Unified Viscous-to-inertial Scaling in Liquid Droplet Coalescence

TL;DR

This paper develops a unified theory for liquid droplet coalescence that accurately describes the transition from viscous to inertial regimes, matching experimental data without free parameters.

Contribution

It introduces an exact, parameter-free solution for the neck radius evolution during droplet coalescence, unifying viscous and inertial scaling laws.

Findings

Exact solution reproduces known scaling laws

Unified description captures viscous-inertial crossover

Data collapse onto a single curve across viscosities

Abstract

This letter presents a theory on the coalescence of two spherical liquid droplets that are initially stationary. The evolution of the radius of a liquid neck formed upon coalescence was formulated as an initial value problem and then solved to yield an exact solution without free parameters, with its two asymptotic approximations reproducing the well-known scaling relations in the viscous and inertial regimes. The viscous-to-inertial crossover observed by Paulsen et al. [Phys. Rev. Lett. 106, 114501 (2011)] is also recovered by the theory, rendering the collapse of data of different viscosities onto a single curve.

Figures

Equations

$$\frac{r_R}{R}\approx \frac{R}{2R_0}\ll 1,$$

$$\rho (u_z{\partial }_z{u}_r+u_r{\partial }_r{u}_r)=-{\partial }_{r}p+\mu \left[{\partial }_z^2{u}_r+{\partial }_r^2{u}_r+{\partial }_r(\frac{u_r}{r})\right],$$

$$\begin{array}{rl}& {\int }_{1\to 2}\left[\frac{1}{2}\rho {\partial }_r{u}_r^2+{\partial }_{r}p-\mu \left({\partial }_z^2{u}_r+{\partial }_r^2{u}_r+{\partial }_r(\frac{u_r}{r})\right)\right]\mathrm{d}r\\ & =\frac{1}{2}\rho U^2+p_2-p_1-\mu \left({\int }_0^R{\partial }_z^2{u}_r\mathrm{d}r+({\partial }_r{u}_r){|}_2+\frac{U}{R}\right)=0,\end{array}$$

$$\begin{array}{rl}& \frac{1}{2}\rho U^2-\sigma \left(\frac{1}{r_R}-\frac{1}{R}+\frac{2}{R_0}\right)\\ & -\mu \left({\int }_0^R{\partial }_z^2{u}_r\mathrm{d}r+({\partial }_z{u}_z){|}_2+\frac{2U}{R}\right)=0.\end{array}$$

$${\partial }_z^2{u}_r\approx \frac{({\partial }_z{u}_r){|}_{z=r_R}-({\partial }_z{u}_r){|}_{z=0}}{r_R}=\frac{({\partial }_r{u}_z+\omega ){|}_{z=r_R}}{r_R},$$

$${\int }_0^R{\partial }_z^2{u}_r\mathrm{d}r\approx \frac{1}{r_R}\left((u_z){|}_4^3+{\int }_0^R\omega {|}_{z=r_R}\mathrm{d}r\right),$$

$${\omega }_0(r^{\prime })=\frac{U}{r_v}{e}^{-{\left(\frac{r^{\prime }}{r_v}\right)}^2},$$

$$-\frac{U}{r_R}{\int }_0^{\frac{R}{r_v}}{e}^{-{\left(\frac{r^{\prime }}{r_v}\right)}^2}\mathrm{d}\left(\frac{r^{\prime }}{r_v}\right)=-\frac{\sqrt{\pi }U}{2r_R}\mathrm{erf}\left(\frac{R}{r_v}\right)\approx -\frac{\sqrt{\pi }U}{2r_R},$$

$$\frac{1}{2}\rho U^2-\sigma \left(\frac{1}{r_R}-\frac{1}{R}+\frac{2}{R_0}\right)-\mu \left(-\frac{\sqrt{\pi }U}{2r_R}+\frac{2U}{R}\right)=0.$$

$$\rho U^2-\frac{4\sigma R_0}{R^2}+\frac{2\sqrt{\pi }\mu R_{0}U}{R^2}=0,$$

$$\frac{\rho {\dot{R}}^{\ast 2}{L}^2}{T^2}-\frac{2\sigma D_0}{R^{\ast 2}{L}^2}+\frac{\sqrt{\pi }\mu D_0{\dot{R}}^{\ast }}{R^{\ast 2}LT}=0,$$

$$\frac{\rho L^2}{T^2}=\frac{\sigma D_0}{L^2}=\frac{\mu D_0}{LT},$$

$${\dot{R}}^{\ast 2}-\frac{2}{R^{\ast 2}}+\frac{\sqrt{\pi }{\dot{R}}^{\ast }}{R^{\ast 2}}=0.$$

$$\begin{array}{r}{t}^{\ast }=\frac{\sqrt{\pi }{R}^{\ast }}{4}+\frac{\sqrt{\pi }}{8}\left[R^{\ast }\sqrt{\frac{8R^{\ast 2}}{\pi }+1}+\frac{\sqrt{\pi }}{2\sqrt{2}}{\sinh }^{-1}\left(\frac{2\sqrt{2}{R}^{\ast }}{\sqrt{\pi }}\right)\right].\end{array}$$

$$t^{\ast }\approx \frac{R^{\ast 2}}{2\sqrt{2}}+O(R^{\ast }).$$

$$t^{\ast }\approx \frac{\sqrt{\pi }}{4}\left[\frac{3R^{\ast }}{2}+\frac{\sqrt{\pi }}{4\sqrt{2}}\ln \left(\frac{2\sqrt{2}{R}^{\ast }}{\sqrt{\pi }}+1\right)\right]\approx \frac{\sqrt{\pi }{R}^{\ast }}{2}+O(R^{\ast 2}).$$

Full text

Droplet Coalescence Eggers et al. (1999); Aarts et al. (2005); Thoroddsen et al. (2005); Paulsen et al. (2011); Xia et al. (2017) is a ubiquitous phenomenon involving impact or contact of dispersed two-phase flows Yarin (2006); Zhang and Law (2011); Thoraval et al. (2012); Tran et al. (2013); Kavehpour (2015); Tang et al. (2016); He et al. (2019). Among the various relevant problems, the initial coalescence of two liquid droplets has been of core interest. The first quantitative analysis of sphere coalescence was provided by Frenkel Frenkel (1945) based on the assumption of internal Stokes flow; however, the result was commented as “misleading” by Hopper Hopper (1993), who gave an analytical solution for the coalescence of two cylindrical droplets of radius $R_{0}$ for viscous sintering. His studies Hopper (1990, 1992) show that the time evolution of the radius $R$ of the neck (or bridge) between the droplets approximately satisfies $t\sim-R/\ln{R^{\ast}}$ where $R^{\ast}=R/R_{0}$. Later, Eggers et al. Eggers et al. (1999) considered the three-dimensional coalescence and attained $R^{\ast}\sim-t^{\ast}\ln{t^{\ast}}$ for $R^{\ast}<0.03$, where $t^{\ast}=t/\tau_{v}$ ($\tau_{v}=\mu R_{0}/\sigma$ with $\mu$ and $\sigma$ being the dynamic viscosity of the liquid and the surface tension coefficient, respectively). For larger $R^{\ast}$, they Eggers et al. (1999); Duchemin et al. (2003) argued that the neck flow goes beyond the Stokes regime to the inertial (or inviscid) regime, and further arrived at the $1/2$ power-law scaling, $R^{\ast}\sim(t/\tau_{i})^{1/2}$ with the time scale being $\tau_{i}=(\rho R_{0}^{3}/\sigma)^{1/2}$, where $\rho$ is the liquid density.

Recent advances in the high-speed digital imaging Aarts et al. (2005); Thoroddsen et al. (2005); Yao et al. (2005), state-of-art probing techniques Case and Nagel (2008); Fezzaa and Wang (2008); Paulsen et al. (2011), and numerical simulation Sprittles and Shikhmurzaev (2012, 2014) enabled researchers to scrutinize the early stages of drop coalescence when $R^{\ast}\ll 1$. As a result, the $1/2$ power-law scaling was confirmed by many experimental Wu et al. (2004); Aarts et al. (2005); Thoroddsen et al. (2005); Burton and Taborek (2007); Fezzaa and Wang (2008); Case (2009) and numerical Duchemin et al. (2003); Eiswirth et al. (2012); Pothier and Lewis (2012); Gross et al. (2013); Sprittles and Shikhmurzaev (2012) studies. The same scaling was also observed for droplet coalescence on substrate Ristenpart et al. (2006); Hernández-Sánchez et al. (2012); Eddi et al. (2013). However, the experiments of Aarts et al. Aarts et al. (2005) and Thoroddsen et al. Thoroddsen et al. (2005) indicate that the viscous regime is well predicted by the linear scaling of $R^{\ast}\sim t^{\ast}$, noting that most of their data were in the $R^{\ast}>0.03$ range. This linear correlation was also corroborated by other studies Yao et al. (2005); Burton and Taborek (2007); Paulsen et al. (2011).

More recently, research interests have been directed towards the crossover (or transition) between the viscous and inertial regimes. The first direct evidence of the crossover from $R^{\ast}\sim t^{\ast}$ to $R^{\ast}\sim(t^{\ast})^{1/2}$ was reported by Burton and Taborek Burton and Taborek (2007). By equating the characteristic velocities from the two scaling laws, they derived the crossover length, $l_{c}\sim\mu(R_{0}/\rho\sigma)^{1/2}$, which was later confirmed by Paulsen et al. Paulsen et al. (2011); Paulsen (2013), who further obtained the crossover time, $\tau_{c}\sim\mu^{2}(R_{0}/\rho\sigma^{3})^{1/2}$. With these time and length scales, Paulsen et al. applied a fitting curve, $(R/l_{c})^{-1}\sim(t/\tau_{c})^{-1}+(t/\tau_{c})^{-1/2}$, to collapse the neck evolutions of distinct viscosities, which points to a universality in droplet coalescence. To theoretically explore this universality, we derived a scaling model for the viscous-to-inertial combined coalescence regime Xia et al. (2018), with two scaling constants determined by fitting experimental data.

In this letter, we present a theory that significantly rigorizes the previous scaling model Xia et al. (2018) and contains no empirical constant. A schematic of the neck between two merging droplets of initial radius $R_{0}$ is shown in Fig. 1. The neck radius, $R$, is defined as the minimum radial distance from the $z$-axis to the neck. Under capillary pressure difference, the neck expands out at a speed of $U(t)$. We assume the flow to be

(i) Quasi-steady, meaning the flow acceleration is mainly associated with the convection induced by the neck movement.

(ii) Quasi-radial, meaning the neck region can be treated as a ring of radius $R$ and width $2r_{R}$, which is driven by a distributed and quasi-radially directed capillary force Eggers et al. (1999). The capillary force is related to two principle curvatures, $1/R$ and $1/r_{R}$ Gross et al. (2013), with the latter being the effective curvature in the $zr$-plane.

(iii) Localized, meaning the significant velocity gradients are restricted to the vicinity of the neck as illustrated in Fig. 1. This assumption accords with the finding of Paulsen et al. Paulsen et al. (2011) that the flow extends over a length comparable to the neck width rather than the neck radius. It follows that the main vortical structure has a length of $O(r_{R})$, and the origin (point 1) is considered as the far field where the velocity gradients are effectively zero.

(iv) Geometrical self-similar, so that the neck width satisfies the simple geometric relation, $r_{R}/R=\tan{(\theta/2)}$. Under the coalescence regime of $R\ll R_{0}$, we have $\tan{(\theta/2)}\approx\theta/2\approx R/(2R_{0})$ and, consequently,

[TABLE]

which is consistent with other studies Paulsen et al. (2011); Gross et al. (2013), but different from Eggers et al. Eggers et al. (1999) who assumed $r_{R}/R$ to be higher-order small. This could be responsible for their logarithmic scaling law, which is believed to occur at very early stage of droplet coalescence Xia et al. (2018), beyond the resolution of existing experiments.

For the axisymmetric and quasi-steady flow, the $r$-direction N-S equation is expressed as

[TABLE]

where $u_{z}$ and $u_{r}$ are the velocity components in the $z$- and $r$- directions, respectively, and $p$ is the pressure. Along the $r$-axis, $u_{z}$ and $\partial_{z}u_{r}$ are all zeros owing to the condition of symmetry, so the term $u_{z}\partial_{z}u_{r}$ vanishes in Eq. 2. We now integrate Eq. 2 along the $r$-axis from point 1 ($r=0,z=0$) to point 2 ($r=R,z=0$) as

[TABLE]

where the subscripts 1 and 2 denote the quantities associated with point 1 and 2, respectively. In attaining Eq. LABEL:eq:2, we have also applied $(u_{r})|_{1}=0$ according to the axisymmetric condition, $(\partial_{r}u_{r})|_{1}=0$ following assumption (iii), and $(u_{r})|_{2}=U(t)$. As the present theory concerns the coalescence of liquid droplets in a gaseous environment, the liquid-gas interface can be treated as a free surface, across which the capillary pressure jump is $p_{\infty}-p=-2\mu\boldsymbol{n}\cdot\boldsymbol{S}\cdot\boldsymbol{n}+\sigma\kappa$ (Tryggvason et al., 2011), where $p_{\infty}$ is the ambient gas pressure, $\boldsymbol{n}$ and $\kappa$ are the unit normal vector and curvature of the interface, respectively, and $\boldsymbol{S}$ is the rate-of-strain tensor. Accordingly, the pressures at the far-side droplet and the neck satisfy $p_{\infty}-p_{1}=-2\sigma/R_{0}$ and $p_{\infty}-p_{2}=-2\mu(\partial_{r}u_{r})|_{2}+\sigma(1/r_{R}-1/R)$, respectively. Here, $p_{1}$ serves as the pressure at the far-side droplet according to assumption (iii). Subtracting the two equations yields $p_{2}-p_{1}=-\sigma(1/r_{R}-1/R+2/R_{0})+2\mu(\partial_{r}u_{r})|_{2}$, which can be plugged into Eq. LABEL:eq:2 to obtain

[TABLE]

Note that the continuity equation, $\partial_{z}u_{z}+\partial_{r}u_{r}+u_{r}/r=0$, has been used in the above derivation.

The quasi-radial assumption (ii) implies $u_{z}=0$ around point 2 and further $(\partial_{z}u_{z})|_{2}=0$ in Eq. LABEL:eq:2-1. Furthermore, $\partial_{z}^{2}u_{r}$ can be expressed as

[TABLE]

with $(\partial_{z}u_{r})|_{z=0}=0$ by axisymmetry and $\omega=\partial_{z}u_{r}-\partial_{r}u_{z}$ being the vorticity. Eq. 5 essentially gives a leading-order approximation based on linearizing the strain rate near the plane of symmetry. Integrating Eq. 5 from point 4 ($r=0,z=r_{R}$) to point 3 ($r=R,z=r_{R}$) yields

[TABLE]

with $(u_{z})|_{3}=0$ by assumption (ii) and $(u_{z})|_{4}=0$ by assumption (iii).

Lacking a priori knowledge of the vorticity field, we seek an approximation of $\omega|_{z=r_{R}}$ based on the computational observation that in the $orz$ plane the vortex-dynamical effect of the neck movement induces two opposite-sign vortices that are centered around the two edges of the neck, as illustrated in Fig. 2. This physical picture is also consistent with assumption (iii). The radial vorticity decay displayed in Fig. 2 further implies that the vortex is analogous to a Batchelor vortex Batchelor (1964) and has a Gaussian vorticity distribution as

[TABLE]

where $r^{\prime}=R-r$ is the radial location relative to the vortex center located at the neck interface, and $r_{v}$ is an effective radius of the vortex core. Eq. 7 is also similar to the Oseen-Lamb vortex Wu et al. (2006), which is an analytical solution to the vorticity diffusion equation. Recognizing that $\omega_{0}(r^{\prime})=\omega(r)=\omega(R-r^{\prime})$ for $z=r_{R}$ and $0\leq r\leq R$, Eq. 6 can be further derived as

[TABLE]

with $R/r_{v}\gg 1$ given by assumption (iii) and Eq. 1.

We now plug in Eq. 8 to cast Eq. LABEL:eq:2-1 in the form,

[TABLE]

Applying Eq. 1 and balancing the leading-order terms of Eq. 9 yields

[TABLE]

which can be combined with $\dot{R}=\mathrm{d}R/\mathrm{d}t=U$ to derive

[TABLE]

where $D_{0}=2R_{0}$, $R^{\ast}=R/L$, $\dot{R}^{\ast}=\dot{R}/U$, and $T=L/U$, with $L$, $U$, and $T$ being the characteristic length, velocity, and time scales, respectively.

The experimental studies of Paulsen et al. Paulsen et al. (2011); Paulsen (2013) imply the existence of a unified formula for the neck movement given the length and time are scaled properly. Let Eq. 11 be such a formula, we have

[TABLE]

yielding $L=OhD_{0}$ and $T=\mu OhD_{0}/\sigma$ where $Oh=\mu/\sqrt{\rho\sigma D_{0}}$ is the Ohnesorge number. Note that $L$ and $T$ match exactly with the viscous-to-inertial crossover scales found by previous studies Burton and Taborek (2007); Paulsen et al. (2011); Paulsen (2013). Accordingly, Eq. 11 takes the dimensionless form,

[TABLE]

We can integrate Eq. 13 with the initial condition $R^{\ast}(t^{\ast}=0)=0$, where $t^{\ast}=t/T$, to obtain the exact solution,

[TABLE]

Eq. 14 readily dictates the asymptotic behaviors associated with the viscous and inertial regimes. For the inertial regime, $R^{\ast}\gg\sqrt{2\pi}/4$, Eq. 14 yields

[TABLE]

For the viscous regimes, $R^{\ast}\ll\sqrt{2\pi}/4$, Eq. 14 yields

[TABLE]

Eq. 16 can be also reduced to the form of $R\sim t\sigma/\mu$, which is void of any characteristic length. This can be interpreted that the physics of the viscous regime is intermediate self-similar Barenblatt (1996). To evaluate our theory, we first write Eq. 15 in the dimensional form of $R/R_{0}\approx c_{1}(t/\tau_{i})^{1/2}$ with $c_{1}=2$, which recovers the $1/2$ power-law scaling for the inertial regime. Similarly, Eq. 16 can be expressed in the dimensional form of $R/R_{0}\approx c_{2}t/\tau_{v}$ with $c_{2}=2/\sqrt{\pi}$, which gives the linear scaling relation observed from experiments of high-viscosity droplets Aarts et al. (2005). As a reference, the fitting coefficients $c_{1}=1.68$ and $c_{2}=1$ were obtained by Paulsen Paulsen (2013) although different values were reported by others Aarts et al. (2005); Thoroddsen et al. (2005); Wu et al. (2004).

Fig. 3 shows existing experimental data of various $Oh$, corresponding to a variety of fluid types, such as water, silicon oil, and glycerol-salt-water mixture, that are of distinct fluid properties as summarized in the Supplementary Materials See Supplementary Materials for more numerical details and a parameter list of the experimental data in Fig. 3. . It is observed that all data tend to collapse onto a single curve, well predicted by the current theory. Considering the assumptions and approximations made in the derivation, the agreement between theory and experiment is quite satisfactory. The theory also captures the asymptotic behaviors of the data in the viscous and inertial regimes. Specifically, the $R^{\ast}\sim t^{\ast}$ and $R^{\ast}\sim\sqrt{t^{\ast}}$ scaling relations show up as $R^{\ast}\ll 1$ and $R^{\ast}\gg 1$, respectively, whereas a clear inflection point can be identified around $R^{\ast}\sim O(1)$ and $t^{\ast}\sim O(1)$, marking the transition from viscous to inertial. It should be emphasized that, although empirical Paulsen (2013) and semi-empirical Xia et al. (2018) models exist previously, this letter presents the first theory that resolves the unified scaling in the viscous-to-inertial combined coalescence process.

Next, we provide further validation of the theory against droplet coalescence simulations of various viscosities. The simulation setup is specified in the Supplementary Materials See Supplementary Materials for more numerical details and a parameter list of the experimental data in Fig. 3. . The neck interface evolution for a representative case of $Oh=0.0016$ is shown in the inset plot of Fig. 4. Similar simulations were conducted for $Oh=$ 0.0082, 0.0179, 0.0718, 0.1795, 0.8975, and 4. The corresponding neck radius evolutions are presented in the main plot of Fig. 4. It is seen that each simulation data set originates from a finite neck radius, causing the simulated evolution to deviate from the theory. Nevertheless, the later-stage coalescence behavior is less affected by the simulation onset, as each neck evolution curve gradually approaches and then follows its designated scaling, showing that the overall trend of the simulation curves are still captured by the theory. Similar neck evolution behaviors were also observed from previous simulations Sprittles and Shikhmurzaev (2012, 2014).

Last, this theory suggests that $R^{\ast}=R/(OhD_{0})$ is a criterion segmenting the different coalescence regimes. Although it involves both $R/D_{0}$ and $Oh$, for different fluids, $Oh$ is the parameter that eventually decides whether the inertial regime could arrive. This is evident from both Fig. 3 and Fig. 4 that data in the inertial regime generally corresponds to smaller $Oh$ and vice versa. This criterion has important practical use. For example, Aarts et al. Aarts et al. (2005) considered Data 3 (20 mPa s silicon oil) and Data 4 (50 mPa s silicon oil) to be within the inertial regime, whereas Fig. 3 clearly shows that Data 3 mainly covers the crossover regime and Data 4 extends from the viscous regime to the crossover regime.

To summarize, this letter presents a theory for the neck evolution during initial coalescence of binary liquid droplets. We have derived and validated a unified solution that applies to the viscous, viscous-to-inertial crossover, and inertial regimes of droplet coalescence. This provides a fundamental framework to support the prominent scaling laws as well as the crossover behaviors observed from previous experimental studies.

We would like to acknowledge the support from the Hong Kong RGC/GRF (PolyU 152217/14E and PolyU 152651/16E) and the “Open Fund” of State Key Laboratory of Engines (Tianjin University, No. K2018-12).