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% Chapter 1  The Governing Equations
%            1-10 SOME ENERGY RELATIONS
%
% 2002/06/17 ���� ����   ��������
% 2002/06/21 ���� ����   ����
% 2002/08/02 ���� ����   2-10-C �ر�ư���ͥ륮���γȻ����ɲ�
%                        2-10-D �إե�å����ե�������ɲ�
%                        2-10-E �ذ��֥��ͥ륮���Υե�å����ե�������ɲ�
%                        2-10-F �ر�ư���ͥ륮���Υե�å����ե�������ɲ�
% 2002/08/03 ���� ����   2-10-C �ر�ư���ͥ륮���γȻ��ٽ���
%                        2-10-D �إե�å����ե�����ٽ���
%                        2-10-E �ذ��֥��ͥ륮���Υե�å����ե�����٤�
%                        2-10-F �ر�ư���ͥ륮���Υե�å����ե������
%                        ��2�Ĥ� 2-10-E��P,I���줾���...�٤�
%                        �ޤȤ�, ����. (App 6 ���˴�)
% 2002/08/09 ���� ����   2-10-F �ر�ư, ����, �������ͥ륮�����Ѵ����ɲ�
% 2002/08/22 ���� ����   ���ߤǻ�Ŧ���줿�Ȥ�������
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%%%% title and date setting %%%%
\markboth{2-10 ���ͥ륮���δط���}{2-10 ���ͥ륮���δط���}
%\markright{1-10 ���ͥ륮���δط���}
%\Ddate{2002/08/22}
%\Dfile{hw-01-10.tex}

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\section{���ͥ륮���δط���}

�絤���ݤ��ϳ�Ū���Ϥˤ�����,
�͡��ʾ���򥨥ͥ륮�������鸦�椹�뤳�Ȥ�
���˽��פǤ���.
���ͥ륮���δط����Ͽ��͵���ͽ��ˤȤäƤ���פʤ�ΤǤ���.
���ͥ륮�����Ѵ�����¸��ʬ�Ϥ���褦����ħ����
���ͥ�ǥ��, ʪ��������ɽ��������,
�����θ������ɵᤷ���ꤹ�뺹ʬ����������߷פ���Τ�
������ƻ��Ȥʤ�.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   2-10-1 Kinetic Energy   %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{��ư���ͥ륮��}

ñ�̼��̤�����α�ư���ͥ륮���˴ؤ��뼰�ϱ�ư��������
$\Dvect{V}$��ݤ��뤳�ȤǴ�ñ��Ƴ�ФǤ���.
�㤨��, ���Ϻ�ɸ�ϤǤα�ư������(2-58)�����Ѥ�������
�ʲ��μ���������
\footnote{
�ܤ����� {\bf 2-10-A}����.
}.
%
\begin{eqnarray}
 dK/dt = \partial K / \partial t
         + \Dvect{V} \cdot \nabla K
	 + \omega \partial K / \partial p
       = - \Dvect{V} \cdot \nabla_{p} \phi
         + \Dvect{V} \cdot \Dvect{F}.
\end{eqnarray}
%
������$K$��$K=V^2/2$�Ǥ���,
$\Dvect{V} \cdot \Dvect{F}$�ϱ�ư���ͥ륮���γȻ��Ⱦû�
\footnote{
��ư���ͥ륮���γȻ��Ⱦû��ˤĤ��Ƥ� {\bfseries 2-10-C}�򻲾�.
}��ɽ��.
($\Dvect{V}$��$\nabla$�Ͽ�ʿ�̤�®�٤���ʬ�黻�ҤǤ���).
Ϣ³�μ�(2-60)
\footnote{
\begin{eqnarray*}
 \nabla_p \cdot \Dvect{V} + \partial \omega / \partial p =0 .
\end{eqnarray*}
}
��$K$��ݤ�, (2-80)�˲ä��뤳�Ȥ�
��(2-80)��{\bfseries �ե�å����ե�����}({\itshape flux form})
\footnote{
{\bfseries �ե�å����ե�����}({\itshape flux form})�˴ؤ��Ƥ�
{\bfseries 2-10-D}�򻲾�.
}
�ˤʤ�.
�ºݤ˷׻���������줿�ե�å����ե������
%
\begin{eqnarray}
 \partial K /\partial t
  + \nabla \cdot K \Dvect{V}
  + \partial (K \omega) / \partial p
 =
  - \nabla \cdot (\phi \Dvect{V})
  + \phi \nabla \cdot \Dvect{V}
  + \Dvect{V} \cdot \Dvect{F}
\end{eqnarray}
%
�Ǥ���.
Ʊ�ͤ�, Ϣ³�μ���$\phi \nabla \cdot \Dvect{V}$�ι��
ɽ���Ǥ���.
%
\begin{eqnarray*}
 \phi \nabla \cdot \Dvect{V}
  =
 - \phi \partial \omega / \partial p
  =
 - \partial (\phi \omega)/\partial p
 + \omega \partial \phi / \partial p.
\end{eqnarray*}
%
�Ǹ��, �ſ尵�μ�(2-59)�Ⱦ������������
$\partial \phi / \partial p = - \alpha = -RT/p$
�ʤΤ�(2-81)�ϰʲ��Τ褦�ˤʤ�.
%
\begin{eqnarray}
 \DP{K}{t}
 + \nabla \cdot [(K + \phi) \Dvect{V}]
 + \DP{[(K + \phi) \omega]}{p}
  =
 - \frac{RT \omega}{p}
 + \Dvect{V} \cdot \Dvect{F}.
\end{eqnarray}
%

��2�����3���ή���ȯ����Ǥ���,
������4��ϱ�ư���ͥ륮����ͯ���Ф���,
��5��ϵۤ����ߤ�ɽ��.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   2-10-2 Potential Energy
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{���֥��ͥ륮��}

ñ�����Ѥ���ɽ�����絤��ü�ޤǤι⤵����ı����
���Ϥˤ����֥��ͥ륮����ͤ���.
���ΰ��֥��ͥ륮��$P$ [J]�ϰʲ�����ʬ��������.
%
\begin{eqnarray*}
 P = \int^{\infty}_{0} gz \rho \ dz
   = \int^{p_s}_{0} \phi \frac{dp}{g}
   = \frac{\phi p}{g} \bigg|^{p_s}_{0}
    - \frac{1}{g} \int p \ d\phi.
\end{eqnarray*}
%
�����ǤϽ��ϲ�®��$g$�α�ľ�������Ѳ���̵�뤷,
��ʬ��ʬ���ſ尵�μ����Ѥ��뤳�Ȥˤ�ä�
�Ǹ�η���Ƴ�ФǤ���
\footnote{
�ѷ��˴ؤ��Ƥ�{\bf 2-10-B}����.
}.
�ſ尵�μ���Ƥ��Ѥ����,
%
\begin{eqnarray}
 P = \phi _{s} p_{s} / g + \frac{1}{g}
     \int^{p_s}_{0} R T \ dp
\end{eqnarray}
%
�Ȥ������Ȥʤ�
\footnote{
{\bf 2-10-B}����.
}.

Ʊ������Ǥ�{\bf �������ͥ륮��}({\it internal energy\/})
$I$ [J]��
%
\begin{eqnarray}
 I = \int^{\infty}_{0} c_v T \rho \ dz
   = \int^{p_s}_{0} \frac{c_v T \ dp}{g}
\end{eqnarray}
%
�Ȥʤ�
\footnote{
�������ͥ륮��$I$��{\bf 1-5�� Ǯ�ϳ����ˡ§}��
�о줷, �����Ǥ�ñ�����Ѥ�������������ͥ륮�����Ѳ�Ψ
$dI/dt = c_v dT/dt$�Ȥ����ط������Ȥ�줿.
}.
�ط���$c_v + R = c_p$��Ȥ���,
�������ͥ륮���Ƚ��Ϥΰ��֥��ͥ륮�����Ȥ߹�蘆�ä��Ǹ��
����������.
�����������ͥ륮���Ƚ��Ϥΰ��֥��ͥ륮�����Ȥ߹�蘆�ä���Τ�
{\bf �����֥��ͥ륮��}({\it total potential energy\/})
�ȸƤ�.
����ϰʲ��μ���ɽ�����.
%
\begin{eqnarray}
 P + I =
 g^{-1} \int^{p_s}_{0} E \ dp
 + g^{-1} \phi_{s} p_{s}.
\end{eqnarray}
%
������$E= c_p T$ [J/kg]��{\bf ���󥿥�ԡ�}({\it enthalpy\/})�Ǥ���.

���η�̤�, �絤�α���������֥��ͥ륮����
���󥿥�ԡ�����ʬ�ι��
��ɽ�̤γ��̤���ι⤵�����㤷��������㤷�Ƥ��뤳�Ȥ�
�����Ƥ���.
���Τ���, �����֥��ͥ륮���δط���������ˤ�
Ǯ�ϳؤμ�(2-57b)��Ȥ�.
$E=c_p T$�Ǥ��뤳�Ȥ���,
%
\begin{eqnarray}
 \DP{E}{t} + \Dvect{V} \cdot \nabla E
  + \omega \DP{E}{p}
 = \frac{RT}{p} \omega + Q - \Dvect{V} \cdot \Dvect{F}
\end{eqnarray}
%
�Ȥʤ�.
�����Ǥ�$\Dvect{V}$�����$\nabla$�Ͽ�ʿ�̤Ǥ�®�٤ȱ黻�ҤǤ���,
$\omega = dp/dt$��, $dE/dt$������ʬ��Ÿ������Ƥ���.

����Ϣ³�μ�(2-60)��$E$��ݤ�, (2-86)��­����碌���
�ʲ��Υե�å����ե���������뤳�Ȥ��Ǥ���
\footnote{
 ���μ��ϰ��֥��ͥ륮�����������ͥ륮�������­����碌��
 �����֥��ͥ륮���Υե�å����ե��������,
 ���֥��ͥ륮�����������ͥ륮���Ȥ��줾��Υե�å����ե�����˴ؤ��Ƥ�
 �ͤ��Ƥߤ�.
 �ܺ٤ˤĤ��Ƥ� {\bfseries 2-10-E}��
 ���ȤΤ���.
}.
%
\begin{eqnarray}
 \DP{E}{t} + \nabla \cdot (E \Dvect{V})
  + \DP{(E \omega)}{p}
 = \frac{RT \omega}{p} + Q  - \Dvect{V} \cdot \Dvect{F}.
\end{eqnarray}
%

(2-82)��(2-87)����Ӥ����,
ξ���μ���$RT \omega /p$�Ȥ����ब
����, ȿ�Ф���椬�Ĥ��Ƥ��뤳�Ȥ��狼��.
���֥��ͥ륮�������ι�αƶ��Ƕɽ�Ū�����ä������
��ư���ͥ륮���ϵդ˸�������.
�Ĥ����ϤǤ��ι����ʬ������Τ�
��ư���ͥ륮���Ȱ��֥��ͥ륮�����Ѵ���ɽ��
\footnote{
��ư���ͥ륮��, ���֥��ͥ륮��, �������ͥ륮���֤Υ��ͥ륮����
�Ѵ��˴ؤ��Ƥ� {\bfseries 2-10-F}�򻲾�.
}.
(2-82)��(2-87)��­����碌��Ȥ���
{\bf �Ѵ��ؿ�}({\it transformation function\/})
�Ͼä�,
�ʲ��Τ褦��$K+E$�ζɽ��Ѳ�����������������.
%
\begin{eqnarray}
 \DP{(K+E)}{t}
 + \nabla \cdot [(K+E+\phi) \Dvect{V}]
 + \DP{}{p}[(K+E+\phi) \omega]
 = Q.
\end{eqnarray}
%

����, �������������絤�������̤���ʬ����.
̵�¾����Ǥ�$\rho \ dx \ dy \ dz = - dx \ dy \ dp /g$
�Ȥʤ�
\footnote{
�ſ尵�μ������.
}.
%
\begin{eqnarray}
 \frac{1}{g} \Dtint^{p_s}_{0} \DP{}{t} (K+E) dpdxdy
 + \frac{1}{g} \Dtint^{p_s}_{0} \nabla \cdot [(K+E+\phi) \Dvect{V}] dpdxdy
 \nonumber \\
 + \frac{1}{g} \Ddint[(K+E+\phi) \omega ]_{s} dxdy
 =
 \overline{Q}.
\end{eqnarray}
%
$\overline{Q}$�ι��, �絤������䲼����ɽ�̤����Ф���,
�⤷���Ϥ���餫��ۼ�������Ǯ�̤�ɽ��
\footnote{
 �������ɽ�̤Ȥ��໤�ˤ�ä�ȯ������Ǯ�ʤɤ�ޤ᤿Ǯ�̤Ǥ���.
 �ʤ�$[(K+E+\phi) \omega]_{s}$��$p_s$�ˤ�����
 $[(K+E+\phi) \omega]$��ɽ��.
}.
��ľ��������ʬ��3���ܤ���ʬ�����������.
������ʬ�黻��$\partial / \partial t$��$\nabla \cdot$
��$p$����ʬ�γ��˽Ф�.
���κݤˤϥ饤�ץ˥åĤθ���
\footnote{
 �ʲ����饤�ץ˥åĤθ����Ǥ���.
 %
 \begin{eqnarray*}
  (fg)^{(n)} =
   \sum_{k=0}^{n}
   \left(
    \begin{array}{c}
     n \\
     k
    \end{array}
  \right)
   f^{(n-k)} g^{(k)}.
 \end{eqnarray*}
 %
 ��ä�, �ʲ��μ�������Ω��.
 %
 \begin{eqnarray*}
  \int^{p_s}_{0} \DP{}{t} A(p)\ dp
   =
   \DP{}{t} \int^{p_s}_{0} A(p) \ dp
   - \DP{p_s}{t} A(p_s)
 \end{eqnarray*}
 %
}
�����Ѥ���.
�ޤ�, �ѿ��ξ��$p_s$���θ�ˤ����.
��̤�,
%
\begin{eqnarray}
 \Ddint \frac{1}{g}
  \Bigg\{
   \DP{}{t} \int^{p_s}_{0} (K+E)dp
   - \DP{p_s}{t} (K+E)_{s}
   + \nabla \cdot \int^{p_s}_{0} [(K+E+\phi) \Dvect{V}] dp
   \nonumber \\
   - \nabla p_s \cdot [(K+E+\phi) \Dvect{V}]_{s}
   + [(K+E+\phi) \omega]_{s}
  \Bigg\}
  dxdy
  =
  \overline{Q}
\end{eqnarray}
%
�Ȥʤ�.
������(��)�ϻ�����ʬ��ɽ��.

�����$\partial /\partial t$����ʬ����γ��˽Ф����Ȥ��Ǥ���.
������, ����Ǥ����ϵ�ǤϿ�ʿ�������˶�����̵��
($p$-�̤��ڤ뻳̮�ʤɤ�̵�뤹��)����,
��ʿ�̤�ή���ȯ����ޤ���3��Ͼä���.
���η�̤��ʲ��μ��Ǥ���
\footnote{
 $K$�λ�����ʬ$\overline{K}$���������ǫ�˽񤯤�,
 %
 \begin{eqnarray}
  \overline{K} = \Dtint K \rho \ dx \ dy \ dz 
   = \frac{1}{g} \Dtint K dx \ dy \ dp
 \end{eqnarray}
 %
 �Ǥ���. $E$��Ʊ�ͤǤ���.
}.
%
\begin{eqnarray}
 \DP{}{t}(\overline{K} + \overline{E})
 - \Ddint \Bigg\{ \frac{1}{g} (K+E)_{s} \DP{p_s}{t}
 + \frac{1}{g} \nabla p_s \cdot [(K+E+\phi) \Dvect{V}]_{s}
 \nonumber \\
 - \frac{1}{g} [(K+E+\phi)_{s} \omega_{s}]
 \Bigg\} dx dy
 =
 \overline{Q}.
\end{eqnarray}
%
��ɽ�̶������(2-62),
$\partial p_s / \partial t + \Dvect{V} \cdot \nabla p_s = \omega_s$
���$(K+E)_s$��ޤ����Ǥ��ä����.
�ޤ�������$\phi _s (\omega_s - \Dvect{V} \cdot \nabla p_s)$��
$\phi _s \partial p_s / \partial t$
���֤������뤳�Ȥ��Ǥ���.
���Τ褦�ˤ����, �����ͥ륮���μ�(2-91)��
�ʲ��Τ褦�ʴ�ñ�ʼ��ˤʤ�.
%
\begin{eqnarray}
 \DP{}{t} [\overline{K} + \overline{E}
  + \frac{1}{g} \overline{\phi_s p_s}]
  = \overline{Q}.
\end{eqnarray}
%
������$\overline{\phi_s p_s}$��2��������ʬ�Ǥ��뤳�Ȥ�
���դ����ߤ���.

(2-93)�κ�¦������ư���ͥ륮����
(2-85)������������褦�������֥��ͥ륮����
�����Ѳ�(�ºݤ�����ʬ$d/dt$)�Ǥ���
\footnote{
���դ���3����Ϸ����ˤ�ä�ͭ���ˤʤ��Ǥ���.
�Ϸ���������̵���Ȥ����$\phi_s$�Ͼ��$0$�Ȥʤ뤿��
���ι��̵�뤵���.
}.
����, ��¦�ϷϤ˲ä���줿�����ͥ륮��
(�໤�ʤɤˤ�������륨�ͥ륮����ޤ�)��ɽ��.
��Ǯ����, �໤��̵����ﲼ�Ǥ�(2-92)�α�¦��
�ä�, �����ͥ륮������¸�����.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%             �ʲ���Appendix             %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%           ���ֹ�Υ������饤��         %%%%%%%% 
\setcounter{equation}{0}
\def\theequation{A-\arabic{equation}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\begin{center}
% {\bf 
% --------------------------------------
% ��ʸ�����ޤ�
% --------------------------------------
% } 
%\end{center}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-10-A      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-10-A \qquad ��ư���������饨�ͥ륮���μ��ؤ��ѷ�}
\setcounter{equation}{0}
\def\theequation{2-10-A-\arabic{equation}}


$p$-��ɸ�ϤǤα�ư������(2-58)
\footnote{
\begin{eqnarray*}
 \DD{\Dvect{V}}{t}
  = - \nabla_{p} \phi
      - f \Dvect{k} \times \Dvect{V} + \Dvect{F}.
\end{eqnarray*}
}
��$\Dvect{V}$��ݤ����,
%
\begin{eqnarray}
 \Deqlab{���ͥ륮���ط���Ƴ��1}
 \Dvect{V} \cdot \DD{\Dvect{V}}{t}
  &=& - \Dvect{V} \cdot \nabla_{p} \phi
      - f \Dvect{V} \cdot (\Dvect{k} \times \Dvect{V})
      + \Dvect{V} \cdot \Dvect{F} 
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{���ͥ륮���ط���Ƴ��1}����1���
%
\begin{eqnarray}
 \Deqlab{���ͥ륮���ط���Ƴ��2}
 \Dvect{V} \cdot \DD{\Dvect{V}}{t}
  =
 \DD{(\Dvect{V}^2 / 2)}{t}
  =
 \DD{K}{t}
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{���ͥ륮���ط���Ƴ��1}����3���
%
\begin{eqnarray}
 \Dvect{V} \cdot (\Dvect{k} \times \Dvect{V})
  &=&
 \Dvect{V} \cdot (- f v \Dvect{i} + f u \Dvect{j})
 \nonumber \\
  &=&
 - f v u + f u v
 \nonumber \\
 \Deqlab{���ͥ륮���ط���Ƴ��3}
  &=&
  0
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{���ͥ륮���ط���Ƴ��2}��\Deqref{���ͥ륮���ط���Ƴ��3}
���\Deqref{���ͥ륮���ط���Ƴ��1}��(2-80)�����ѷ��Ǥ���.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-10-B      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-10-B \qquad ���֥��ͥ륮���μ����ѷ�}
\setcounter{equation}{0}
\def\theequation{2-10-B-\arabic{equation}}


���֥��ͥ륮���μ�
%
\begin{eqnarray}
 \Deqlab{hw-02-10-���֥��ͥ륮���μ�1}
 P = \int^{\infty}_{0} gz \rho \ dz
\end{eqnarray}
%
����ʸ�Ǥ��ѷ��˴ؤ��Ƥ⤦�����ܺ٤���������.

�ſ尵�μ����, �ʲ��δط���������Ω��.
%
\begin{eqnarray}
 \Deqlab{hw-02-10-�ſ尵�δط���}
 \rho \ dz = - dp/g .
\end{eqnarray}
%
�ޤ�, $p$��$z$�ϰʲ��Τ褦�ʴط��ˤ���.
%
\begin{eqnarray}
 p=p_s \hspace{10mm} at \hspace{10mm} z=0 \hspace{2mm},
 \nonumber \\
 \Deqlab{hw-02-10-p��z�δط���}
 p=0   \hspace{12mm} at \hspace{10mm} z= \infty .
\end{eqnarray}
%
\Deqref{hw-02-10-�ſ尵�δط���}��
\Deqref{hw-02-10-p��z�δط���}��
$\phi = gz$���,
\Deqref{hw-02-10-�ſ尵�δط���}
�ϰʲ��Τ褦���ѷ��Ǥ���.
%
\begin{eqnarray}
 \Deqlab{hw-02-10-���֥��ͥ륮���μ�2}
 P = - \int^{0}_{p_s} \phi \frac{dp}{g}
   = \int^{p_s}_{0} \phi \frac{dp}{g}.
\end{eqnarray}
%
���μ�����ʬ��ʬ�����,
%
\begin{eqnarray}
 \Deqlab{hw-02-10-���֥��ͥ륮���μ�3}
 P = \int^{p_s}_{0} \phi \frac{dp}{g}
   = \frac{\phi p}{g} \bigg|^{p_s}_{0}
    - \frac{1}{g} \int^{p_s}_{0} p \DP{\phi}{p} \ dp
\end{eqnarray}
%
�Ȥʤ�
(��ľ����$p$�ˤ�$g$���Ѳ����ʤ��Ȳ��ꤷ�Ƥ���).
���κǸ�ι��$(\partial \phi / \partial p) \ dp = d \phi$
�Ȥ��뤳�Ȥ���ʸ�μ���������.
%

{\bf (2-84)����Ƴ��}

³����(2-84)����Ƴ�Ф�Ԥ�.
$p$-��ɸ�ϤǤ��ſ尵�μ�(2-59)��
�������������,
%
\begin{eqnarray}
 \Deqlab{hw-02-10-�ſ尵�μ��Ⱦ���������}
 \DP{\phi}{p} = - \alpha = - \frac{RT}{p}
\end{eqnarray}
%
�Ȥʤ�.
\Deqref{hw-02-10-�ſ尵�μ��Ⱦ���������}��
\Deqref{hw-02-10-���֥��ͥ륮���μ�3}
�����������,
%
\begin{eqnarray}
 P = \phi _{s} p_{s} / g + \frac{1}{g}
     \int^{p_s}_{0} R T \ dp
\end{eqnarray}
%
�Ȥʤ�.
�ʤ�,
$\phi_{s} = \phi (x,y,p_s,t) = gz(x,y,p_s,t)$
�Ǥ���.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-10-C      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-10-C\qquad ��ư���ͥ륮���γȻ�}
\setcounter{equation}{0}
\def\theequation{2-10-C-\arabic{equation}}


$\Dvect{F} = \nabla^2 \Dvect{V}$�Ȥ����Ϥ�Ư���Ƥ���Ȥ���.
���ΤȤ�$\Dvect{V} \cdot \Dvect{F}$��
%
\begin{eqnarray}
 \Dvect{V} \cdot \Dvect{F}
  &=& \Dvect{V} \cdot \left( \nabla^2 \Dvect{V} \right)
  \nonumber \\
 &=&
  v_i \DP[2]{v_i}{x_j}
  \nonumber \\
 &=&
  \DP[]{}{x_j} \left\{ v_i \DP{v_i}{x_j} \right\}
  - \left( \DP{v_i}{x_j} \right)^2
  \nonumber \\
 &=&
  \DP[]{}{x_j} \left\{\frac{1}{2} \DP{v_i^2}{x_j} \right\}
  - \left( \DP{v_i}{x_j} \right)^2
  \nonumber \\
 &=&
  \DP[2]{}{x_j} \left\{\frac{1}{2} v_i^2 \right\}
  - \left( \DP{v_i}{x_j} \right)^2
  \nonumber \\
 &=&
  \nabla^2 K
  - \left( \DP{v_i}{x_j} \right)^2
  \nonumber
\end{eqnarray}
%
�Ȥʤ�.
���λ�, ���դ���1���ư���ͥ륮���γȻ�,
��2���ư���ͥ륮���ξû��ȸƤ�.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-10-D      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-10-D\qquad �ե�å����ե�����}
\setcounter{equation}{0}
\def\theequation{2-10-D-\arabic{equation}}


�ʲ��Τ褦�ʼ��η���{\bfseries �ե�å����ե�����}
({\itshape flux form})�ȸƤ�.
%
\begin{eqnarray}
 \Deqlab{hw-02-10-�ե�å����ե�����}
 \DP{A}{t} + \Ddiv (B \Dvect{V}) = C.
\end{eqnarray}
%
�ե�å����ե�����������Ϥ����̤���¸�ˤĤ��Ƥ��줤��
ɽ�����Ȥ��Ǥ������Ǥ���.
$C = 0$�Ȥ���, \Deqref{hw-02-10-�ե�å����ե�����}��
�������$V$����ʬ�����,
%
\begin{eqnarray}
 \int_{V} \DP{A}{t} dV + \int_{V} \Ddiv (B \Dvect{V}) dV
  &=& 0
  \nonumber
\end{eqnarray}
%
�Ȥʤ�, ��������ˡ§�ˤ�ä�,
%
\begin{eqnarray}
 \DP{}{t} \int_{V} A dV + \int_{S} B \Dvect{V} \cdot d\Dvect{S}
  &=& 0
  \nonumber, \\
 \DP{}{t} \int_{V} A dV + \int_{S} B \Dvect{V} \cdot \Dvect{n} dS
  &=& 0
  \nonumber 
\end{eqnarray}
%
��ɽ�����Ȥ��Ǥ���.
($\Dvect{n}$�϶���$V$��ɽ��$S$�˿�ľ��ñ�̥٥��ȥ�Ǥ���).
���μ���, �������$V$�������Τ�����$A$��$V$��ɽ��$S$��������ꤹ��
$BV_n$($V_n = \Dvect{V} \cdot \Dvect{n}$)���̤ˤ�ä�
��ޤ뤳�Ȥ��̣���Ƥ���.

�⤦����
\Deqref{hw-02-10-�ե�å����ե�����}�򸫤Ƥߤ�.
$B\Dvect{V}$��{\bfseries �ե�å���}({\itshape flux})�ȸƤФ���̤�,
$\Ddiv (B \Dvect{V})$��ή��ή�Фκ����ˤ��$A$�μ�«ȯ���̤Ǥ��뤳�Ȥ�
�����ۤɤ��㤫���狼�������.
���ʤߤ�$C$���������ǹ�Ȥ���, �ե�å����ǤϽ񤱤ʤ���ν��ޤ�Ǥ���.

�����ǰʲ���Ϣ³�μ�(2-6)��פ��Ф��Ƥߤ褦.
%
\begin{eqnarray}
 \DP{\rho}{t} + \Ddiv (\rho \Dvect{V}) = 0.
  \nonumber
\end{eqnarray}
%
�����ե�å����ե�����ˤʤäƤ���Τ��狼�������.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-10-E      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-10-E\qquad $P$, $I$���줾��Υե�å����ե�����}
\setcounter{equation}{0}
\def\theequation{2-10-E-\arabic{equation}}


��ʸ�Ǥϰ��֥��ͥ륮�����������ͥ륮����­����蘆�줿
���󥿥�ԡ��Ȥ�����
�ǥե�å����ե�����μ����ä�.
�Ǥ�, ���֥��ͥ륮�����������ͥ륮���Ǥ��줾��ե�å����ե������
��ä���ɤ��ʤ�Τ�������?
�����Ǥϰ��֥��ͥ륮��$P$���������ͥ륮��$I$�Ǥ��줾��
�ե�å����ե��������, �ͻ����Ƥߤ�.





\subsubsection*{2-10-E-a \quad ���֥��ͥ륮��$P$�Υե�å����ե�����}

Ϣ³�μ�(2-6)
\footnote{
\begin{eqnarray*}
 \DP{\rho}{t} + \Ddiv \left( \rho \Dvect{V} \right) = 0.
\end{eqnarray*}
}
���Ѥ��ư��֥��ͥ륮���Υե�å����ե������Ƴ��.
�����ǰ��֥��ͥ륮����$P = g z \rho$ [J m$^{-3}$]
�Ȥ��ƹͤ���.
%
\begin{eqnarray}
 \DP{\rho}{t} + \Ddiv \left( \rho \Dvect{V} \right)
  &=& 0
  \nonumber , \\
 z \DP{\rho}{t} + z \Ddiv \left( \rho \Dvect{V} \right)
  &=& 0
  \nonumber , \\
 \DP{}{t} \left( z \rho \right)
  + \Ddiv \left( z \rho \Dvect{V} \right)
  - \rho
  \left\{
   \DP{z}{t} + \Dvect{V} \cdot \nabla z
  \right\}
  &=& 0
  \nonumber , \\
 \DP{}{t} \left( g z \rho \right)
  + \Ddiv \left( g z \rho \Dvect{V} \right)
  &=& 
  g \rho \DD{z}{t} .
  \nonumber
\end{eqnarray}
%
�ſ尵ʿ�դμ������Ѥ����
\footnote{
�����Ǥ�
%
\begin{eqnarray*}
 \DD{z}{t} = \DP{z}{p} \DD{p}{t}
\end{eqnarray*}
%
�Ȥ��Ƥ���.
},
%
\begin{eqnarray}
 \DP{P}{t} 
  + \Ddiv \left( P \Dvect{V} \right)
  &=& 
  g \rho \DD{z}{t} 
  \nonumber \\
 &=& 
  g \rho \DP{z}{p} \DD{p}{t} 
  \nonumber \\ 
 &=& 
  g \rho \left( - \frac{\alpha}{g} \right) \DD{p}{t} 
  \nonumber \\ 
 &=& 
  - \DD{p}{t} 
  \nonumber \\ 
 &=& 
  - \DD{}{t} \left( \rho R T \right) 
  \nonumber \\ 
 &=& 
  - R T \DD{\rho}{t}
  - R \rho \DD{T}{t}
  \nonumber \\ 
 &=& 
  - R T \DD{\rho}{t}
  - \rho \left( c_p - c_v \right)
    \frac{1}{c_p} \left( Q + \frac{1}{\rho} \DD{p}{t} \right)
  \nonumber \\
 \Deqlab{hw-02-10-���֥��ͥ륮���Υե�å����ե�����}
 &=& 
  - R T \DD{\rho}{t}
  - \rho Q
  + \rho \frac{c_v}{c_p} Q
  - \frac{R}{c_p} \DD{p}{t}
\end{eqnarray}
%
�Ȥʤ�.






\subsubsection*{2-10-E-b \quad �������ͥ륮��$I$�Υե�å����ե�����}

���󥿥�ԡ��Υե�å����ե�����(2-86)
\footnote{
 \begin{eqnarray*}
  \DP{E}{t} + \Dvect{V}_H \cdot \nabla_H E
   + \omega \DP{E}{p}
   &=& \frac{RT}{p} \omega + Q,
   \\
  \DP{E}{t} + \Dvect{V} \cdot \nabla E
   &=& \alpha \DD{p}{t} + Q,
   \\
  \DD{E}{t}
   &=& \alpha \DD{p}{t} + Q.
 \end{eqnarray*}
}
���Ѥ����������ͥ륮��$I$�Υե�å����ե������Ƴ��.
�ʤ�, $z$��ľ��ɸ�ϤǤ��������ͥ륮��$I = c_v T \rho$ [J m$^{-3}$]��ͤ���.
�ޥ��䡼�δط���$c_p = c_v + R$���
$E = c_p T = (c_v + R) T = c_p T + p \alpha$�Ȥʤ�Τ�
(2-86)��
%
\begin{eqnarray}
 \DD{}{t} \left( c_v T \right)
  &+& \DD{}{t} \left( p \alpha \right)
  = \alpha \DD{p}{t} + Q
  \nonumber , \\
 \DD{}{t} \left( c_v T \right)
  &=& - \DD{}{t} \left( \frac{p}{\rho} \right)
  + \frac{1}{\rho} \DD{p}{t} + Q
  \nonumber , \\
 &=&
  - p \DD{}{t} \left( \frac{1}{\rho} \right) + Q
  \nonumber
\end{eqnarray}
%
�Ȥʤ�, ���դ�����ʬ��Ÿ�������,
%
\begin{eqnarray}
 \DP{}{t} \left( c_v T \right)
  + \Dvect{V} \cdot \nabla \left( c_v T \right)
  &=&
  - p \DD{}{t} \left( \frac{1}{\rho} \right) + Q
  \nonumber , \\ 
  &=&
  \frac{p}{\rho^2} \DD{\rho}{t} + Q
  \nonumber , \\ 
 \rho \DP{}{t} \left( c_v T \right)
  + \rho \Dvect{V} \cdot \nabla \left( c_v T \right)
  &=&
  \frac{p}{\rho} \DD{\rho}{t} + \rho Q
  \nonumber , \\ 
 \DP{}{t} \left( c_v T \rho \right)
  + \Ddiv \left( c_v T \rho \Dvect{V} \right)
  - c_v T
    \left\{
     \DP{\rho}{t} + \Ddiv \left( \rho \Dvect{V} \right)
    \right\}
  &=&
  \frac{p}{\rho} \DD{\rho}{t} + \rho Q
  \nonumber
\end{eqnarray}
%
�Ȥʤ�.
Ϣ³�μ�(2-6)
\footnote{
\begin{eqnarray*}
 \DP{\rho}{t} + \Ddiv \left( \rho \Dvect{V} \right) = 0.
\end{eqnarray*}
}
�����3���ܤϾ��Ǥ���,
%
\begin{eqnarray}
 \DP{}{t} \left( c_v T \rho \right)
  + \Ddiv \left( c_v T \rho \Dvect{V} \right)
  &=&
  \frac{p}{\rho} \DD{\rho}{t} + \rho Q
  \nonumber , \\
 \Deqlab{hw-02-10-�������ͥ륮���Υե�å����ե�����}
  \DP{I}{t} 
  + \Ddiv \left( I \Dvect{V} \right)
  &=&
  R T \DD{\rho}{t} + \rho Q
\end{eqnarray}
%
�Ȥʤ�. ���줬�������ͥ륮���Υե�å����ե�����Ǥ���.




\subsubsection*{2-10-E-c \quad $P + I$�Υե�å����ե�����}

\Deqref{hw-02-10-���֥��ͥ륮���Υե�å����ե�����}��
\Deqref{hw-02-10-�������ͥ륮���Υե�å����ե�����}��
­����碌���,
%
\begin{eqnarray}
 \DP{(P + I)}{t} 
  + \Ddiv \left\{ \left( P + I \right)  \Dvect{V} \right\}
  &=& 
  \rho \frac{c_v}{c_p} Q
  - \frac{R}{c_p} \DD{p}{t}
\end{eqnarray}
%
�Ȥʤ�.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-10-F      %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection*{2-10-F\qquad ��ư, ����, �������ͥ륮�����Ѵ�}
\setcounter{equation}{0}
\def\theequation{2-10-F-\arabic{equation}}


��ʸ�Ǥϱ�ư���ͥ륮��$K$�Ȱ��֥��ͥ륮��$\phi$�μ�(2-82)��
���󥿥�ԡ�$E$
\footnote{
���ݶȳ��Ǥ��������ͥ륮��$I$������˥��󥿥�ԡ�$E$���Ѥ���
���Ȥ�¿��.
�ʤ�, �������ͥ륮���ȥ��󥿥�ԡ��Ȥδ֤ˤϰʲ��δط�����
����Ω��.
%
\begin{eqnarray*}
 E &=& I + RT
  \\
 &=& I + p \alpha.
\end{eqnarray*}
%
���δط������Ѥ���ȥ��󥿥�ԡ�$E = c_p T$�Υե�å����ե�����(2-87)����
�������ͥ륮��$I = c_v T$�Υե�å����ե�����μ������뤳�Ȥ�Ǥ���.
�ܤ�����{\bfseries 2-10-F-1}�򻲾ȤΤ���.
}
�μ�(2-87)�Ȥδ֤ǤΥ��ͥ륮�����Ѵ��ˤĤ��Ƥ�
���ڤ��Ƥ��뤬, 3�ĤΥ��ͥ륮���Τ��줾����Ѵ��˴ؤ��ƤϿ���Ƥ��ʤ���
�Ǥ����ǵ������Ƥߤ�.


\subsubsection*{2-10-F-a \quad ��ư���ͥ륮���Υե�å����ե�����}

��ư���ͥ륮���Υե�å����ե��������ʸ��(2-81)�Ǥ���ʲ��μ��Ǥ���.
%
\begin{eqnarray}
 \DP{K}{t} + \Ddiv (K \Dvect{V}) + \DP{(K \omega)}{p}
  &=& - \Ddiv (\phi \Dvect{V}) + \phi \Ddiv \Dvect{V}
  + \Dvect{V} \cdot \Dvect{F}
  \nonumber \\
 \Deqlab{hw-02-10-��ư���ͥ륮���ΤߤΥե�å����ե�����}
  &=& - \Dvect{V} \cdot \nabla \phi + \Dvect{V} \cdot \Dvect{F}.
\end{eqnarray}
%


\subsubsection*{2-10-F-b \quad ���֥��ͥ륮���Υե�å����ե�����}

ñ�̼��̤�����ΰ��֥��ͥ륮���ϥ����ݥƥ󥷥��$\phi$�ǹͤ��뤳�Ȥ�
�Ǥ���.
��ʸ�μ�(2-82)
\footnote{
\begin{eqnarray}
 \DP{K}{t}
 + \nabla \cdot [(K + \phi) \Dvect{V}]
 + \DP{[(K + \phi) \omega]}{p}
  =
 - \frac{RT \omega}{p}
 + \Dvect{V} \cdot \Dvect{F}.
 \nonumber
\end{eqnarray}
}
���鼰(2-81)
\footnote{
\begin{eqnarray}
 \partial K /\partial t
  + \nabla \cdot K \Dvect{V}
  + \partial (K \omega) / \partial p
 =
  - \nabla \cdot (\phi \Dvect{V})
  + \phi \nabla \cdot \Dvect{V}
  + \Dvect{V} \cdot \Dvect{F}.
  \nonumber 
\end{eqnarray}
}
������Ȱʲ��Υ����ݥƥ󥷥��Υե�å���
�ե�����μ���������
\footnote{
$\partial \phi / \partial t$�ϥ����ʤΤ�����ϵ��Ҥ���ʤ�.
}.
%
\begin{eqnarray}
 \Ddiv (\phi \Dvect{V}) + \DP{(\phi \omega)}{p}
  &=& - \frac{R T \omega}{p} + \Ddiv (\phi \Dvect{V}) - \phi \Ddiv \Dvect{V},
  \nonumber \\
 \Deqlab{hw-02-10-�����ݥƥ󥷥��ΤߤΥե�å����ե�����}
  \DP{\phi}{t} + \Ddiv (\phi \Dvect{V}) + \DP{(\phi \omega)}{p}
  &=& - \frac{R T \omega}{p} + \Dvect{V} \cdot \nabla \phi.
\end{eqnarray}
%
���ʤߤˤ��μ���Ϣ³�μ��˥����ݥƥ󥷥��$\phi$��ݤ�����
(��(2-81)�ȼ�(2-82)�δ֤μ�)�������Ǥ���.


\subsubsection*{2-10-F-c \quad ���󥿥�ԡ��Υե�å����ե�����}

���󥿥�ԡ��Υե�å����ե��������ʸ�μ�(2-87)���Τ�ΤǤ���.
%
\begin{eqnarray}
 \Deqlab{hw-02-10-���󥿥�ԡ��ΤߤΥե�å����ե�����}
  \DP{E}{t} + \nabla \cdot (E \Dvect{V})
  + \DP{(E \omega)}{p}
  = \frac{RT \omega}{p} + Q .
\end{eqnarray}
%


\subsubsection*{2-10-F-d \quad ���ͥ륮�����Ѵ���}

\Deqref{hw-02-10-��ư���ͥ륮���ΤߤΥե�å����ե�����},
\Deqref{hw-02-10-�����ݥƥ󥷥��ΤߤΥե�å����ե�����},
\Deqref{hw-02-10-���󥿥�ԡ��ΤߤΥե�å����ե�����}
��3���α��դ����ܤ��Ƥۤ���.
\Deqref{hw-02-10-��ư���ͥ륮���ΤߤΥե�å����ե�����}��
\Deqref{hw-02-10-�����ݥƥ󥷥��ΤߤΥե�å����ե�����}��
���ˤ�$\Dvect{V} \cdot \nabla \phi$�ιब����, ��椬�դˤʤäƤ��뤳�Ȥ�
�狼��.
�Ĥޤ�, $z$�����ä���б�ư���ͥ륮����������������˰��֥��ͥ륮����
���ä�, �դ�$z$����������б�ư���ͥ륮�������ä��������
���֥��ͥ륮���ϸ�������.

����
\Deqref{hw-02-10-�����ݥƥ󥷥��ΤߤΥե�å����ե�����}��
\Deqref{hw-02-10-���󥿥�ԡ��ΤߤΥե�å����ե�����}�ˤ�ξ����
$R T \omega /p$�ιब���뤳�Ȥ����ܤ���.
�����Ǥ��㤨�ж��������徺���뤳�Ȥ�ͤ���.
�徺����Ȥ������Ȥ�$\omega < 0$
\footnote{
$p$-��ɸ�Ϥ�($z$-��ɸ�ϤǸ����Ȥ�����)�������ʤΤǾ徺ή��
$\omega < 0$�Ǥ���.
}
�Ǥ��뤳�Ȥ����դ��Ƥۤ���.
�徺����Ȥ������Ȥϰ��֥��ͥ륮�������ä���.
�ޤ�, �徺����Τ�ȼ�����٤ϲ�����Τǥ��󥿥�ԡ�$E$��
��������.
�����������ߤ��������������ȵդΤ��Ȥ�������.
���Τ��Ȥ�
\Deqref{hw-02-10-�����ݥƥ󥷥��ΤߤΥե�å����ե�����}��
\Deqref{hw-02-10-���󥿥�ԡ��ΤߤΥե�å����ե�����}�ˤ�����
$R T \omega /p$�ιब�ڤܤ��ƶ��Ǥ���.

�Ĥޤ�, ��ư���ͥ륮���Ȱ��֥��ͥ륮����$- \Dvect{V} \cdot \nabla \phi$
�ι���Ѵ�����, ���֥��ͥ륮���ȥ��󥿥�ԡ�(�Ĥޤ���������ͥ륮��)��
$R T \omega /p$�ι���Ѵ������.
����3�ĤΥ��ͥ륮����­����碌���, ���������ι���Ǥ��ä���ä�
�ʤ��ʤ�櫓�Ǥ���.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      2-10-F-1    %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{2-10-F-e \qquad �������ͥ륮���Υե�å����ե�����}

ñ�̼��̤�������������ͥ륮��$I = c_v T$ [J/kg]�Υե�å����ե������
(2-87)����Ƴ�Ф��Ƥߤ�.
�ޥ��䡼�δط���$c_p = c_v + R$���
$E = c_p T = (c_v + R) T = c_p T + p \alpha$�Ȥʤ�Τ�
(2-86)��
%
\begin{eqnarray}
 \DD{E}{t} &=& \frac{R T \omega}{p} + Q
  \nonumber , \\
 \DD{}{t} \left( c_v T \right)
  + \DD{}{t} \left( p \alpha \right)
  &=& \alpha \DD{p}{t} + Q
  \nonumber , \\
 \DD{}{t} \left( c_v T \right)
  &=& - \DD{}{t} \left( \frac{p}{\rho} \right)
  + \frac{1}{\rho} \DD{p}{t} + Q
  \nonumber , \\
 &=&
  - p \DD{}{t} \left( \frac{1}{\rho} \right) + Q
  \nonumber , \\
 \DP{I}{t} + \Dvect{V} \cdot \nabla I + \omega \DP{I}{p}
  &=& - p \DD{}{t} \left( \frac{1}{\rho} \right) + Q
  \nonumber , \\
 \DP{I}{t} + \Ddiv (I \Dvect{V})+ \DP{(I \omega)}{p}
  - I \left\{
       \Ddiv \Dvect{V} + \DP{\omega}{p}
      \right\}
  &=& - p \DD{}{t} \left( \frac{1}{\rho} \right) + Q.
  \nonumber
\end{eqnarray}
%
�Ȥʤ�. Ϣ³�μ�(2-60)
\footnote{
\begin{eqnarray}
 \Ddiv \Dvect{V} + \DP{\omega}{p} = 0.
\end{eqnarray}
}
���,
%
\begin{eqnarray}
 \DP{I}{t} + \Ddiv (I \Dvect{V})+ \DP{(I \omega)}{p}
  &=& - p \DD{}{t} \left( \frac{1}{\rho} \right) + Q
  \nonumber 
\end{eqnarray}
%
�Ȥʤ�.
���줬�������ͥ륮���Υե�å����ե�����Ǥ���.
���μ��ȥ��󥿥�ԡ��Υե�å����ե��������٤Ƥߤ��,
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