Viscoelasticity Breaks the Symmetry of Impacting Jets

A jet of a Newtonian liquid impacting on a wall at right angle spreads as a thin liquid sheet which preserves the radial symmetry of the jet. We report that for a viscoelastic jet (solution of polyethylene glycol in water) this symmetry can break; close to the wall, the jet cross section becomes faceted and radial steady liquid films (wings) form, which connect the cross-section vertices to the sheet. The number of wings increases with increasing the viscoelastic relaxation time of the solution, but also with increasing jet velocity and decreasing distance from the jet nozzle to the wall. We propose a mechanism for this surprising destabilization of the jet shape, which develops perpendicularly to the direction expected for a buckling mechanism, and explain these dependencies. We also discuss the large-scale consequences of the jet destabilization on the sheet spreading and fragmentation, which show through the faceting of hydraulic jumps and of suspended (Savart) sheets.

Figure 1

Impact of a viscoelastic jet (PEO $4\times {10}^6\mathrm{g}/\mathrm{mol}$ at $2\mathrm{g}/\mathrm{L}$ in water, $a=0.71\mathrm{mm}$, mean velocity $u=8.46\mathrm{m}/\mathrm{s}$) extruded from a Pasteur pipette onto a glass plate. (a),(c),(d) At small plate-to-nozzle distance $l$, steady triangular radial liquid films (wings) connect the jet to the sheet that spreads on the plate. (a) Side view showing three of the five wings (the jet image at the bottom is a reflection on the sheet). (c),(d) Bottom view through the plate. Close to the plate, the jet has a cusped pentagonal cross section (appearing in black). The five equally spaced wings initiate at the cross-section vertices and generate a wake downstream on the sheet. (b) For larger $l$ no wing forms.

Figure 2

Typical evolution of the number $n$ and size $s$ of the wings versus, (a) the mean jet velocity [$a=1.38\mathrm{mm}$, $l=3.75\mathrm{mm}$, and $\lambda =9.6\mathrm{ms}$. The red solid line is the prediction for $n$ from Eq. (2)], and (b) the plate-to-nozzle distance $l$ ($a=1.38\mathrm{mm}$, $u=3.07\mathrm{m}/\mathrm{s}$, and $\lambda =8.2\mathrm{ms}$).

Figure 3

(a),(b) Angle and perpendicular view of the transverse destabilization of a liquid sheet (elongated jet) impacting on a plate. The dashed line indicates the plate surface. (c) Schematics of the destabilization mechanism. The liquid is sheared at the nozzle wall and develops an elastic tension ${\sigma }_{\theta \theta }$ along the streamlines. On impact, where the air-liquid interface curves ($R_0>0$) due to the deflection by the plate, ${\sigma }_{\theta \theta }$ promotes the destabilization of the interface in the perpendicular direction ${\mathit{e}}_{\mathit{z}}$. By contrast, the velocity $u_s$ at the jet surface (set by the viscous relaxation of the shear, as the liquid travels from the nozzle to the plate) stabilizes the interface due to an inertial effect.

Figure 4

Dimensionless number of wings $n/N$ versus dimensionless jet velocity $u/U$, where $N$ and $U$ are computed from the parameter-free expressions of Eq. (3). All the experiments (with $a=0.77–1.38\mathrm{mm}$, $u=1.33–16.3\mathrm{m}/\mathrm{s}$, $l=0.5-15.0\mathrm{mm}$ and $\lambda =7.3-17.6\mathrm{ms}$) are presented. The solid line represents Eq. (2). The filled (hollow) circles stand for straight (convergent) nozzles. The color code represents the absolute number $n$ (from 0, dark blue, to the maximal value 34, dark red).

Figure 5

Large-scale consequences of the jet symmetry-breaking for $n=4$ to 6. (a),(b),(c) Suspended (Savart) liquid sheets formed by the impact on a disk twice as large as the jet (represented by the white disk in the center). The sheet radius (hence the sheet momentum) is minimal in the wings directions, and the liquid is ejected at the vertices of the sheet. (d),(e),(f) Hydraulic jumps formed by the impact on a large plate. The radius of the hydraulic jump (hence the sheet flow rate) is also minimal in the wings directions.