\varpi ,z)$, or of the specific angular momentum, $j(\varpi ,z)={\varpi }^2\dot{\phi }(\varpi ,z)$.

After setting the radial and vertical velocities to zero, we see that the $1^{\mathrm{s}\mathrm{t}}$ (continuity) and $4^{\mathrm{t}\mathrm{h}}$ (first law of thermodynamics) equations are trivially satisfied while the $2^{\mathrm{n}\mathrm{d}}$ & $3^{\mathrm{r}\mathrm{d}}$ (Euler) and $5^{\mathrm{t}\mathrm{h}}$ (Poisson) give, respectively,

$[\frac{1}{\rho }\frac{\partial P}{\partial \varpi }+\frac{\partial \mathrm{\Phi }}{\partial \varpi }]-\frac{j^2}{{\varpi }^3}$	$=$	$0$
$[\frac{1}{\rho }\frac{\partial P}{\partial z}+\frac{\partial \mathrm{\Phi }}{\partial z}]$	$=$	$0$
$\frac{1}{\varpi }\frac{\partial }{\partial \varpi }\left[\varpi \frac{\partial \mathrm{\Phi }}{\partial \varpi }\right]+\frac{{\partial }^2\mathrm{\Phi }}{\partial z^2}$	$=$	$4\pi G\rho .$

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 $P$ and $\rho$.

Spherical Coordinate Base

We begin with an Eulerian formulation of the principle governing equations written in spherical coordinates for an axisymmetric configuration, namely,

Equation of Continuity

$\frac{\partial \rho }{\partial t}+[\frac{1}{r^2}\frac{\partial (\rho r^2\dot{r})}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(\rho \dot{\theta }r\sin \theta \left)\right]$

$=$

$0$

The Two Relevant Components of the
Euler Equation

${\hat{e}}_r$:	$\{\frac{\partial \dot{r}}{\partial t}+[\dot{r}\frac{\partial \dot{r}}{\partial r}]+[\dot{\theta }\frac{\partial \dot{r}}{\partial \theta }\left]\right\}-r{\dot{\theta }}^2$	=	$-[\frac{1}{\rho }\frac{\partial P}{\partial r}+\frac{\partial \mathrm{\Phi }}{\partial r}]+\left[\frac{j^2}{r^3{\sin }^2\theta }\right]$
${\hat{e}}_{\theta }$:	$r\{\frac{\partial \dot{\theta }}{\partial t}+[\dot{\theta }\frac{\partial \dot{\theta }}{\partial r}]+[\dot{\theta }\frac{\partial \dot{r}}{\partial \theta }\left]\right\}+2\dot{r}\dot{\theta }$	=	$-[\frac{1}{\rho r}\frac{\partial P}{\partial \theta }+\frac{1}{r}\frac{\partial \mathrm{\Phi }}{\partial \theta }]+\left[\frac{j^2}{r^3{\sin }^3\theta }\right]\cos \theta$

Adiabatic Form of the
First Law of Thermodynamics

$\{\frac{\partial ϵ}{\partial t}+[\dot{r}\frac{\partial ϵ}{\partial r}]+[\dot{\theta }\frac{\partial ϵ}{\partial \theta }\left]\right\}+P\left\{\frac{\partial }{\partial t}\right(\frac{1}{\rho })+[\dot{r}\frac{\partial }{\partial r}\left(\frac{1}{\rho }\right)]+[\dot{\theta }\frac{\partial }{\partial \theta }\left(\frac{1}{\rho }\right)\left]\right\}$

$=$

$0$

Poisson Equation

$\frac{1}{r^2}\frac{\partial }{\partial r}\left[r^2\frac{\partial \mathrm{\Phi }}{\partial r}\right]+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial \mathrm{\Phi }}{\partial \theta })$

$=$

$4\pi G\rho$

where the pair of "relevant" components of the Euler equation have been written in terms of the specific angular momentum,

$j(r,\theta )\equiv (r\sin \theta {)}^2\dot{\phi }$,

which is a conserved quantity in axisymmetric systems.

Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, $\dot{r}$ and $\dot{\theta }$, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:

Spherical Coordinate Base

Poisson Equation $\frac{1}{r^2}\frac{\partial }{\partial r}\left[r^2\frac{\partial \mathrm{\Phi }}{\partial r}\right]+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial \mathrm{\Phi }}{\partial \theta })$ $=$ $4\pi G\rho$ The Two Relevant Components of the Euler Equation ${\hat{e}}_r$:     $0$ = $-[\frac{1}{\rho }\frac{\partial P}{\partial r}+\frac{\partial \mathrm{\Phi }}{\partial r}]+\left[\frac{j^2}{r^3{\sin }^2\theta }\right]$ ${\hat{e}}_{\theta }$:     $0$ = $-[\frac{1}{\rho r}\frac{\partial P}{\partial \theta }+\frac{1}{r}\frac{\partial \mathrm{\Phi }}{\partial \theta }]+\left[\frac{j^2}{r^3{\sin }^3\theta }\right]\cos \theta$

See Also

• Part I of Axisymmetric Configurations: Simplified Governing Equations