this is for holding javascript data
Daniel D'Orazio edited untitled.tex
over 9 years ago
Commit id: ec99b92979b00f56120ebe26c5c22b4fade9e995
deletions | additions
diff --git a/untitled.tex b/untitled.tex
index a603730..74bbc4b 100644
--- a/untitled.tex
+++ b/untitled.tex
...
\begin{equation}
\left(\mathbf{v} \cdot \nabla \right)\mathbf{v} + \frac{1}{\rho}\nabla P + \nabla \Phi_G -\nu\left[\nabla^2\mathbf{v} + \frac{1}{3}\left(\nabla \cdot \mathbf{v} \right) \right] = 0
\end{equation}
where it is understoond that gravitational potential $\Phi_G$ and the coeficcient of kinematic viscosity $\nu$ are time independent. Now use the idenity $\left(\mathbf{v} \cdot \nabla \right)\mathbf{v} = \frac{1}{2} \nabla \left( \mathbf{v} \cdot \mathbf{v}\right) - \mathbf{v} \times \left( \nabla \times \mathbf{v}\right)$
Integrate neglect viscosity for now, and integrate the momentum equation
for a steady, nonviscous, flow along a streamline
from a refernce point to the point of evaluation
\begin{equation}
\int{\mathbf{ds}\cdot
(\rm{momentum equation})} \nabla \left( v^2\right) - \mathbf{v} \times \left( \nabla \times \mathbf{v}\right) + \frac{1}{\rho}\nabla P + \nabla \Phi_G -\nu\left[\nabla^2\mathbf{v} + \frac{1}{3}\left(\nabla \cdot \mathbf{v} \right) \right]}
\end{equation}
you get
\begin{equation}