Meandering Fluid Streams in the Presence of Flow-Rate Fluctuations

We report that meandering of a rivulet flowing down a nonerodible, partially wetting incline is triggered by flow-rate fluctuations and sustained by external noise forcing. In our experiments, the former is provided by an electronically controlled valve, and the latter is due to fluid droplets left on the surface by previous meanderings. We observe power-law behavior of the averaged spectrum of the deviations of the stream from its center line, which rules out the existence of a preferred wavelength in ongoing meandering. We derive a simple theoretical model of rivulet meandering from first principles, incorporating stream dynamics and external noise forcing. The model provides an accurate statistical description of the stream deviation from a nonmeandering path.

Figure 1

Laboratory rivulet flows, view from above. (a) Nonmeandering flow (constant flow rate) of a 50%-50% water-glycerine mix. (b) Meandering flow of the same fluid with the same mean flow rate, where fluctuations in the flow rate are induced by a pulsed electromagnetically operated valve. Coordinate axis $x$ corresponds to the center line of the nonmeandering rivulet, and $h(x)$ represents the deviation of the meandering rivulet from this center line. (c) Example of the three regimes present in transient downstream flow. 1: Steady nonmeandering flow; 2: meandering flow; 3: stream breakup.

Figure 2

(a) Growth of average amplitude of oscillations $⟨|h(x)|⟩$ with downstream distance. Data for three surfaces (same tilt angle and discharge rate) are shown. (b) Dimensionless spatial growth rate of $⟨|h(x)|⟩$ vs the static contact angle of the surface. (c) A histogram of droplet size distribution for three surfaces used in experiment. The horizontal scale is nondimensionalized by the cross-sectional area of the supply tube. The vertical scale uses the droplet count on the acrylic surface (highest among the three) for normalization. This figure demonstrates the relationship between contact angle, drop formation, and meandering.

Figure 3

Power spectra of the stream deviation from the center line on different substrates (as labeled in the plot) and for the model (the latter scaled by an arbitrary number). The solid line represents the $\alpha =-5/2$ power law. The noise amplitude in the model (a universal constant for all experiments) is fixed to achieve agreement at small wave numbers $k$. This agreement continues over all $k$ having physical relevance.

Figure 4

Averaged power spectrum of the time dependence of $h(x,t)$ for fixed $x$ (see text for details).