% start
%
\global\count12 = 0
\def\refnum{\global\advance\count12 by 1 {[{\the\count12}]}}
\def\refnums #1#2{{[{#1}]}\global\advance\count12 by #2}
\def\mybibliography{
   \list
   {[\arabic{enumi}]}{\settowidth\labelwidth{[99]}
   \leftmargin\labelwidth \advance\leftmargin\labelsep
   \usecounter{enumi}}
   \def\newblock{\hskip .11em plus .07em minus -.07em}
   \sloppy \sfcode`\.=1000\relax }
\let\endthebibliography=\endlist
\let\num=\item
%
\def\centhead#1{ \typeout{#1}
  \smallskip\centerline{\bf #1}
  \smallskip
  \addcontentsline{toc}{section}{\hspace{3em}#1}}
\def\Centhead#1{{\smallskip\flushleft{\underline{{#1}}}}}
%
% end
%
%*****************************************************
%
%     Beginning of text
% macros for the special operator used in SIMPLER
\def\maxl{\lbrack\!\lbrack}
\def\maxr{\rbrack\!\rbrack}
%
\def\xb{{\bf x}}
\def\Xb{{\bf X}}
\def\xdot{\dot{\xb}}
\def\Kdot{\dot K}
\def\Udot{\dot U}
\def\ub{{\bf u}}
\def\to{t_{o}}
\def\ti{t_{1}}
\def\tfi{t_{2}}
\def\tf{t_{f}}
\def\xbt{\xb(t)}
\def\ubt{\ub(t)}
\def\xbto{\xb(\to)}
\def\xbtf{\xb(\tf)}
\def\psitf{{\bf{\psi}}(\xbtf,\tf)}
\def\phitf{\phi(\xbtf,\tf)}
\def\lamr{\lambda_{r}}
\def\lamu{\lambda_{u}}
\def\lamv{\lambda_{v}}
\def\lamth{\lambda_{\theta}}
\def\sq{^{2}}
\def\cub{^{3}}
%% \def\mycite#8{$^{\ref{#8}$}
\def\refer#1{(\ref{#1})}
\def\refchap#1{Chapter~\ref{#1}}
\def\reffig#1{Figure~\ref{#1}}
\def\reftab#1{Table~\ref{#1}}
\def\refeq#1{Eq.~\refer{#1}}
\def\rdot{\dot{r}}
\def\udot{\dot{u}}
\def\vdot{\dot{v}}
\def\thdot{\dot{\theta}}
\def\lamrdot{\dot{\lamr}}
\def\lamudot{\dot{\lamu}}
\def\lamvdot{\dot{\lamv}}
\def\lamthdot{\dot{\lamth}}
\def\omegdot{\dot{\omega}}
\def\mui{\mu_{1}}
\def\ri{r_{1}}
\def\di{d_{1}}
\def\costh{\cos\theta}
\def\sinth{\sin\theta}
\def\dicos{\di\costh}
\def\disin{\di\sinth}
\def\rdicos{\left(r-\dicos\right)}
\def\muiri#1{\frac{\mui}{\ri^{#1}}}
\def\muidi#1{\frac{\mui}{\di^{#1}}}
\def\dridr{\frac{\partial\ri^{2}}{\partial r}}
\def\dridth{\frac{\partial\ri^{2}}{\partial\theta}}
\def\refs{_{ref}}
\def\new{_{new}}
\def\thetai{\theta_1}
\def\ii{_{1}}
\def\norm#1#2{\left|#1\right|_{#2}}
\def\mod#1{\left|#1\right|}
\def\rhat{\hat{r}}
\def\rihat{\hat{\ri}}
\def\dihat{\hat{\di}}
\def\thhat{\hat{\theta}}
\def\rvec{\vec{r}}
\def\rivec{\vec{\ri}}
\def\divec{\vec{\di}}
\def\omvec{\vec{\omega}}



%
%    Fluid Mechanics  ..................................
%

\def\vvec#1{\vec#1}
\def\uvec#1{\hat{e}_#1}
% for double ubsecription
\def\dbvr#1{V_{R_{#1}}}
\def\dbvt#1{V_{\Phi_{#1}}}
\def\dbvz#1{V_{Z_{#1}}}
%    Thermodynamics .........................................
\def\dc {$^\circ$C}
\def\dk {$^\circ$K}
\def\df {$^\circ$F}
\def\dr {$^\circ$R}
\def\sV {$-$\llap V}
\def\sv {{-}\llap v}
\def\ein {{e_{int}}}
\def\spcvt {$C_{\sv}$}
\def\spcpt {$C_p$}
%    Mathematics    .......................................
\def\parte{{\partial\over\partial t}}
\def\parthe{{\partial\over\partial \bfee}}
\def\parre{{\partial\over\partial \bR}}
\def\parze{{\partial\over\partial \bZ}}
\def\part#1{{\partial #1\over\partial t}}
\def\parth#1{{\partial#1\over\partial \bfee}}
\def\parr#1{{\partial#1\over\partial \bR}}
\def\parz#1{{\partial#1\over\partial \bZ}}
\def\px{{\partial\over\partial x}}
\def\py{{\partial\over\partial y}}
\def\pz{{\partial\over\partial z}}
\def\pax#1{{\partial#1\over\partial x}}
\def\pay#1{{\partial#1\over\partial y}}
\def\paz#1{{\partial#1\over\partial z}}
\def\pxx{{\partial\over\partial x_1}}
\def\pxy{{\partial\over\partial x_2}}
\def\pxz{{\partial\over\partial x_3}}
\def\paxx#1{{\partial#1\over\partial x_1}}
\def\paxy#1{{\partial#1\over\partial x_2}}
\def\paxz#1{{\partial#1\over\partial x_3}}
\def\ohx#1{{#1\over h_1}}
\def\ohy#1{{#1\over h_2}}
\def\ohz#1{{#1\over h_3}}
\def\deldot{{\nabla\bullet}}
\def\del{{\nabla}}
\def\half {{1\over2}}
% Unit vectors
\def\un#1{ \hat{e}_#1}
%    Gasdynamics    .......................................
%\def\TOT #1#2{$T_{#2}\overT_{\bf#1}$}
%\def\ToT #1#2{$T_{#2}/T_{\bf#1}$}
%\def\T3T1 {T_3/T_1}
\def\ps #1{$p^*_#1$}
\def\pones {\ps 1}
\def\ptwos {\ps 2}
\def\ponets {\ps 1t}
\def\ptwots {\ps 2t}
\def\As #1{$A^*_#1$}
\def\Aones {\As 1}
\def\Atwos {\As 2}
%\def\A2A1 {A_2/A_1}
\def\as #1{$a^*_#1$}
\def\aones {\as 1}
\def\atwos {\as 2}
\def\aonets {\as 1t}
\def\atwots {\as 2t}
\def\ro #1{\rho_#1}
\def\rone {\ro 1}
\def\rtwo {\ro 2}
\def\ronet {\ro 1t}
\def\rtwot {\ro 2t}
\def\rs #1{\rho^*_#1}
\def\rones {\rs 1}
\def\rtwos {\rs 2}
\def\ronets {\rs 1t}
\def\rtwots {\rs 2t}
%##############################################################
% ... define special hyphenations heere to avoid overfull boxes
%
\hyphenation{axi-sym-metric}
\hyphenation{di-men-sion-al}
\hyphenation{non-dimen-sion-alized}
\hyphenation{equa-tion}
\hyphenation{time-de-pen-dent}
\hyphenation{shock-cap-tur-ing}
\hyphenation{non-osc-il-la-to-ry}
\hyphenation{com-pu-ta-tion-al}
\hyphenation{nav-ier-stokes}
\hyphenation{character-is-tics}
\hyphenation{con-ju-gate}
\hyphenation{gra-di-ent}
%\hyphenation{ap-prx-i-mate-ly}
\hyphenation{fac-tored}
\hyphenation{lin-e-a-rized}
\hyphenation{im-pli-cit}
\hyphenation{es-sen-ti-al-ly}
\hyphenation{Shock-Cap-tur-ing}
\hyphenation{two-di-men-sion-al}
\hyphenation{Nu-mer-i-cal}
\hyphenation{nu-mer-i-cal}
\hyphenation{Rey-nold}
%####################################################
%----------------------------------------------------
% macros for eqalign and eqalignno that LATeX doesn't support
%
\def\openup{\afterassignment\Apenup\dimen0=}
\def\mAth{\mathsurround=0pt}
\def\Apenup{\advance\lineskip\dimen0
  \advance\baselineskip\dimen0 \advance \lineskiplimit\dimen0}
\def\eqalign#1{\null\,\vcenter{\openup1\jot \mAth
  \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
      \crcr#1\crcr}}\,}
%\def\ialign{\everycr={}\tabskip=0pt \halign} % initialized halign
\newif\ifdtAp
\def\displAy{\global\dtAptrue \openup1\jot \mAth
  \everycr{\noalign{\ifdtAp \global\dtApfalse
      \vskip-\lineskiplimit \vskip 0pt
      \else \penalty 100 \fi}}}
\def\Align{\tabskip=0pt\everycr={}} % restore inside \displAy
\def\displaylines#1{\displAy
  \halign{\hbox to\displaywidth{$\hfil\Align\displaystyle##\hfil$}\crcr
    #1\crcr}}
%
\def\eqalignno#1{\displAy \tabskip=0pt plus 1000pt minus 1000pt
  \halign to\displaywidth{\hfil$\Align\displaystyle{##}$\tabskip=0pt
    &$\Align\displaystyle{{}##}$\hfil\tabskip=0pt plus 1000pt minus 1000pt
   &\llap{$\Align##$}\tabskip=0pt\crcr
   #1\crcr}}

%-----------------------------------------------------------------









