Slippage effect on interfacial destabilization driven by standing surface acoustic waves under hydrophilic conditions

In the present work, we theoretically analyze the influence of the slippage phenomenon on the atomization via surface acoustic waves of a millimeter-order water drop deposited over a hydrophilic substrate. The analysis is conducted by considering, in the first place, a standing surface acoustic wave acting at the free surface of the parent drop. Subsequently, the lubrication theory is applied to the flow field governing equations to derive an evolution equation of the air-liquid interface in terms of the acoustic capillary number and the Navier-slip coefficient. Such an equation's numerical solution leads to a simplified drop model, depicting the spatiotemporal deformation of the free surface under the influence of slippage phenomenon and predicts the atomization threshold once the instability length at the induced capillary waves is achieved. Our numerical simulations show that the high-frequency acoustic excitation under consideration leads to the development of a standing wave at the free surface, which oscillates at a viscous-capillary resonance frequency on order ${10}^4$ Hz. Moreover, a spreading phenomenon on the fluid drop is induced, strongly linked to the magnitude of the acoustic capillary number. In this scenario, the slippage under hydrophilic conditions has a noticeable impact on the free surface dynamics, causing smaller aerosol characteristic diameters in comparison with the no-slip case. In this context, the present study provides an analytical expression that calculates the droplet diameter in terms of the slip coefficient. In the process, we postulate the slippage phenomenon as a valuable means to control the parent drop's deformation mechanism and, therefore, the aerosol characteristic diameter.

Figure 1

Evolution of the free surface $h(x,t)$, of a slender drop (with $H\ll L$) due to the excitation forcing ${\mathrm{\Pi }}_{ac}(x,t)$ by a standing SAW that propagates from the substrate to the interface, acting in the direction of the unit normal vector ($\mathbf{n}$). An initial drop profile, $h(x,0)=Hexp[-20{(x/L)}^2]$ [18], with a contact angle ${\theta }_c$ below ${90}^{\circ }$ is considered. The slender drop is subject to the Navier slip condition at the solid-liquid boundary once acoustic excitation begins (See inset $A$). This scenario traduces in implementing a hydrophilic material, placed between the substrate and the drop, which allows a nonzero tangential fluid velocity $u$ at the wall.

Figure 2

Spatiotemporal evolution surface $F$ that depicts the deformation process of a single drop profile along the dimensionless time $\overline{t}$. Case (a) shows a stress ratio quantified at $\mathcal{C}=10.0$ and results in a quasistable deformation process that keeps away the drop of being atomized, i.e., acoustic forcing is nearly balanced respect capillary forces at the interface. However, case (b) with $\mathcal{C}=50.0$, suggests a strong dominance of the acoustic stress over the surface tension stabilizing effect that results in a drop profile susceptible to be atomized.

Figure 3

Spatiotemporal evolution of an air-liquid interface subjected to SAW excitation, in terms of $\overline{h}=h(\overline{x},\overline{t})$. Three values of $\mathcal{C}$ are considered (10.0, 30.0, 50.0), distributed in three columns (a)–(c). Each column contains a sequence of four subfigures (i)–(iv), portraying the first cycle of the induced standing wave at the free surface. In this context, each subfigure shows three drop profiles developed under a particular slip coefficient: —–${\beta }_0=0.0$, $\_\_\_$ ${\beta }_0=0.05,$ and $͟$ ${\beta }_0=0.1$. All of the profiles have been obtained under an acoustic excitation with $\overline{\mathrm{\Omega }}={10}^4$ and $\overline{\kappa }\sim \mathcal{O}\left(10\right)$ (see Sec. 2).

Figure 4

Spatiotemporal evolution of an air-liquid interface subjected to SAW excitation, in terms of $\overline{h}=h(x,\overline{t})$. Three values of $\mathcal{C}$ are considered (10.0, 30.0, 50.0), distributed in three columns (a), (b), (c). Each column contains a sequence of three subfigures (i)–(iii), portraying the influence of slippage phenomenon at the drop's aspect ratio, $\varepsilon$. In this context, each subfigure shows three drop profiles developed under a particular slip coefficient: —–${\beta }_0=0.0$, $\_\_\_$ ${\beta }_0=0.05$ and $͟$ ${\beta }_0=0.1$. All the drop shapes have been obtained under an acoustic excitation with $\overline{\mathrm{\Omega }}={10}^4$ and $\overline{\kappa }\sim \mathcal{O}\left(10\right)$.

Figure 5

Temporal evolution of the aspect ratio $\varepsilon$ under the influence of four different slip coefficients—${\beta }_0=0.0,0.05,0.1,$ and 0.2—and two particular acoustic capillary numbers: $\mathcal{C}=10$ and $\mathcal{C}=50$. We note panel (a) illustrates the temporal evolution of the aspect ratio obtained by keeping fixed $\mathcal{C}$ at 10 and varying the slip coefficient in the range referred to above. Moreover, panel (b) has been developed in a similar manner, however, the acoustic capillary number is now fixed at 50. On both cases, our numerical estimations have been plotted using square symbols of different colors—each color is associated with a particular slip coefficient value—whereas the continuous lines depict a regression fit of our numerical data, based on the least-squares method. In this context, a power-law relationship, in the form $\varepsilon =B{\overline{t}}^n$, can be used to analytically describe the dependence between $\varepsilon$ and $\overline{t}$, except for the time interval $0\le \overline{t}\lesssim 2$ at the no-slip case depicted in panel (a). During such time interval the drop's aspect ratio results to be constant, as can be appreciated in more detail at the inset $\mathbf{A}$. Finally, let us note that panels (a) and (b) have been developed under a dimensionless excitation frequency $\overline{\mathrm{\Omega }}\sim \mathcal{O}\left({10}^4\right)$ and a wave number $\overline{\kappa }\sim \mathcal{O}\left(10\right)$.

Figure 6

The aspect ratio ($\varepsilon$) evolution of a drop exposed to SAW atomization against the slip coefficient (${\beta }_0$) is shown in panel (a) under different acoustic capillary numbers. The numerical estimations, obtained for our drop model are represented by geometric shapes whereas the dashed lines represent a regression fit of our data. In this context we have found the relationship $\varepsilon =K{\beta }_0^{-b}$ for ${\beta }_0\gtrsim 0.01$ (let us note the regression coefficients $K$ and $b$—obtained under several acoustic capillary numbers— are specified on Table 1). Panel (b) shows several droplet diameter ($d$) estimations, with and without the presence of slippage, for the SAW atomization of a parent drop. Specifically, our numerical estimations for $d$—developed under different values for $\mathcal{C}$—are plotted as geometric shapes whereas the regression fit of our numerical data is plotted through a blue dashed line. The regression fit suggest $d$ scales linearly respect the quantity ${\left(\varepsilon \mathcal{C}\right)}^{2/3}{f}^{-1}$. Let us note the inset $\mathbf{A}$ offers a major resolution of our numerical data which have been plotted using different colors—each one of them associated to a specific value of $\mathcal{C}$— and various shapes which are related to a particular value of the slip coefficient.

Figure 7

Numerical measurement of the dimensionless contact line speed, $\overline{V}$, of our drop model under different values of the slip coefficient and the acoustic capillary number. Specifically, $\overline{V}$ is the absolute value of the speed at which the two contact lines of our two-dimensional model symmetrically draw apart from the $\overline{z}$ axis as time progresses. In this context, let us recall we are considering a dimensionless frequency $\overline{\mathrm{\Omega }}\sim \mathcal{O}\left({10}^4\right)$ and a wave number $\overline{\kappa }\sim \mathcal{O}\left(10\right)$. Such values, in physical units, correspond to typical excitation parameters with $f_e\left(\mathrm{Hz}\right)\sim {10}^7$ and $\lambda \left(\mathrm{m}\right)\sim {10}^{-4}$. Specifically, the contact line speed has been obtained under four different slip coefficients: ${\beta }_0=0.0,0.05,0.1,$ and 0.2 and under four acoustic capillary numbers.