Coiling of a viscoelastic fluid filament

A viscous filament falling from a certain height onto a surface can form coils resulting from a buckling instability. This coiling phenomenon has been extensively studied for Newtonian fluids. However, the effect of liquid viscoelasticity has not been fully explored. Here we present experimental measurements of the coiling performance using Newtonian fluids and viscoelastic Boger fluids. Compared to Newtonian fluids with a comparable viscosity, the onset of viscoelastic fluid coiling is delayed; the coiling frequency is found to be smaller for viscoelastic fluids under the same experimental conditions. We show that these differences in coiling performance are due to the prevalence of extensional viscosity in viscoelastic fluids. Moreover, we find that the coiling frequency curves of different Newtonian fluids can be collapsed using a gravitational frequency and length scaling. This frequency and length scaling can also be used to collapse the data for Boger fluids considering the extensional viscosity instead of shear viscosity, confirming the importance of this property on filament coiling phenomenon for viscoelastic liquids.

Figure 1

(a) Scheme of the experimental setup: viscous fluids released from a nozzle (diameter, $d_n$) located above the bottom plate. Typical images of a viscous filament coiling: (b) Boger fluid and (c) Newtonian fluid.

Figure 2

Shear stress $\tau$ and normal stress $N_1$ for glucose-water-PAA Boger fluids B1. Shear stress data are fitted to a power-law model (blue line); the viscosity is nearly constant since $n\in$ [0.9, 1]. Normal stress data are fitted to a quadratic normal stress behavior (red line).

Figure 3

Coiling frequency as a function of falling height for different experiments with Newtonian fluids (a) using the scaling from Ribe [10] and (b) using the scaling of Eq. (6) and single pendulum frequency. The nondimensional group $({\mathrm{\Pi }}_1,{\mathrm{\Pi }}_2)$ for the present Newtonian fluids are (103.3, 0.61), (71.7, 0.56), and (21.1, 0.41), respectively.

Figure 4

(a) Filament coiling frequency and (b) filament radius as a function of falling height for both Newtonian and viscoelastic fluids; the falling height, $H$, is normalized by the kinematic shear viscosity $\nu$.

Figure 5

(a) Filament coiling frequency and (b) filament radius as a function falling height for both Newtonian and viscoelastic fluids: the falling height is normalized by the proposed extensional viscosity (characterized by Trouton ratio, Tr) listed in the figure.

Figure 6

(a) Independently measured Trouton ratio (circles) as a function of Hencky Strain; the horizontal dash lines show the proposed Trouton ratio for the corresponding fluids, respectively; (b) the Trouton ratio as a function of Weissenberg number.

Figure 7

(a) Coiling radius for both Newtonian and Boger fluids using extensional viscosity in Fig. 5. Filament clouds (overlapping images) near the frequency jump [(3) to (4)]: Newtonian fluids N3 (b), N1 (c), and Boger fluid B1 (d), respectively. The falling height increases from left to right.

Figure 8

Comparison of (a) falling height and (b) frequency at the jump between experimental observations ($H_j$ and ${\mathrm{\Omega }}_j$) and predictions ($H_1$ and ${\mathrm{\Omega }}_1$) using Eqs. (10) and (11).

Figure 9

(a) Sequence of sample images showing the thinning fluid bridge between the nozzle and the substrate sessile drop; (b) radius evolution of the fluid bridge in time for three tested fluids, filament radius normalized by nozzle radius $r_0$.