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Given velocity vector 

LaTeX Math Inline
body{\vec {u}}=(u_x,v,wu_y,u_z)
 the pathlines are defined as  is a bunch of curves 
LaTeX Math Inline
body{\vec {r}}_s (t) = \big( \, x(_s), \, y(_s), \, z(_s ) \, \big) \in \mathbb{R}^3
, parametrized by parameter the stem points
LaTeX Math Inline
body{\vec {r}}_{s0}
 at zero-time
LaTeX Math Inline
bodyst_0
, and solving the following equation:

LaTeX Math Block
alignmentleft
\begin{cases}

\displaystyle 

{\frac {d{\vec {xr}}_{P}}s(t)}{dt}}(t)={\vec {u}}_{P}s({\vec {xr}}_{P}(t)s,t)
\\[1.2ex]

{\vec {xr}}_{P}s(t_{0})={\vec {xr}}_{P0s0}

\end{cases}