Governing equations on differential form: Difference between revisions

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==== Conservation of Mass ====
==== Conservation of Mass ====


\noindent If we apply the substantial derivative operator to density we get\\
If we apply the substantial derivative operator to density we get


\[\frac{D\rho}{Dt}=\frac{\partial \rho}{\partial t}+\mathbf{v}\cdot\nabla\rho\]\\
<math display="block">
\frac{D\rho}{Dt}=\frac{\partial \rho}{\partial t}+\mathbf{v}\cdot\nabla\rho
</math>


\noindent From before we have the continuity equation on differential form as\\
From before we have the continuity equation on differential form as


\[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0\]\\
<math display="block">
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0
</math>


\noindent which can be rewritten as\\
which can be rewritten as


\[\frac{\partial \rho}{\partial t} + \rho(\nabla\cdot\mathbf{v}) + \mathbf{v}\cdot\nabla\rho=0\]\\
<math display="block">
\frac{\partial \rho}{\partial t} + \rho(\nabla\cdot\mathbf{v}) + \mathbf{v}\cdot\nabla\rho=0
</math>


\noindent and thus\\
and thus


\begin{equation}
<math display="block">
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0
\label{eq:governing:cont:non}
</math>
\end{equation}\\


\noindent Eqn. \ref{eq:governing:cont:non} says that the mass of a fluid element with a fixed set of fluid particles is constant as the element moves in space.\\
Eqn. \ref{eq:governing:cont:non} says that the mass of a fluid element with a fixed set of fluid particles is constant as the element moves in space.


==== Conservation of Momentum ====
==== Conservation of Momentum ====
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