# Fluid rope trick investigated

Original entry by Joerg Fritz, AP225 Fall 2009

## Contents

## Source

"Fluid rope trick investigated"

L. Mahadevan, W.S. Ryu, A.D.T. Samuel: *Nature*, 1998, 392, pp 140 to 141

## Keywords

Buckling, Viscosity, Surface tension, Instability

## Summary

This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.

## The Math

What sets the speed <math>\Omega</math> with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the
flow rate *Q*, the gravity constant *g*, the filament radius *r*, and the height *h* from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.

Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).

Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. The viscous torque scales like <math>f_v = \int \sigma r dA \approx \mu U r^4 / R^2</math> where R is a characteristic radius of curvature, which is approximately the radius of the coil structure on the toast. The inertial forces scale like <math>f_i \approx T\rho \Omega r^2 R^3</math> and the torque due to this force is <math>f_i \ r^2 R^2</math>. A balance of the two torques together with the continuity equation leads to

<math>\Omega \approx Q^{4/3} r^{-10/3} (\mu/\rho)^{-1/3}</math>

Experimental results, shown in figure 2 agree very well with this scaling, indicating that all the other neglected effects, like surface tension, air drag and non-Newtonian effects are infact not important in this problem.

## Conclusion

Soft matter is everywhere. And even for apparently very complicated phenomena there is sometimes an elegant scaling that gives the desired answer without involved numerical simulations. The beauty of this result is that it is testable on a daily basis. We can derive three quantitative predictions that can be tested during the next weekend breakfast over toast and honey:

- The coiling starts for a critical height for the falling filament of honey
- Above this height a further increase of height has no influence on the rotation speed
- An increase in the flow rate should lead to a marked increase in coiling velocity.