Editing Apps/Korycansky Papaloizou 1996 (section)

==Two-Dimensional Planar Flow==

In keeping with their objective to study steady-state flows in infinitesimally thin disks, {{ KP96hereafter }} imposed one additional important constraint on the set of governing equations.  

<font color="darkblue">Assumption #3:</font> {{ KP96hereafter }} set <math>v_z = 0</math> everywhere.

Then, in order to determine the steady-state spatial distribution of the three principal physical variables <math>\rho(x,y)</math>, <math>v_x(x,y)</math>, and <math>v_y(x,y)</math>, they looked for solutions that would simultaneously satisfy the following three PDEs:
* The z-component of the curl of the steady-state Euler equation, that is,
<div align="center">
<math>
[\nabla\times\vec{A}]_z =  0 ;
</math>
</div>
* The steady-state continuity equation, that is, 
<div align="center">
<math>
\vec{v}\cdot\nabla\rho + \rho\nabla\cdot\vec{v} = 0 ; 
</math>
</div>
and,
* The scalar product of <math>\vec{v}</math> with the steady-state Euler equation, that is,
<div align="center">
<math>
\vec{v} \cdot \nabla F_B  = 0 .
</math>
</div>
The algebraic equation of state that they used to supplement this coupled set of governing PDEs is identified in their paper, in the discussion associated with their Eq. (23).  

Drawing from our [[PGE/RotatingFrame#Nonlinear_Velocity_Cross-Product|accompanying discussion]] of how the curl of <math>\vec{A}</math> behaves when <math>v_z = 0</math>, the first of these PDEs takes the form,
<div align="center">
<math>
[\nabla\times\vec{A}]_z = \frac{\partial}{\partial x}\biggl[ (\zeta_z + 2\Omega) v_x  \biggr] + \frac{\partial}{\partial y}\biggl[ (\zeta_z + 2\Omega) v_y \biggr] = 0 
</math><br />

<math>
\Rightarrow ~~~~~\vec{v}\cdot\nabla(\zeta_z + 2\Omega) + (\zeta_z + 2\Omega)\nabla\cdot\vec{v} = 0 .
</math>
</div>
This last expression appears as Eq. (2) in {{ KP96hereafter }}.  (For this last expression to be valid it must be understood that, for the inherently 2D problem under investigation by {{ KP96hereafter }}, <math>\nabla</math> is only operating in x and y.) Rewriting this equation, we conclude that,
<div align="center">
<math>
\nabla\cdot\vec{v} =-\vec{v} \cdot \biggl[ \frac{\nabla(2\Omega + \zeta_z)}{(2\Omega + \zeta_z)} \biggr] 
= -\vec{v} \cdot \nabla[\ln(2\Omega + \zeta_z)].
</math>
</div>
But from the steady-state continuity equation we also know that,
<div align="center">
<math>
\nabla\cdot\vec{v} = -\vec{v}\cdot\biggl[\frac{\nabla\rho}{\rho} \biggr] = -\vec{v} \cdot \nabla[\ln\rho] .
</math>
</div>
Hence, combining the two relations, we conclude that,
<div align="center">
<math>
\vec{v} \cdot \nabla[\ln(2\Omega + \zeta_z)]  = \vec{v} \cdot \nabla[\ln\rho] ,
</math>
</div>
that is,
<div align="center">
<math>
\vec{v} \cdot \nabla\ln\biggl[ \frac{(2\Omega + \zeta_z)}{\rho} \biggr]  = 0  .
</math>
</div>
This is essentially {{ KP96hereafter }}'s Eq. (3).  It tells us that, in the steady-state flow whose spatial structure we are seeking, the velocity vector (and also the momentum density vector <math>\rho\vec{v}</math>) must everywhere be tangent to contours of constant <math>(2\Omega + \zeta_z)/\rho</math> &#8212; a scalar quantity that {{ KP96hereafter }} refer to as ''<font color="red">vortensity</font>.''  

<font color="darkblue"><b>Introduce stream function:</b></font>
The constraint implied by the continuity equation also suggests that it might be useful to define a stream function in terms of the momentum density &#8212; instead of in terms of just the velocity, which is the natural treatment in the context of incompressible fluid flows.  {{ KP96hereafter }} do this. They define the stream function, <math>\Psi</math>, such that (see their Eq. 4),
<div align="center">
<math>
\rho\vec{v} = \nabla\times(\hat{k}\Psi)  .
</math>
</div>
in which case,
<div align="center">
<math>
v_x = \frac{1}{\rho} \frac{\partial \Psi}{\partial y} ~~~~~\mathrm{and}~~~~~  
v_y = - \frac{1}{\rho} \frac{\partial \Psi}{\partial x} .
</math>
</div>
This implies as well that the z-component of the fluid vorticity can be expressed in terms
of the stream function as follows (see Eq. 5 of {{ KP96hereafter }}):
<div align="center">
<math>
\zeta_z = - \nabla\cdot \biggl( \frac{\nabla\Psi}{\rho} \biggr) = - \frac{\partial}{\partial x} \biggl[ \frac{1}{\rho} \frac{\partial\Psi}{\partial x} \biggr] - \frac{\partial}{\partial y} \biggl[ \frac{1}{\rho} \frac{\partial\Psi}{\partial y} \biggr].
</math>
</div>

Since, by design, streamlines defined by the momentum-density vector field must trace out lines of constant <math>\Psi</math> and, according to the conclusion drawn above, the momentum density vector must everywhere be tangent to contours of constant ''vortensity'', we can conclude &#8212; as did {{ KP96hereafter }} &#8212; that the ''vortensity'' <math>(\zeta_z + 2\Omega)/\rho</math> must be constant along streamlines.  The ''vortensity'' is therefore a function of <math>\Psi</math> alone, so we can write,
<div align="center">
<math>
\frac{\zeta_z + 2\Omega}{\rho} = g(\Psi) .
</math>
</div>

By the same token, the condition obtained from the scalar product of <math>\vec{v}</math> with the steady-state Euler equation,
<div align="center">
<math>
\vec{v} \cdot \nabla F_B  = 0 ,
</math>
</div>
implies that the Bernoulli function must also be expressible as a function of <math>\Psi</math> alone.  Hence, we can write,
<div align="center">
<math>
\frac{1}{2}v^2 + H + \Phi - \frac{1}{2}|\Omega\hat{k} \times \vec{x}|^2  = F_B(\Psi) .
</math>
</div>
{{ KP96hereafter }} then go on to demonstrate that the relationship between the functions <math>g(\Psi)</math> and <math>F(\Psi)</math> is,
<div align="center">
<math>
\frac{dF}{d\Psi} = -g(\Psi) ,
</math>
</div>
which allows the determination of <math>F</math> up to a constant of integration.