Instabilities and elastic recoil of the two-fluid circular hydraulic jump

Abstract

The two-fluid circular hydraulic jump, also called “rinsing flow,” is a common process where a jet of one liquid impinges upon a layer of a second liquid. We present an experimental analysis of rinsing flows using a high-speed camera and model fluids to decouple the effect of shear-thinning and elasticity. Varying the rheology of the coating fluid produced several types of instabilities at both the air–liquid interface and liquid–liquid interface. Layered “stepped jumps” and “crowning” on the rim of the jumps were both suppressed by fluid elasticity, while Saffman–Taylor fingering patterns showed strong dependence on both shear-thinning and normal stresses. In addition, the hydraulic jump evolution was quantitatively determined using a laser triangulation technique, and “recoil” of the jump front resulting from fluid elasticity was observed. Our work shows that the non-Newtonian two-fluid circular hydraulic jump is very complex, and the instabilities that arise also introduce additional complications when developing theoretical models.

Appendix: Cone-like jumps

In the previous sections, all the experiments used a jet of water to impinge on a layer of coating liquid with significantly higher viscosity (viscosity ratio ∼200). We now turn our attention to the case where the jet fluid has a higher viscosity than the coating fluid (viscosity ratio <1). Experiments were performed where the jet liquids were either a Newtonian glycerol–water (25 wt% glycerol) solution, a 0.0080 wt% xanthan gum solution, a 0.10 wt% polyacrylamide solution (molecular weight 15 × 10⁶ g/mol), or a Boger fluid (0.005 wt% polyacrylamide molecular weight 15 × 10⁶ g/mol and 0.24 wt% glycerol in water) impinging on water as the coating fluid. Under these lower concentrations, the density and surface tension of all the test fluids were found to be approximately equal to water (ρ _j and σ _j ). The viscosities of all the jet liquids were approximately $0.003 \,\hbox{Pa}\,\hbox{s}$, making the viscosity ratio of these trials ∼0.33. However, because the shear viscosity values of these liquids were below the instrument limit of the shear and extensional rheometer, their viscosities were determined using a Cannon–Fenske glass capillary viscometer (Cannon Instrument). Because the capillary viscometer cannot accurately measure the viscosity of shear-thinning fluids, the results presented in this section are qualitative. The flow rate of the jet (U _j ) was increased to compensate for the increased viscosity of the jet fluid so that the Reynolds number of the trials was the same as the previous trials. The system’s jet Reynold’s number (Re _j ), jet Ohnesorge number (Oh _j ), jet Weber number (We _j ), and characteristic shear rate $\dot{\gamma}$ are found to be 6,900, 5.7 × 10⁻³, and 1,500, respectively.

Figure 12 shows screen shots of these experiments. The most striking property of these flow profiles is that in all cases, the height of the hydraulic jumps was much higher than the previous cases when the viscosity ratio was much greater than one, because the coating liquids were not able to dissipate the inertial energy from the impinging jet. All trials reached maximum jump heights ∼5 times the diameter of the jet. Another notable feature is the “cone-like” jumps exhibited by the non shear-thinning glycerol–water fluid and the Boger fluid. For the two shear-thinning fluids, the jumps were more cylindrical-like, and peripheral jetting could be seen on the hydraulic jump rims. For the inelastic xanthan gum solution, the jetting created a “splash” and many small individual drops, while for the elastic polyacrylamide solution, the secondary drops were connected to the rim of the hydraulic jump due to elongational properties of the liquid. Similar observation was made by Roux et al. (2003) in their drop impact experiments onto a film of the same liquid, where they saw thin liquid threads connecting the upper rim of the crown and the secondary drops for Boger fluids.

Fig. 12

Screen shots of the rinsing flows with the testing jet fluid being a glycerol–water, b polyacrylamide, c xanthan gum, and d Boger15M impinging on water. All of the screen shots were taken ∼0.01 s after the rinsing flow began