\documentstyle[12pt]{article} \pagestyle{plain} \topmargin 0in \headheight 0pt \headsep 0pt \raggedbottom \oddsidemargin 0in \textheight 9.25in \textwidth 6.0in \parskip 5pt plus 1pt minus 1pt \def \baselinestretch {1.0} % single spaced \setlength {\unitlength} {1.0in} \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}