I have imagined a vector field describing helices moving at different speeds:
$V(x, y, z, z_0) = <cz_0, a \ sin(y\ k\ z_0), b \ cos(z\ k\ z_0)> \ \forall x,y,z \ s.t. y^2+(z_0-z)^2=1, $
I did my best to describe the vector speed $V$ of the flow in a field of many helixes. My trick was to consider each helix separately The helical flow increases with speed as $z$ increases. $x$ is horizontal distance from the origin and $z_0$ measures the centerline of the helix in terms of vertical distance (the $z$ direction) from the origin. The horizontal speed is proportional to the vertical position: $\frac{\delta V_x}{\delta z_0}=c$. Similarly, I did my best to model the swirling part of the helical motion as being proportional to the height. The idea is that helixes which are lower should generally move slower that the helixes which are higher up.
I am having difficulty applying Stoke's theorem to this problem. I am interested in the net flux associated with some control volume surrounding these helixes. Here, I try to use a spherical ($\theta=arccos(\frac{x}{y})$ and $\phi=arccos(\frac{x}{z})$) control volume:
$ \int V \bullet\hat{dr} = \int_0^{2\pi} \int_0^{2\pi} \Delta \times V d\phi d\theta = $
$ \int_0^{2\pi} \int_0^{2\pi} <\frac{\delta }{\delta x}, \frac{\delta }{\delta y}, \frac{\delta }{\delta z}> \times <cz_0, a \ sin(y\ k\ z_0), b \ cos(z\ k\ z_0)> d\phi d\theta =$
$\int_0^{2\pi} \int_0^{2\pi} $
$<(\frac{\delta }{\delta y} b \ cos(z\ k\ z_0) - \frac{\delta }{\delta z} a \ sin(y\ k\ z_0)), (\frac{\delta }{\delta z} cz_0 - \frac{\delta }{\delta x} b \ cos(z\ k\ z_0)), (\frac{\delta }{\delta x} a \ sin(y\ k\ z_0) - \frac{\delta }{\delta y} cz_0) > d\phi d\theta$
I am not sure how to proceed from here. Is there a better way to formulate this problem? Am I applying Stoke's theorem correctly? While an analytical solution would be ideal, I am not sure how to even solve this numerically.
Edit Really I only modeled one of these cylinders. I want to model many (infinite) cylinders with these velocity properties. I think a cylinder is a good surrogate for a helix here. I think I can do that by integrating across $z_0$ assuming a maximum helix centerline height $Z$.
$V(x, y, z) = \int_0^Z <cz_0, a \ sin(y\ k\ z_0), b \ cos(z\ k\ z_0)> dz_0\ \forall x,y,z \in R$
$V(x,y,z) = <c \frac{Z^2}{2}, \frac{a_2}{y} \ (cos(y\ k\ Z)-1), \frac{b_2}{z} \ sin(z\ k\ Z)> $