Breakup of liquid jets: Thermodynamic perspectives

Abstract

Breakup of a liquid jet into a chain of droplets is common in nature and industry. Previous researchers developed profound mathematic and fluid dynamic models to address this breakup phenomenon starting from tiny perturbations. However, the morphological evolution of the jets at large amplitude perturbations is still an open question. Here, we report a concise thermodynamic model based on the surface area minimization principle. Our results demonstrate a reversible breakup transition from a continuous jet via droplets towards a uniform-radius cylinder, which has not been found previously, but is observed in our simulations. This new observation is attributed to a geometric constraint, which was often overlooked in former studies. We anticipate our model to be a shortcut to tackle many similar highly nonlinear morphological evolutions without solving abstruse fluid dynamic equations for inviscid fluids.

Figures (3)

• Figure 1: Morphological evolution of liquid jets. (a) Formation of droplets when water trickles down from a water-tap. (b) 2D projection of a perturbed jet whose surface follows $r_o=R_0^o+a_1^o\cos(2\pi z/\lambda)$, where $a_1^o$ and $\lambda$ are the amplitude and the wavelength of the initial harmonic perturbation, respectively. The volume of the jet is $\int_0^\lambda \pi r_o^{2} dz=\pi [(R_0^{o})^2+\frac{1}{2}(a_1^{o})^2]\lambda$ and we define $R_u^2=(R_0^{o})^2 +\frac{1}{2}(a_1^{o})^2$ as the mean radius of the jet. (c), (d), (e), Morphological evolution of jets with different amplitudes and wavelengths via phase-field simulations. All the results in (c) (tiny perturbations) coincide with Rayleigh's theory Rayleigh1879. Conversely, in (d)-ii, the breakup wavelength deviates from Rayleigh's prediction and this deviation also appears in (e)-ii. In (e)-iii, we observe an unusual breakup: a perturbed jet $\rightarrow$ dispersed droplets $\rightarrow$ continuous cylinder.
• Figure 2: Surface area landscape. (a), (b), (c), Surface area of jets as function of all the possible values of the first two Fourier coefficients ($a_1/R_0$ and $a_2/R_0$) with normalized wavelengths $\lambda/(2\pi R_u)=0.37$ (Fig. \ref{['fig:1']}iii), $1.05$ (Fig. \ref{['fig:1']}i), $0.80$ (Fig. \ref{['fig:1']}ii), respectively. The black/green circles denote the evolution routes of $a_1/R_0$ and $a_2/R_0$ from the phase-field simulations. The black/green dashed curves represent the evolution paths from the gradient descent method. The gray circles in (c) embrace a barrier interval along the horizontal dot-dashed line $a_2/R_0=0$. The hatched and dotted regions in (c) are partitioned by the isolines (magenta lines) of the saddle point of the surface area landscape.
• Figure 3: Stability diagram. (a) The normalized critical breakup wavelength $\lambda_{\text{crit}}/(2\pi R_\text{u})$ as a function of the scaled initial amplitude $a_1^o/R_0^o$. The red and blue squares depict the results from the gradient descent method and the Allen-Cahn model, respectively. The red dashed line is the fitting curve for the red squares. The green diamonds demonstrate the simulation results from a fluid dynamics model (see supplemental document), which are in good agreement with our model. The dot-dashed line denotes the geometric criterion. The gray shaded region illustrates all the barrier intervals shown in Fig. \ref{['fig:2']}(c) for different wavelengths. (b) A reversible separation in II from a continuous jet via separated ellipsoid-shaped droplets towards a uniform-radius cylinder. (c) Regular breakup in III, where $R_s$ is the radius of the resulting spheroids.