Editing ThreeDimensionalConfigurations/Challenges (section)

====Equation of Continuity====
In steady state, <math>~\partial (\rho\bold{u})/\partial t = 0</math>. Hence the rotating-frame-based continuity equation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla \cdot (\rho\bold{u})</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math>
  </td>
</tr>
</table>
If we insist that the momentum-density vector be expressible in terms of the curl of a vector &#8212; for example,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho\bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\nabla \times \boldsymbol{\mathfrak{J}} \, ,</math>
  </td>
</tr>
<tr><td align="center" colspan="3">[https://digitalcommons.lsu.edu/gradschool_disstheses/6650/ Saied W. Andalib (1998)], &sect;4.1, p. 80, Eq. (4.1)
</td></tr>
</table>

then satisfying this steady-state continuity equation is guaranteed because the [https://en.wikipedia.org/wiki/Vector_calculus_identities#Divergence_of_curl_is_zero divergence of a curl is always zero].   "<font color="orange">The task</font> of satisfying the steady-state equation of continuity <font color="orange">then shifts to identifying an appropriate expression for the vector potential, <math>~\boldsymbol{\mathfrak{J}} \, .</math></font>"  In the most general case, in terms of this vector potential the three Cartesian components of the momentum-density vector are,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho u_x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \mathfrak{J}_z}{\partial y}
-
\frac{\partial \mathfrak{J}_y}{\partial z} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho u_y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \mathfrak{J}_x}{\partial z}
-
\frac{\partial \mathfrak{J}_z}{\partial x} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho u_z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \mathfrak{J}_y}{\partial x}
-
\frac{\partial \mathfrak{J}_x}{\partial y} \, .
</math>
  </td>
</tr>
</table>
Here, <font color="red">we will follow Andalib's lead and only look for fluid flows with no vertical motions</font>.  That is to say, we will set <math>~\rho u_z = 0</math>, in which case this last expression establishes the constraint,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial \mathfrak{J}_y}{\partial x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \mathfrak{J}_x}{\partial y} \, .
</math>
  </td>
</tr>
</table>
"<font color="orange">A general solution to this equation can be found only if there exists a scalar function <math>~\Gamma(x, y, z)</math> such that &hellip;</font>"

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathfrak{J}_y = \frac{\partial \Gamma}{\partial y}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\mathfrak{J}_x = \frac{\partial \Gamma}{\partial x} \, ;</math>
  </td>
</tr>
</table>
note that this adopted functional behavior works because the constraint becomes,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\partial^2 \Gamma}{\partial x \partial y}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\partial^2 \Gamma}{\partial y \partial x} \, .</math>
  </td>
</tr>
</table>

Hence, the expressions for the x- and y-components of the momentum-density vector may be rewritten, respectively, as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho u_x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \mathfrak{J}_z}{\partial y}
-
\frac{\partial}{\partial z} \biggl[ \frac{\partial \Gamma}{\partial y} \biggr]
=
+ \frac{\partial}{\partial y}\biggl[ \mathfrak{J}_z - \frac{\partial\Gamma}{\partial z} \biggr]
\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\rho u_y</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial }{\partial z}\biggl[ \frac{\partial \Gamma}{\partial x} \biggr]
-
\frac{\partial \mathfrak{J}_z}{\partial x} 
=
- \frac{\partial}{\partial x}\biggl[ \mathfrak{J}_z - \frac{\partial\Gamma}{\partial z} \biggr]
\, .
</math>
  </td>
</tr>
</table>

If we again <font color="red">follow Andalib's lead and only look for models in which the x-y-plane flow is independent of the vertical coordinate, z</font>, then, <font color="orange"><math>~\mathfrak{J}_z</math> and <math>~\partial\Gamma/\partial z</math> must be functions of x and y only.  Therefore, <math>~\mathfrak{J}_z</math> is independent of z and <math>~\Gamma</math> is at most linear in z.</font>  Now, rather than focusing on the determination of <math>~\Gamma(x,y)</math>, we can just as well define the scalar function, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Psi(x,y)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\mathfrak{J}_z - \frac{\partial\Gamma}{\partial z} \, ,
</math>
  </td>
</tr>
</table>
in which case "<font color="orange">&hellip; the components of the momentum density may be written as:</font>"
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho \bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \frac{\partial \Psi}{\partial y} - \boldsymbol{\hat\jmath} \frac{\partial \Psi}{\partial x} \, .
</math>
  </td>
</tr>
</table>
It is straightforward to demonstrate that this expression for the momentum-density vector does satisfy the steady-state continuity equation.  "<font color="orange">The function <math>~\Psi(x, y)</math> will serve a similar role as the velocity potential for incompressible fluids.</font>"