Editing Appendix/Ramblings/HybridSchemeImplications (section)

====Inertial Frame====

As viewed from a cylindrical-coordinate-based <math>~(\varpi, \varphi, z)</math> inertial reference frame, we are interested in specifying the location, 
<div align="center">
<math>~\bold{x} = \mathbf{\hat{e}}_\varpi \varpi + \bold{\hat{k}} z \, ,</math><br />

[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 646, Appendix &sect;1.B.2, Eq. (1B-18)
</div>
of a Lagrangian fluid element at time <math>~t = 0</math> &#8212; hereafter denoted by the subscript, <math>~0</math> &#8212; as well as at later times.  Although the position vector, <math>~\bold{x}</math>, does not explicitly display a dependence on the azimuthal coordinate angle, <math>~\varphi</math>, it is important to realize that the orientation in space of the unit vector, <math>~\bold{\hat{e}}_\varpi</math>, does depend on the value of this coordinate angle.

At any point in time, the instantaneous velocity of this Lagrangian fluid element will correspond precisely with the (total) time-derivative of its instantaneous position vector, that is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{v}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{d\bold{x}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varpi \frac{d\varpi}{dt} + \bold{\hat{k}} \frac{dz}{dt} + \varpi \frac{d \bold{\hat{e}}_\varpi}{dt}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varpi \frac{d\varpi}{dt} + \bold{\hat{k}} \frac{dz}{dt} + \varpi \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \bigg] \, .</math>
  </td>
</tr>
<tr><td align="center" colspan="5">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 647, Appendix &sect;1.B.2, Eq. (1B-23)
</td></tr>
</table>

In carrying out this time differentiation, the last term on the right-hand-side accounts for the aforementioned dependence of <math>~\bold{\hat{e}}_\varpi</math> on <math>~\varphi</math>.  Similarly, the following component breakdown of the Lagrangian fluid element's acceleration takes into account the dependence of <math>~\bold{\hat{e}}_\varphi</math> on <math>~\varphi</math>:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\bold{a}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{d\bold{v}}{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bold{\hat{e}}_\varpi \frac{d^2\varpi}{dt^2} + \bold{\hat{k}} \frac{d^2z}{dt^2} 
+ \bold{\hat{e}}_\varphi \biggl[\frac{d\varpi}{dt} \cdot  \frac{d\varphi}{dt} + \varpi \frac{d^2\varphi}{dt^2}\biggr]
+ \varpi \frac{d\varphi}{dt}  \biggl[ \frac{d\bold{\hat{e}}_\varphi}{dt} \biggr]
+ \frac{d\varpi}{dt}  \biggl[ \frac{d\bold{\hat{e}}_\varpi}{dt} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bold{\hat{e}}_\varpi \frac{d^2\varpi}{dt^2} + \bold{\hat{k}} \frac{d^2z}{dt^2} 
+ \bold{\hat{e}}_\varphi \biggl[\frac{d\varpi}{dt} \cdot  \frac{d\varphi}{dt} + \varpi \frac{d^2\varphi}{dt^2}\biggr]
+ \varpi \frac{d\varphi}{dt}  \biggl[- \bold{\hat{e}}_\varpi \frac{d\varphi}{dt}  \biggr]
+ \frac{d\varpi}{dt}  \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right" colspan="3">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bold{\hat{e}}_\varpi \biggl[\frac{d^2\varpi}{dt^2} - \varpi \biggl(\frac{d\varphi}{dt}\biggr)^2 \biggr]  
+ \bold{\hat{e}}_\varphi \biggl[ 2 \biggl( \frac{d\varpi}{dt} \cdot  \frac{d\varphi}{dt} \biggr) + \varpi \frac{d^2\varphi}{dt^2}\biggr]
+ \bold{\hat{k}} \frac{d^2z}{dt^2} \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="5">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 647, Appendix &sect;1.B.2, Eq. (1B-24)
</td></tr>
</table>

Let's rewrite the velocity vector as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="1">
<math>~\bold{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\bold{\hat{e}}_\varpi \dot\varpi + \bold{\hat{e}}_\varphi \varpi \dot\varphi  + \bold{\hat{k}} \dot{z} \, ,</math>
  </td>
</tr>
</table>
and (the second line of) this acceleration expression as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="left">
<math>~~\bold{a} \equiv \frac{d\bold{v}}{dt} =
\bold{\hat{e}}_\varpi \frac{d \dot\varpi}{dt} + \bold{\hat{e}}_\varphi \frac{d}{dt}\biggl[\varpi \dot\varphi \biggr] + \bold{\hat{k}} \frac{d \dot{z}}{dt} 
+ \underbrace{
\dot\varpi \biggl[ \bold{\hat{e}}_\varphi \frac{d\varphi}{dt} \biggr] 
- \varpi \dot\varphi  \biggl[\bold{\hat{e}}_\varpi \frac{d\varphi}{dt}  \biggr]
}_\text{curvature terms}\, .
</math>
  </td>
</tr>
</table>

Now, if <math>~\bold{B}</math> is a vector quantity that characterizes some property of a fluid element &#8212; such as momentum density, velocity, or vorticity &#8212; the difference between the Lagrangian and Eulerian time-derivatives of that vector quantity is given by the expression,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d\bold{B}}{dt} - \frac{\partial \bold{B}}{\partial t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(\bold{v} \cdot \bold\nabla)\bold{B} \, ,</math>
  </td>
</tr>
</table>
where the various elements of this right-hand-side mathematical operator can be obtained by replacing <math>~\bold{A}</math> with <math>~\bold{v}</math> in the so-called ''convective operator.''

<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
<div align="center">'''Convective Operator in Cylindrical Coordinates'''</div>

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

<tr>
  <td align="right">
<math>~(\bold{A} \cdot \bold\nabla) \bold{B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bold{\hat{e}}_\varpi \biggl[
A_\varpi \frac{\partial B_\varpi}{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial B_\varpi}{\partial \varphi} + A_z \frac{\partial B_\varpi}{\partial z} - \frac{A_\varphi B_\varphi}{\varpi}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \bold{\hat{e}}_\varphi \biggl[
A_\varpi \frac{\partial B_\varphi}{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial B_\varphi}{\partial \varphi} + A_z \frac{\partial B_\varphi}{\partial z} + \frac{A_\varphi B_\varpi}{\varpi}
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \bold{\hat{e}}_z \biggl[
A_\varpi \frac{\partial B_z}{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial B_z}{\partial \varphi} + A_z \frac{\partial B_z}{\partial z} 
\biggr] \, .
</math>
  </td>
</tr>
<tr><td align="center" colspan="3">
[<b>[[Appendix/References#BT87|<font color="red">BT87</font>]]</b>], p. 651, Appendix &sect;1.B.3, Eq. (1B-54)
</td></tr>
</table>

We will adopt the following, more compact notation:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\bold{A} \cdot \bold\nabla) \bold{B}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_A B_\varpi- \frac{A_\varphi B_\varphi}{\varpi} \biggr]
+ \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_A B_\varphi + \frac{A_\varphi B_\varpi}{\varpi} \biggr]
+ \bold{\hat{e}}_z \biggl[ \mathcal{L}_A B_z \biggr] 
\, ,
</math>
  </td>
</tr>
</table>
where the operator,

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

<tr>
  <td align="right">
<math>~\mathcal{L}_A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[
A_\varpi \frac{\partial }{\partial \varpi} + \frac{A_\varphi }{\varpi}\frac{\partial }{\partial \varphi} + A_z \frac{\partial }{\partial z} 
\biggr] \, .
</math>
  </td>
</tr>
</table>

</td>
</table>

In particular, if we are examining the behavior of the fluid velocity <math>~(\bold{B} \rightarrow \bold{v} )</math>, we find that, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right" colspan="2">
<math>~\frac{d\bold{v}}{dt} - \frac{\partial \bold{v}}{\partial t}</math>
  </td>
  <td align="left">
<math>~=~(\bold{v} \cdot \bold\nabla)\bold{v} </math>
  </td>
</tr>

<tr>
  <td align="right" colspan="2">
&nbsp;
  </td>
  <td align="left">
<math>~=~
\bold{\hat{e}}_\varpi \biggl[ \mathcal{L}_v \dot\varpi \biggr]
+ \bold{\hat{e}}_\varphi \biggl[ \mathcal{L}_v (\varpi \dot\varphi) \biggr]
+ \bold{\hat{e}}_z \biggl[ \mathcal{L}_v \dot{z} \biggr] 
~+ \underbrace{\bold{\hat{e}}_\varphi (\dot\varphi \dot\varpi )
-~
\bold{\hat{e}}_\varpi ( \varpi {\dot\varphi}^2 ) }_\text{curvature terms} \, ,
</math>
  </td>
</tr>
</table>
where the operator,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{L}_v</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\biggl[
v_\varpi \frac{\partial }{\partial \varpi} + \frac{v_\varphi }{\varpi}\frac{\partial }{\partial \varphi} + v_z \frac{\partial }{\partial z} 
\biggr] \, .
</math>
  </td>
</tr>
</table>

Notice that the pair of "curvature terms" that appear in this expression are identical to the pair of curvature terms that appear in the acceleration expression, above.  We conclude, therefore, that for each of the three separate (cylindrical-coordinate-based) components of the vector acceleration, the relationship between the Lagrangian (total) and Eulerian (partial) time derivative is, respectively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right"><math>~\bold{\hat{e}}_\varpi</math>:&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{d\dot\varpi}{dt}  - \varpi {\dot\varphi}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \dot\varpi}{\partial t} + \biggl[\mathcal{L}_v \dot\varpi \biggr] 
- \varpi {\dot\varphi}^2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\bold{\hat{e}}_\varphi</math>:&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{d (\varpi \dot\varphi ) }{dt} + \dot\varpi \dot\varphi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial (\varpi \dot\varphi ) }{\partial t} + \biggl[
\mathcal{L}_v (\varpi \dot\varphi )
\biggr] 
+ \dot\varpi \dot\varphi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\bold{\hat{k}}</math>:&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~\frac{d \dot{z} }{dt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\partial \dot{z} }{\partial t} + \biggl[
\mathcal{L}_v \dot{z}
\biggr] \, .
</math>
  </td>
</tr>
</table>