Sagging(s) of Viscous Filaments
Jacopo Seiwert
Merie Le Merrer
David Quere
PMMH-ESPC
Lay-language version of "An other viscous catenary"
Liquid threads are commonly found in nature (spider webs, saliva, honey threads) and exhibit remarkably subtle properties between those of viscous fluids and elastic threads. We discuss here the different shapes observed while stretching horizontally a viscous liquid such as honey. Two different families of shapes are observed: either a catenary, as if the liquid were solid (Figure 1 and Video 1), or a U-shape (Figure 2 and Video 2), depending on the initial dimensions of the filament.
The experiment is simple: a thread is formed by placing a given quantity of viscous fluid between two walls, which are rapidly pulled apart by a few centimeters, resulting in a cylinder of liquid of length L and diameter D bridging the two sides.
If L and D are both large the filament stretches uniformly: its shape is a catenary at any time. Conversely, thin and short filaments evolve in an unexpected way: their shape tends towards a “U”, with a thin horizontal portion joining two thicker vertical threads. Even more striking is the fact that U-shaped filaments descend very little (typically one centimeter) and then literally stop despite gravity, in contrast with catenaries that descend to depths much larger than L.
The shape of the filament is dictated by the repartition of liquid along its length. Indeed, the liquid inside the thread can drain towards the edges. If the liquid thread falls before draining, its diameter remains uniform and we observe a falling catenary. On the other hand, if it drains before falling, a non-uniform thread is observed, leading to a U shape.
To illustrate this point with a solid rope analog, we photographed two strings of the same length with fishing plumbs attached (see fig. 3). In the first case, the plumbs are equally spaced and the string has a catenary shape, whereas in the second case the mass is concentrated on the edges, leading to a U shape.
The drainage of fluid towards the edges is due to surface tension. Indeed, a liquid/air interface is associated with an energy cost proportional to the interface area, the factor γ being the surface tension of the liquid. This is the reason why small drops (rain drops, for example) have a spherical shape, which minimizes their surface. At a larger scale, gravity flattens drops when their radius is greater than the capillary length a of the liquid (a few millimeters for most liquids), defined by a² = γ/ρg.
In our system, the liquid cylinder thins to minimize its area in a typical time Tc = ηD/γ: the larger the surface tension γ, the greater the capillary effects then the smaller the drainage time. On the other hand, the greater the viscosity η of the liquid, the slower the capillary drainage.
To discuss the shape selection, we need to compare this drainage time with the time of fall Tf = η/ρgL. The ratio Tc/Tf writes: LD/a². If this quantity is greater than 1, the liquid does not have time to drain during the fall and we expect a catenary shape to be observed. If, however, this ratio is smaller than 1, a U form is predicted.
It is worth mentioning that this criterion does not involve the viscosity of the liquid, because both characteristic times are proportional to it. The final shape is only determined by the initial dimensions L, D and by the liquid characteristic length a.
This study belongs to a more general framework of liquid morphogenesis. When using non-Newtonian liquids, one can obtain a Y-shaped filament (video 3): instead of migrating towards the edges, the liquid now moves towards the center. Another cirumstance that we currently study is the inverse of the one presented here: a small filament is formed, when the walls are approached at a constant speed. If the speed is high enough, the filament buckles and forms a loop (video 4). When the latter experiment is repeated with a rotating wall, a twisted loop results, much like a twisted string (see video 5): indeed, the behavior of viscous filaments is similar to the one of elastic threads (as intuited long ago by Rayleigh).
Legends for the Videos
Video 1 and Video 2: Evolution of a viscous thread hanging between two plates 3 cm apart. Silicon oil 10,000 times more viscous than water was used in this experiment, and the movie is 10 times slower than real time. In the first video, the viscous filament is “thick” and long enough to hold its catenary shape throughout the fall, whereas in the second video the filament is thin enough to evolve towards a U-shape, then to break.
Video 3: Evolution of a viscous thread hanging between two plates 3 cm apart. We used here a non Newtonian fluid, which leads to a migration of fluid towards the center and a “Y” shape. The video is played at real time.
Video 4: A viscous filament (approximately 1.5 cm long) of very viscous silicon oil (1,000,000 times more viscous than water) is formed, then its edges are joined at a constant speed. For a high enough velocity, the filament buckles and forms an out of axis loop. The video is three times slower than real time.
Video 5: A viscous filament (approximately 1.5 cm long) of very viscous silicon oil (1,000,000 times more viscous than water) is formed, then its edges are joined at a constant speed. Moreover, its upper edge is rotated, which leads to the formation of a twisted loop. The video is three times slower than real time.