Question

Velocity vector of a fluid in a cylindrical vessel is of the form v=ω×r, where ω is the constant rotation vector. Show that div V = 0?

Answer 1

Could you please clarify what you mean by "div V"?

Answer 2

divergence of V

Answer 3

Ah... further in Calc III than I know... sorry. :P

Answer 4

not a problem. Thanks for looking

Answer 5

\[\nabla\cdot\vec{v}=\nabla\cdot(\vec\omega\times\vec{r})=(\nabla\times\vec{\omega})\cdot\vec{r}-(\nabla\times\vec{r})\cdot\vec{\omega}\]
\[\vec\omega-constant\quad\Rightarrow\quad\nabla\times\vec{\omega}=0\]
\[\vec{r}-only\enspace radial\quad\Rightarrow\quad\nabla\times\vec{r}=0\]
\[\Rightarrow\quad\nabla\cdot\vec{v}=0\]