Crocco's equation

\section{Crocco's Equation}

\noindent The momentum equation without body forces\\

\[\rho\frac{D\mathbf{v}}{Dt}=-\nabla p\]\\

\noindent Expanding the substantial derivative\\

\[\rho\frac{\partial \mathbf{v}}{\partial t}+\rho\mathbf{v}\cdot\nabla\mathbf{v}=-\nabla p\]\\

\noindent The first and second law of thermodynamics gives\\

\[T\nabla s =\nabla h-\frac{\nabla p}{\rho}\]\\

\noindent Insert $\nabla p$ from the momentum equation\\

\[T\nabla s =\nabla h+\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\]\\

\noindent Definition of total enthalpy ($h_o$)\\

\[h_o=h+\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\Rightarrow \nabla h=\nabla h_o-\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)\]\\

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\noindent The last term can be rewritten as\\

\[\nabla\left(\frac{1}{2}\mathbf{v}\cdot\mathbf{v}\right)=\mathbf{v}\times(\nabla\times\mathbf{v})+\mathbf{v}\cdot\nabla\mathbf{v}\]\\

\noindent which gives\\

\[\nabla h=\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\mathbf{v}\cdot\nabla\mathbf{v}\]\\

\noindent Insert $\nabla h$ in the entropy equation gives\\

\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})-\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}+\frac{\partial \mathbf{v}}{\partial t}+\cancel{\mathbf{v}\cdot\nabla\mathbf{v}}\]\\

\[T\nabla s =\nabla h_o-\mathbf{v}\times(\nabla\times\mathbf{v})+\frac{\partial \mathbf{v}}{\partial t}\]\\