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最后几行未显示在 pdf 中

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最后几行未显示在 pdf 中

在此处输入图片描述

\documentclass{article}
\usepackage{amsmath}   
\providecommand\pdv[2]{\frac{\partial #1}{\partial #2}}

\begin{document}
\begin{align}
%%\begin{split}
\pdv{}{\hat{t}}\bigg[
\bigg(\frac{1}{\gamma *\gamma-1*M_{\infty}^{2}}\bigg)\bigg(\overline{\rho}T'+\rho'\overline{T}\bigg)+\frac{1}{2}\bigg\{2\overline{\rho}\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)&+\rho'\bigg(\overline{u}\overline{u}+\overline{v}\overline{v}+\overline{w}\overline{w}\bigg)\bigg\}\bigg]&+
\pdv{}{\hat{x}}\bigg[\bigg(\frac{1}{(\gamma-1)M_{\infty}^{2}}\bigg)
\bigg(\overline{\rho}\overline{u}T'+\overline{\rho}u'\overline{T}+\rho'\overline{u}\overline{T}\bigg)+\\
\frac{1}{2}\bigg\{2\overline{\rho}\overline{u}\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)+\bigg(\overline{\rho}u'+\rho'\overline{u}\bigg)\bigg(\overline{u}\overline{u}+\overline{v}\overline{v}+\overline{w}\overline{w}\bigg)\bigg\}&
-\overline{u}\frac{1}{Re_{D}}\bigg\{2\pdv{u'}{\hat{x}}-\frac{2}{3}\bigg(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\bigg)&+\pdv{u'}{\hat{y}}+\pdv{v'}{\hat{x}}+\pdv{u'}{\hat{z}}+\pdv{w'}{\hat{x}}\bigg\}\\
-u'\frac{1}{Re_{D}}\bigg\{2\pdv{\overline{u}}{\hat{x}}-\frac{2}{3}\bigg(\pdv{\overline{u}}{\hat{x}}+\pdv{\overline{v}}{\hat{y}}+\pdv{\overline{w}}{\hat{z}}\bigg)+\pdv{\overline{u}}{\hat{y}}+\pdv{\overline{v}}{\hat{x}}+\pdv{\overline{u}}{\hat{z}}+\pdv{\overline{w}}{\hat{x}}\bigg\}-\bigg\{\bigg(\frac{1}{Pr (\gamma-1)}\bigg)\bigg(\frac{1}{M_{\infty}^{2}}\bigg)\bigg(\frac{1}{Re_{D}}\bigg)\pdv{T'}{\hat{x}}\bigg\}\bigg]\dots\\
\vspace{50mm}
+\pdv{}{\hat{y}}\bigg[
\bigg(\frac{1}{(\gamma-1)M_{\infty}^{2}}\bigg)\bigg(\overline{\rho}\overline{v}T'+\overline{\rho}v'\overline{T}+\rho'\overline{v}\overline{T}\bigg)+\dots\\\frac{1}{2}\bigg\{2\overline{\rho}\overline{v}\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)+\bigg(\overline{\rho}v'+\rho'\overline{v}\bigg)\bigg(\overline{u}\overline{u}+\overline{v}\overline{v}+\overline{w}\overline{w}\bigg)\bigg\}\dots\\-\overline{v}\frac{1}{Re_{D}}\bigg\{\
\pdv{v'}{\hat{x}}+\pdv{u'}{\hat{y}}+2\pdv{v'}{\hat{y}}-\frac{2}{3}\bigg(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\bigg)+\pdv{v'}{\hat{z}}+\pdv{w'}{\hat{y}}\bigg\}\dots \\
-v'\frac{1}{Re_{D}}\bigg\{\pdv{\overline{v}}{\hat{x}}+\pdv{\overline{u}}{\hat{y}}+2\pdv{\overline{v}}{\hat{y}}-\frac{2}{3}\bigg(\pdv{\overline{u}}{\hat{x}}+\pdv{\overline{v}}{\hat{y}}+\pdv{\overline{w}}{\hat{z}}\bigg)+\pdv{\overline{v}}{\hat{z}}+\pdv{\overline{w}}{\hat{y}}\bigg\}\dots\\-\bigg(\frac{1}{Pr (\gamma-1)}\bigg)\bigg(\frac{1}{M_{\infty}^{2}}\bigg)\bigg(\frac{1}{Re_{D}}\bigg)\pdv{T'}{y}\bigg]\dots\\
+\pdv{}{\hat{z}}\bigg[\bigg(\frac{1}{(\gamma-1)M_{\infty}^{2}}\bigg)
\bigg(\overline{\rho}\overline{w}T'+\overline{\rho}w'\overline{T}+\rho'\overline{w}\overline{T}\bigg)\dots\\+\frac{1}{2}\bigg\{2\overline{\rho}\overline{w}\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)+\bigg(\overline{\rho}w'+\rho'\overline{w}\bigg)\bigg(\overline{u}\overline{u}+\overline{v}\overline{v}+\overline{w}\overline{w}\bigg)\bigg\}\dots\\-\overline{w}\frac{1}{Re_{D}}\bigg\{\pdv{w'}{\hat{x}}+\pdv{u'}{\hat{z}}+\pdv{w'}{\hat{y}}+\pdv{v'}{\hat{z}}+2\pdv{w'}{\hat{z}}-\frac{2}{3}\bigg(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\bigg)\bigg\}\dots \\
-\bigg(w'\frac{1}{Re_{D}}\bigg)\bigg\{\pdv{\overline{w}}{\hat{x}}+\pdv{\overline{u}}{\hat{z}}+\pdv{ \overline{w}}{\hat{y}}+\pdv{\overline{v}}{\hat{z}}+2\pdv{\overline{w}}{\hat{z}}-\frac{2}{3}\bigg(\pdv{\overline{u}}{\hat{x}}+\pdv{\overline{v}}{\hat{y}}+\pdv{\overline{w}}{\hat{z}}\bigg)\bigg\}\dots\\-\bigg(\frac{1}{Pr (\gamma-1)}\bigg)\bigg(\frac{1}{M_{\infty}^{2}}\bigg)\bigg(\frac{1}{Re_{D}}\bigg)\pdv{T'}{\hat{z}}\bigg]\dots \\
=\pdv{}{\hat{t}}\bigg[-\bigg(\frac{1}{\gamma *\gamma-1*M_{\infty}^{2}}\bigg)
\rho'T'-\frac{1}{2}\bigg\{\overline{\rho}\bigg(u'u'+v'v'+w'w'\bigg)+\dots\\2\rho'\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)+\rho'\bigg(u'u'+v'v'+w'w'\bigg)\bigg\}\bigg]\dots\\
+\pdv{}{\hat{x}}\bigg[
-\bigg(\frac{1}{(\gamma-1)M_{\infty}^{2}}\bigg)\bigg(\overline{\rho}u'T'+\rho'\overline{u}T'+\rho'u'\overline{T}+\rho'u'T'\bigg)\dots\\-\bigg\{\bigg(\overline{\rho}\overline{u}+\overline{\rho}u'+\rho'\overline{u}+\rho'u'\bigg)\bigg(u'u'+v'v'+w'w'\bigg)+2\bigg(\overline{\rho}u'+\rho'\overline{u}+\rho'u'\bigg)\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)+\dots\\\rho'u'\bigg(\overline{u}\overline{u}+\overline{v}\overline{v}+\overline{w}\overline{w}\bigg)\bigg\}+\bigg(u'\frac{1}{Re_{D}}\bigg)\bigg\{2\pdv{u'}{\hat{x}}-\frac{2}{3}\bigg(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\bigg)+\pdv{u'}{\hat{y}}+\pdv{v'}{\hat{x}}+\pdv{u'}{\hat{z}}+\pdv{w'}{\hat{x}}\bigg\}\bigg]\dots \\
+\pdv{}{\hat{y}}\bigg[
-\bigg(\frac{1}{(\gamma-1)M_{\infty}^{2}}\bigg)-\bigg(\overline{\rho}v'T'+\rho'\overline{v}T'+\rho'v'\overline{T}+\rho'v'T'\bigg)-\dots\\ \newline- \bigg\{\bigg(\overline{\rho}\overline{v}+\overline{\rho}v'+\rho'\overline{v}+\rho'v'\bigg)\bigg(u'u'+v'v'+w'w'\bigg)+2\bigg(\overline{\rho}v'+\rho'\overline{v}+\rho'v'\bigg)\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)+\dots\\\linebreak\rho'v'\bigg(\overline{u}\overline{u}+\overline{v}\overline{v}+\overline{w}\overline{w}\bigg)\bigg\}\dots\\+\bigg(v'\frac{1}{Re_{D}}\bigg)\bigg\{
\pdv{u'}{\hat{y}}+\pdv{v'}{\hat{x}}+2\pdv{v'}{\hat{y}}-\frac{2}{3}\bigg(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\bigg)+\pdv{w'}{\hat{y}}+\pdv{v'}{\hat{y}}\bigg\}
\bigg]\dots \\
+\pdv{}{\hat{z}}\bigg[
-C_{p}\bigg(\overline{\rho}w'T'+\rho'\overline{w}T'+\rho'w'\overline{T}+\rho'w'T'\bigg)\dots\\-\bigg\{\bigg(\overline{\rho}\overline{w}+\overline{\rho}w'+\rho'\overline{w}+\rho'w'\bigg)\bigg(u'u'+v'v'+w'w'\bigg)+2\bigg(\overline{\rho}w'+\rho'\overline{w}+\rho'w'\bigg)\bigg(\overline{u}u'+\overline{v}v'+\overline{w}w'\bigg)+\dots\\\rho'w'\bigg(\overline{u}\overline{u}+\overline{v}\overline{v}+\overline{w}\overline{w}\bigg)\bigg\}+\bigg(w'\mu\bigg)\bigg\{
\pdv{u'}{z}+\pdv{w'}{x}+\pdv{v'}{z}+\pdv{w'}{y}+2\pdv{w'}{z}-\frac{2}{3}\bigg(\pdv{u'}{x}+\pdv{v'}{y}+\pdv{w'}{z}\bigg)\bigg\}
\bigg]
%%\end{split}
\end{align}
\end{document}

编译后，最后几行未显示在 pdf 中

答案1

环境align包含几行非常宽的行 - 事实上太宽了，无法容纳在文本块中。但主要问题是材料有太多行，以至于无法放在一页上。我猜想这就是 OP 用“最后几行未显示”这个表达方式要表达的意思。

align在(或) 环境中允许分页符（如果需要）所需的命令align*称为\allowdisplaybreaks。请务必消除无意义的split包装器，因为环境内不允许分页符split。并且，请摆脱看起来奇怪的\vspace{50mm}、\newline和\linebreak指令。

我认为需要紧急解决的另一个问题是，环境align*目前不包含有意义的&对齐点。这是最不幸的，因为结果是所有线都对齐了最后每行的对齐点。几乎任何替代对齐点集都必须比现状更好。\dots每行末尾的指令也并不具有吸引力。

为了解决这个问题，我建议将行改为左对齐，8 个主要行（以 开头的行）不缩进，\pdv{}其他\qquad所有行都缩进。为了提供进一步的视觉分组，我建议分别用\bigg[和替换所有和。\bigg]\Biggl[\Biggr]

最后，也可能是最不重要的一点，考虑用 替换所有 154 个 [!]\overline实例\bar。

在此处输入图片描述

\documentclass{article} % or some other suitable document class
\usepackage{amsmath} % for 'align*' env. and '\allowdisplaybreaks' macro
\providecommand\pdv[2]{\frac{\partial #1}{\partial #2}}

\begin{document}

\begingroup % limit scope of next instruction to current TeX group
\allowdisplaybreaks % enable page breaking 

\begin{align*}
&\pdv{}{\hat{t}}\Biggl[
    \biggl(\frac{1}{\gamma*\gamma-1*M_{\infty}^{2}}\biggr)
    \biggl(\bar{\rho}T'+\rho'\bar{T}\biggr) \\
&\qquad +\frac{1}{2}\biggl\{
    2\bar{\rho}\biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)
    +\rho'\biggl(\bar{u}\bar{u}+\bar{v}\bar{v}+\bar{w}\bar{w}\biggr)
    \biggr\}\Biggr] \\
&+\pdv{}{\hat{x}}\Biggl[
    \biggl(\frac{1}{(\gamma-1)M_{\infty}^{2}}\biggr)
    \biggl(\bar{\rho}\bar{u}T'+\bar{\rho}u'\bar{T} +\rho'\bar{u}\bar{T}\biggr) \\
&\qquad +\frac{1}{2}\biggl\{2\bar{\rho}\bar{u}
    \biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)
    +\biggl(\bar{\rho}u'+\rho'\bar{u}\biggr)
    \biggl(\bar{u}\bar{u}+\bar{v}\bar{v} +\bar{w}\bar{w}\biggr)\biggr\} \\
&\qquad -\bar{u}\frac{1}{Re_{D}}
    \biggl\{2\pdv{u'}{\hat{x}}-\frac{2}{3}
    \biggl(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\biggr)
    +\pdv{u'}{\hat{y}}+\pdv{v'}{\hat{x}}
    +\pdv{u'}{\hat{z}}+\pdv{w'}{\hat{x}}\biggr\} \\
&\qquad -u'\frac{1}{Re_{D}}\biggl\{
    2\pdv{\bar{u}}{\hat{x}}-\frac{2}{3}
    \biggl(\pdv{\bar{u}}{\hat{x}}
    +\pdv{\bar{v}}{\hat{y}}+\pdv{\bar{w}}{\hat{z}}\biggr)
    +\pdv{\bar{u}}{\hat{y}}+\pdv{\bar{v}}{\hat{x}}
    +\pdv{\bar{u}}{\hat{z}}+\pdv{\bar{w}}{\hat{x}}\biggr\} \\
&\qquad -\biggl\{\biggl(\frac{1}{Pr (\gamma-1)}\biggr)
    \biggl(\frac{1}{M_{\infty}^{2}}\biggr)
    \biggl(\frac{1}{Re_{D}}\biggr)\pdv{T'}{\hat{x}}\biggr\}\Biggr] \\
&+\pdv{}{\hat{y}}\Biggl[
    \biggl(\frac{1}{(\gamma-1)M_{\infty}^{2}}\biggr)
    \biggl(\bar{\rho}\bar{v}T'+\bar{\rho}v'\bar{T} +\rho'\bar{v}\bar{T}\biggr) \\
&\qquad +\frac{1}{2}\biggl\{2\bar{\rho}\bar{v}
    \biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)
    +\biggl(\bar{\rho}v'+\rho'\bar{v}\biggr)
    \biggl(\bar{u}\bar{u}+\bar{v}\bar{v}+\bar{w}\bar{w}\biggr)
    \biggr\} \\
&\qquad -\bar{v}\frac{1}{Re_{D}}\biggl\{
    \pdv{v'}{\hat{x}}+\pdv{u'}{\hat{y}}+2\pdv{v'}{\hat{y}}-\frac{2}{3}
    \biggl(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\biggr)
    +\pdv{v'}{\hat{z}}+\pdv{w'}{\hat{y}}\biggr\} \\
&\qquad -v'\frac{1}{Re_{D}}
    \biggl\{\pdv{\bar{v}}{\hat{x}}
    +\pdv{\bar{u}}{\hat{y}}+2\pdv{\bar{v}}{\hat{y}}-\frac{2}{3}
    \biggl(\pdv{\bar{u}}{\hat{x}}
    +\pdv{\bar{v}}{\hat{y}}+\pdv{\bar{w}}{\hat{z}}\biggr)
    +\pdv{\bar{v}}{\hat{z}}+\pdv{\bar{w}}{\hat{y}}\biggr\} \\
&\qquad -\biggl(\frac{1}{Pr(\gamma-1)}\biggr)
    \biggl(\frac{1}{M_{\infty}^{2}}\biggr)
    \biggl(\frac{1}{Re_{D}}\biggr)
    \pdv{T'}{y}\Biggr] \\
&+\pdv{}{\hat{z}}\Biggl[
    \biggl(\frac{1}{(\gamma-1)M_{\infty}^{2}}\biggr)
    \biggl(\bar{\rho}\bar{w}T'
    +\bar{\rho}w'\bar{T}+\rho'\bar{w}\bar{T}\biggr) \\
&\qquad +\frac{1}{2}\biggl\{2\bar{\rho}\bar{w}
    \biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)
    +\biggl(\bar{\rho}w'+\rho'\bar{w}\biggr)
    \biggl(\bar{u}\bar{u}+\bar{v}\bar{v}+\bar{w}\bar{w}\biggr)
    \biggr\} \\
&\qquad -\bar{w}\frac{1}{Re_{D}}
    \biggl\{\pdv{w'}{\hat{x}}+\pdv{u'}{\hat{z}}
    +\pdv{w'}{\hat{y}}+\pdv{v'}{\hat{z}}+2\pdv{w'}{\hat{z}}
    -\frac{2}{3}
    \biggl(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}
    +\pdv{w'}{\hat{z}}\biggr)
    \biggr\} \\
&\qquad -\biggl(w'\frac{1}{Re_{D}}\biggr)
    \biggl\{\pdv{\bar{w}}{\hat{x}}
    +\pdv{\bar{u}}{\hat{z}}+\pdv{ \bar{w}}{\hat{y}} 
    +\pdv{\bar{v}}{\hat{z}}+2\pdv{\bar{w}}{\hat{z}}
    -\frac{2}{3}\biggl(\pdv{\bar{u}}{\hat{x}}
    +\pdv{\bar{v}}{\hat{y}}+\pdv{\bar{w}}{\hat{z}}\biggr)
\biggr\} \\
&\qquad -\biggl(\frac{1}{Pr(\gamma-1)}\biggr)
    \biggl(\frac{1}{M_{\infty}^{2}}\biggr)
    \biggl(\frac{1}{Re_{D}}\biggr)
    \pdv{T'}{\hat{z}}\Biggr] \\[\jot]
&=\pdv{}{\hat{t}}\Biggl[
    -\biggl(\frac{1}{\gamma *\gamma-1*M_{\infty}^{2}}\biggr)
    \rho'T'-\frac{1}{2}\biggl\{\bar{\rho} \biggl(u'u'+v'v'+w'w'\biggr) \\
&\qquad +2\rho'\biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)
    +\rho'\biggl(u'u'+v'v'+w'w'\biggr)\biggr\}\Biggr] \\
&+\pdv{}{\hat{x}}\Biggl[
    -\biggl(\frac{1}{(\gamma-1)M_{\infty}^{2}}\biggr)
    \biggl(\bar{\rho}u'T'+\rho'\bar{u}T' +\rho'u'\bar{T}+\rho'u'T'\biggr) \\
&\qquad -\biggl\{
    \biggl(\bar{\rho}\bar{u}+\bar{\rho}u'+\rho'\bar{u}+\rho'u'\biggr)
    \biggl(u'u'+v'v'+w'w'\biggr) \\
&\qquad+2\biggl(\bar{\rho}u'+\rho'\bar{u}+\rho'u'\biggr)
    \biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)+
    \rho'u'
    \biggl(\bar{u}\bar{u}+\bar{v}\bar{v} +\bar{w}\bar{w}\biggr)\biggr\} \\
&\qquad+\biggl(u'\frac{1}{Re_{D}}\biggr)
    \biggl\{2\pdv{u'}{\hat{x}}-\frac{2}{3}
    \biggl(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\biggr)
    +\pdv{u'}{\hat{y}}+\pdv{v'}{\hat{x}}
    +\pdv{u'}{\hat{z}}+\pdv{w'}{\hat{x}}\biggr\}\Biggr] \\
&+\pdv{}{\hat{y}}\Biggl[
    -\biggl(\frac{1}{(\gamma-1)M_{\infty}^{2}}\biggr)
    -\biggl(\bar{\rho}v'T'+\rho'\bar{v}T' +\rho'v'\bar{T}+\rho'v'T'\biggr)- \\
&\qquad-\biggl\{
    \biggl(\bar{\rho}\bar{v}+\bar{\rho}v'+\rho'\bar{v}+\rho'v'\biggr)
    \biggl(u'u'+v'v'+w'w'\biggr) \\
&\qquad+2\biggl(\bar{\rho}v'+\rho'\bar{v}+\rho'v'\biggr)
    \biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)
    +\rho'v'\biggl(\bar{u}\bar{u}+\bar{v}\bar{v}+\bar{w}\bar{w}\biggr)
    \biggr\} \\
&\qquad+\biggl(v'\frac{1}{Re_{D}}\biggr)
    \biggl\{\pdv{u'}{\hat{y}} +\pdv{v'}{\hat{x}}+2\pdv{v'}{\hat{y}}-\frac{2}{3}
    \biggl(\pdv{u'}{\hat{x}}+\pdv{v'}{\hat{y}}+\pdv{w'}{\hat{z}}\biggr)
    +\pdv{w'}{\hat{y}}+\pdv{v'}{\hat{y}}\biggr\}
    \Biggr] \\
&+\pdv{}{\hat{z}}\Biggl[-C_{p}
    \biggl(\bar{\rho}w'T'+\rho'\bar{w}T' +\rho'w'\bar{T}+\rho'w'T'\biggr) \\
&\qquad-\biggl\{
    \biggl(\bar{\rho}\bar{w}+\bar{\rho}w'+\rho'\bar{w}+\rho'w'\biggr)
    \biggl(u'u'+v'v'+w'w'\biggr) \\
&\qquad+2\biggl(\bar{\rho}w'+\rho'\bar{w}+\rho'w'\biggr)
    \biggl(\bar{u}u'+\bar{v}v'+\bar{w}w'\biggr)
    +\rho'w'\biggl(\bar{u}\bar{u} +\bar{v}\bar{v}+\bar{w}\bar{w}\biggr)\biggr\} \\
&\qquad+w'\mu\biggl\{
    \pdv{u'}{z}+\pdv{w'}{x}+\pdv{v'}{z}+\pdv{w'}{y}+2\pdv{w'}{z}-\frac{2}{3}
    \biggl(\pdv{u'}{x}+\pdv{v'}{y}+\pdv{w'}{z}\biggr)
    \biggr\} \Biggr] \,.
\end{align*}
\endgroup
\end{document}

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