# Viscous losses in loop helical flow

Imagine flow of the following kind. The flow line for each flow particle is a helix with its axis bent so that the beginning attaches to the end.

How can I determine the viscous losses in this flow and how can I minimize such losses?

One such flow is $$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}\sin \left(t\right)\left(3+\cos \left(20t\right)\right)\\\cos \left(t\right)\left(3+\cos \left(20t\right)\right)\\\sin \left(20t\right)\end{bmatrix}$$ looking like

This can also be imagined as a vortex ring rotating around its axis of translational motion.

Another such flow is $$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}\sin \left(t\right)\left(3+\cos \left(.05t\right)\right)\\\cos \left(t\right)\left(3+\cos \left(.05t\right)\right)\\\sin \left(.05t\right)\end{bmatrix}$$ looking like

A flow path with infinite levels of helicity.

• No answers in two weeks leads me to hint (at the risk of biasing) the question. Is there an way to allow zero vorticity. – David Jonsson Jul 6 at 10:37