2
$\begingroup$

I am quite new to fluid dynamics and I cannot seem to understand the concept of vorticity. It is defined as $\nabla\times\vec{v}$ where $\vec{v}$ is the velocity vector. Now (working in cartesian coordinates) velocity: $\vec{v}=\frac{{\rm d} \vec{r}}{{\rm d} t}$ so that $\nabla\times\vec{v}=\nabla\times\frac{{\rm d} \vec{r}}{{\rm d} t}$. And (assuming interchangeable derivatives as is usual) $\nabla\times\vec{v}=\frac{{\rm d}}{{\rm d} t}\nabla\times\vec{r}\equiv 0$.

So because vorticity is even considered as a quantity and thus there have to be cases when it is non-zero, I have made an error. I can think of two sources. One, the spatial curl derivative in $\nabla$ is with respect to different variables (I doubt that). Two, the derivative is not interchangeable. Or is there another reason? Please try to explain in terms of definition of velocity as a time derivative of a position vector, rather than as it is usually done in terms of a given velocity field $(v_x(t,\vec{r}),v_y(t,\vec{r}),v_z(t,\vec{r}))$ without any apparent relation to $\vec{r}$. Thank you.

$\endgroup$
3
$\begingroup$

The answer is that the derivatives are not interchangeable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.