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\begin{document}

\def \ei    {{e^{i(\alpha x - \omega t)}}}
\def \eiax  {{e^{- \alpha_i x}}}
\def \eitax {{e^{-2\alpha_i x}}}

\def \cwt {\cos (\,)}
\def \swt {\sin (\,)}
\def \cqwt {\cos^2 (\,)}
\def \sqwt {\sin^2 (\,)}

\def \thalf {{\textstyle \frac{1}{2}}}

\def \d {\partial}

\def \beqa {\begin{eqnarray*}}
\def \eeqa {\end{eqnarray*}}

\def \strut {\rule{0em}{2.0ex}}

\beqa
\alpha &=& \alpha_r + i \alpha_i  \\
\omega &=& \omega_r
\eeqa
\beqa
\cwt & \equiv & \cos( \alpha_r x - \omega_r t)  \\[0.25em]
\swt & \equiv & \sin( \alpha_r x - \omega_r t)
\eeqa
\beqa
u' \;=\; \Re \left[ (u_r + i u_i ) \, \ei \right] &=&
  \left( u_r \cwt - u_i \swt \strut \right) \eiax \\[0.5em]
v' \;=\; \Re \left[ (v_r + i v_i ) \, \ei \right] &=&
  \left( v_r \cwt - v_i \swt \strut \right) \eiax \\[0.5em]
p' \;=\; \Re \left[ (p_r + i p_i ) \, \ei \right] &=&
  \left( p_r \cwt - p_i \swt \strut \right) \eiax
\eeqa
\beqa
u'v' & = & \left( u_r v_r \cqwt \,+\, u_i v_i \sqwt \:-\:
  (u_r v_i + u_i v_r) \swt \cwt \right) \eitax \\[0.5em]
{u'}^2 & = & \left( u_r^2 \cqwt \,+\, u_i^2 \sqwt \:-\:
  2 u_r u_i \swt \cwt \right) \eitax \\[0.5em]
{v'}^2 & = & \left( v_r^2 \cqwt \,+\, v_i^2 \sqwt \:-\:
  2 v_r v_i \swt \cwt \right) \eitax
\eeqa
\beqa
\overline{ u'v' } &=& \thalf \left( u_r v_r + u_i v_i \strut \right)
  \eitax \\[0.5em]
\overline{{u'}^2} &=& \thalf \left( u_r^2 + u_i^2 \right)
  \eitax \\[0.5em]
\overline{{v'}^2} &=& \thalf \left( v_r^2 + v_i^2 \right)
  \eitax
\eeqa
\beqa
\frac{\d u'}{\d x}  &=&
\Re \!\left[ (i \alpha_r - \alpha_i )
        \left(u_r + i u_i \strut \right) \, \ei \right]
\;=\; \left( -(\alpha_i u_r + \alpha_r u_i) \cwt
        \:-\: (\alpha_r u_r - \alpha_i u_i) \swt \strut \right) \eiax \\[0.5em]
%
\frac{\d v'}{\d x}  &=&
\Re \!\left[ (i \alpha_r - \alpha_i )
        \left(v_r + i v_i \strut \right) \, \ei \right]
\;=\; \left( -(\alpha_i v_r + \alpha_r v_i) \cwt
        \:-\: (\alpha_r v_r - \alpha_i v_i) \swt \strut \right) \eiax \\[0.5em]
%
\frac{\d u'}{\d y}  &=&
\Re \!\left[ \left(Du_r + i Du_i \strut \right) \, \ei \right]
\;=\;  \left( Du_r \cwt - Du_i \swt \strut \right) \eiax \\[0.5em]
\frac{\d v'}{\d y}  &=&
\Re \!\left[ \left(Dv_r + i Dv_i \strut \right) \, \ei \right]
\;=\;  \left( Dv_r \cwt - Dv_i \swt \strut \right) \eiax
\eeqa
\beqa
\nabla^2 u' &=& -\frac{\d \omega'}{\d y}
\;=\;  \left(-D\omega_r \cwt + D\omega_i \swt \strut \right) \eiax \\[0.5em]
\nabla^2 v' &=& \;\; \frac{\d \omega'}{\d x}
\;=\; \left( -(\alpha_i \omega_r + \alpha_r \omega_i) \cwt
        \:-\: (\alpha_r \omega_r - \alpha_i \omega_i) \swt \strut \right) \eiax
\eeqa
\newpage

\beqa
Q &=& \int \thalf \left(\overline{{u'}^2 + {v'}^2} \right) \bar{u} \, dy \;=\;
 \int {\textstyle \frac{1}{4}} 
  \left( u_r^2 + u_i^2 + v_r^2 + v_i^2 \right) \bar{u} \, dy 
          \;\; \eitax \\[0.75em]
\frac{dQ}{dx} &=& \int \left( 
  \overline{u' {\textstyle \frac{\d u'}{\d x}}} \:+\:
  \overline{v' {\textstyle \frac{\d v'}{\d x}}}   \right) \bar{u} \, dy
\;=\; \int \thalf \left[ 
      - u_r ( \alpha_i u_r + \alpha_r u_i )
  \:+\: u_i ( \alpha_r u_r - \alpha_i u_i ) \rule{0ex}{3ex} \right. \\
& & \hspace{29.0ex} \left. \rule{0ex}{3ex}
    -\: v_r ( \alpha_i v_r + \alpha_r v_i )
  \:+\: v_i ( \alpha_r v_r - \alpha_i v_i ) \right] \bar{u} \, dy
\;\; \eitax
\eeqa
\beqa
\epsilon &=& \nu \left[
      2 \overline{\textstyle \left( \frac{\d u'}{\d x} \right)^2 }
\:+\: 2 \overline{\textstyle \left( \frac{\d v'}{\d y} \right)^2 }
\:+\:   \overline{\textstyle \left( \frac{\d u'}{\d y}
                                   +\frac{\d v'}{\d x} \right)^2 }  \right] \\
%
&=& \nu \left[ \rule{0ex}{3ex}
      (\alpha_i u_r + \alpha_r u_i)^2
\:+\: (\alpha_r u_r - \alpha_i u_i)^2
\:+\: (Dv_r)^2 + (Dv_i)^2  \right. \\
& & \hspace{3ex} \left. \rule{0ex}{3ex}
\;+\; \thalf ( Du_r - \alpha_i v_r - \alpha_r v_i)^2
\:+\: \thalf ( Du_i + \alpha_r v_r - \alpha_i v_i)^2  \right]
\;\;\; \eitax
\eeqa
\beqa
{\cal D}_x \;=\; \frac{\d}{\d x} \! \left\{ 
 \overline{ u' \left[ p' + \thalf ( {u'}^2 + {v'}^2 ) \right] } 
                     \right\} &=&
\overline{ \textstyle \frac{\d u'}{\d x} 
          \left[ p' + \thalf ( {u'}^2 + {v'}^2 ) \right] }  \:+\:
\overline{ \textstyle  u' \left[ \frac{\d p'}{\d x}
             + u' \frac{\d u'}{\d x} + v' \frac{\d v'}{\d x} \right] } \\
&=&  \thalf \left[
      - p_r ( \alpha_i u_r + \alpha_r u_i )
  \:+\: p_i ( \alpha_r u_r - \alpha_i u_i ) \rule{0ex}{3ex} \right. \\
& & \left. \hspace{2.5ex}
      - u_r ( \alpha_i p_r + \alpha_r p_i )
  \:+\: u_i ( \alpha_r p_r - \alpha_i p_i ) \rule{0ex}{3ex} \right]
\;\; \eitax
\eeqa
\beqa
\Pi_q \;=\; \nu \left( u' \nabla^2 u' \,+\, v' \nabla^2 v' \right) 
&=& \thalf \, \nu \left[ -u_r D\omega_r - u_i D\omega_i
 \:-\: v_r ( \alpha_i \omega_r + \alpha_r \omega_i )
 \:+\: v_i ( \alpha_r \omega_r - \alpha_i \omega_i ) \rule{0ex}{3ex} \right]
\eeqa

\beqa
\frac{dQ}{dx} &=& \int -\overline{u'v'} \, d\bar{u} 
 \;-\; \int {\cal D}_x \, dy \;+\; \int \Pi_q \, dy
\eeqa
\end{document}
