3.1. Oscillation frequency

The simulation and experimental results for the oscillation frequency of the lateral mode $\omega$ are shown in figure 4. The results of previous experiments by C–K and Sharp, Farmer & Kelly (Reference Sharp, Farmer and Kelly2011), and theoretical models by C–K and B–S are also included for comparison. It is observed that the experimental data for different drop volumes collapse approximately, when $\omega$ is normalized by $\omega _c$. This scaling relation $\omega \sim \omega _c$ indicates $\omega \sim V_d^{3/2}$, which is consistent with previous observations for a minor gravity effect (Noblin et al. Reference Noblin, Buguin and Brochard-Wyart2004). The normalized frequency $\omega /\omega _c$ for different $V_d$ still varies slightly due to the effect of $Bo$. For the experiments of C–K ($0.04\lesssim Bo \lesssim 0.19$) and Sharp ($0.16\lesssim Bo \lesssim 0.66$), the surface is horizontal, $\omega /\omega _c$ increases with $Bo$ (Sakakeeny & Ling Reference Sakakeeny and Ling2021), while for the present experiment ($0.12\lesssim Bo \lesssim 0.16$) that uses vertical surfaces, $\omega /\omega _c$ decreases with $Bo$ (as shown later in figure 7). When $Bo$ decreases, the experimental results for different surface orientations approach to the simulation results for $Bo =0$.

Figure 4.

Present simulation, model and experimental results for the frequency of the lateral oscillation mode as a function of the equilibrium contact angle $\theta _0$ for $Bo =0$, compared with previous models and experiments by C–K, Sharp et al. (Reference Sharp, Farmer and Kelly2011) and B–S. The experiments of C–K and Sharp are for sessile drops on a horizontal surface, for which the normalized frequency increases with $Bo$, while the present experiments are for a vertical surface, where the frequency decreases with $Bo$. The arrows indicate the variation of the normalized frequency as $Bo$ increases. When $Bo$ decreases, the experimental results for different surface orientations approach the simulation and model results for $Bo =0$.

The B–S model assumes $Bo=0$, so the equilibrium drop is a spherical cap. The inviscid linear stability theory is used to predict the frequency of the oscillation. The B–S predictions generally agree well with simulation results, see figure 4. Small discrepancies are observed for $\theta _0\gg {90}^{\circ }$, which may be due to the simplified contact line boundary conditions used in the B–S model (Sakakeeny & Ling Reference Sakakeeny and Ling2020). The C–K model predicts the frequency by a different approach. The shape oscillation of the drop is modelled as a harmonic oscillator, so the frequency is $\omega =\sqrt {k/m_d}$, where $m_d$ is the drop mass. The effective spring constant $k$ is related to surface tension, which serves as a restoring force to bring the drop back to its equilibrium state. The C–K model is more convenient to use since an explicit expression of $\omega$ is given, but it significantly under-predicts the frequencies, as shown in figure 4 and also in the original paper of C–K.

3.2. An inviscid model to predict oscillation frequency

A new inviscid model is developed in this study. The goal is to achieve an explicit expression to accurately predict the oscillation frequency. Furthermore, the new model incorporates the effect of finite $Bo$, which is ignored in B–S and C–K models. We take the equilibrium state for $Bo=0$ as the reference state, where the $y$-coordinate of the centroid and the drop surface area are $y_c= y_{c0}=0$ and $S=S_0$, respectively. The surface energy of the drop is $E_s=\sigma (S-S_0)$. The surface area $S-S_0$ can be expanded as a function of $y_c-y_{c0}$

(3.1)\begin{equation} \frac{S-S_0}{S_0} = \eta\left(\frac{y_c-y_{c0}}{R_0}\right)^2 + \xi\left(\frac{y_c-y_{c0}}{R_0}\right)^4. \end{equation}

Due to symmetry, only the even-order terms remain and the terms higher than fourth order are truncated, assuming small oscillation amplitudes. The relation between $S$ and $y_c$ during oscillation is assumed to be similar to that for a static equilibrium drop under different body forces (Celestini & Kofman Reference Celestini and Kofman2006). As a result, the parameters $\eta$ and $\xi$ depend only on $\theta _0$ and can be obtained from the solution for static drops. The Surface Evolver code was used to obtain the static solutions for different body forces for a given $\theta _0$, which are then used to measure $\eta$ and $\xi$ following (3.1). The procedures are similar to our previous study (Sakakeeny & Ling Reference Sakakeeny and Ling2020) and thus are not repeated here. The results for $\eta$ and $\xi$ are shown in figure 5(a). For convenient use of the results, fitted correlations are provided

(3.2)$$\begin{gather} \eta(\theta_0) = a_0 \left[ \exp \left(a_1 (\cos\theta_0+1) \right)-1\right] , \end{gather}$$

(3.3)$$\begin{gather}-\xi(\theta_0) = b_0 \left[ \exp \left(b_1 (\cos\theta_0+1) \right)-1\right], \end{gather}$$

where the fitting coefficients $[a_0,a_1]=[0.52.0.99]$ and $[b_0,b_1]=[0.20,2.11]$. In the asymptotic limit $\theta _0\to {180}^{\circ }$, both $\eta$ and $\xi$ approach zero, since in that limit the lateral mode does not induce deformation. The static solutions also indicate that $\eta >0$ and $\xi <0$ for all $\theta _0$.

Figure 5.

Model parameters (a) $\eta$ and $\xi$ and (b) $\zeta$ as functions of $\theta _0$; (c) $E_k$ as a function of $E_{kc}$ and (d) temporal evolution of $E_k$ and the approximation $\zeta E_{kc}$ for $\theta _0={140}^{\circ }$ and $Bo =0$.

The gravitational potential energy is expressed as $E_g=m_d g(y_c-y_{c0})$, and the overall potential energy is $E_p=E_s+E_g$. The equilibrium state is located at the minimum of $E_p$, namely $\textrm {d}E_p/\textrm {d}y_c=0$, which yields

(3.4)\begin{equation} 2\eta(y_{c1}/R_0) + 4\xi(y_{c1}/R_0)^3 + Bo _1=0, \end{equation}

where $Bo _1 = mgR_0/\sigma S_0=0.42(2 + \cos \theta _0)^{1/3} (1-\cos \theta _0)^{-1/3} Bo$. The cubic equation can be solved for the equilibrium centroid position $y_{c1}$, near which $E_p$ can be expanded as

(3.5)\begin{equation} E_p=\frac{1}{2}\frac{\textrm{d}^2 E_p}{\textrm{d}y_c^2}(y_c-y_{c1})^2=\sigma S_0 \eta_1\left(\frac{y_c-y_{c1}}{R_0}\right)^2, \end{equation}

where $\eta _1=\eta + 6 \xi (y_{c1}/R_0)^2$. Although the exact solution exists for (3.4), we use the approximation $y_{c1} \approx -{ Bo _1}/{2\eta }$ by neglecting the cubic term. Then we get the approximation

(3.6)\begin{equation} \eta_1\approx\eta + \frac{3\xi}{2\eta^2} Bo_1^2. \end{equation}

The kinetic energy of the drop fluid is $E_k=\rho _l \int _{V_d} |\boldsymbol {u}|^2/2\, \textrm {d}V$, which is evaluated in the simulation by the integral over the domain $E_k = \rho _l \int f (u^2 + v ^2 + w^2)\, \textrm {d}V$, where $f$ is the liquid volume fraction, and $u$, $v$ and $w$ are the velocity components. If the drop moves as a rigid body, the kinetic energy would be $E_{kc}=m_d v_c^2/2$, where $v_c=\textrm {d}y_c/\textrm {d}t$ is the $y$-component of the centroid velocity. Due to the internal flow with respect to the centroid induced by the shape oscillation, $E_k>E_{kc}$ (Sakakeeny & Ling Reference Sakakeeny and Ling2021). Furthermore, as shown in figure 5(c), $E_k$ is approximately a linear function of $E_{kc}$ when the drop oscillates. This is due to the fact that the temporal evolutions of energy of the internal flow and the magnitude of $v_c$ are in phase. As a result, the kinetic energy correction factor, defined as $\zeta =E_k/E_{kc}$, is approximately a constant in time and can be measured from the simulation results as shown in figure 5(c). Then we can approximate

(3.7)\begin{equation} E_k\approx \zeta E_{kc} = \zeta m_d v_c^2/2. \end{equation}

The temporal evolutions of $E_k$ and $\zeta E_{kc}$ are plotted in figure 5(b) and the latter is affirmed to be generally a good approximation for all time. Further tests show that $\zeta$ varies little over $Bo$ and is only a function of $\theta _0$, see figure 5(b). The fitted function for $\zeta$ is given as

(3.8)\begin{equation} \zeta(\theta_0) = c_1\mathrm{erf}\left[(\cos\theta_0+1)/c_2\right]+1, \end{equation}

where the fitting coefficients $[c_1,c_2]=[0.14,0.93]$. In the asymptotic limit $\theta _0\to {180}^{\circ }$, $\zeta \to 1$ since there is no shape oscillation and the lateral mode corresponds to rigid-body translation of the drop.

The total energy $E=E_p+E_k$ is constant in time, so $\textrm {d}E/\textrm {d}t=0$. With (3.5) and (3.7) we obtain

(3.9)\begin{equation} k(y_c-y_{c1})+m_d \frac{\textrm{d}^2 (y_c-y_{c1})}{\textrm{d}t^2} = 0 , \end{equation}

where $k=2\sigma S_0 \eta _1/(\zeta R_0^2)$. The oscillation frequency $\omega =\sqrt {k/m_d}$ can be expressed as

(3.10)\begin{equation} \frac{\omega^2}{\omega_c^2} = \frac{12 \left( \eta + \dfrac{3\xi}{2\eta^2} Bo _1^2\right)}{\zeta(2+\cos\theta_0)(1-\cos\theta_0)} . \end{equation}

In the limit of $Bo =0$, (3.10) reduces to

(3.11)\begin{equation} \frac{\omega^2}{\omega_c^2} = \frac{12 \eta}{\zeta(2+\cos\theta_0)(1-\cos\theta_0)} . \end{equation}

The results of (3.11) are plotted in figure 4, which are shown to agree remarkably well with the simulation results. The C–K model for $Bo=0$ can be written as

(3.12)\begin{equation} \frac{\omega_{CK}^2}{\omega_c^2} = \frac{6 \eta}{(2+\cos\theta_0)(1-\cos\theta_0)} . \end{equation}

Comparing the (3.11) and (3.12) for $Bo=0$, it is shown that the present model yields better predictions due to the incorporation of the contribution of the kinetic energy of the internal flow through $\zeta$.

3.3. Model predictions for finite Bond numbers

Since numerical solutions are required for the model parameters $\eta$, $\xi$ and $\zeta$, the present model, similar to the C–K model, is semi-analytical. Nevertheless, once these parameters are established, as shown in (3.2), (3.3), (3.8), the explicit expression (3.10) is ready to predict the natural frequencies for the lateral oscillation of drops pinned on a vertical surface with different $\theta _0$ and $Bo$.

The predictions of $\omega /\omega _c$ by the present, B–S and C–K models are compared with the present experimental results for finite $Bo$ in table 3. The results are also plotted in figure 6. Since the C–K model neglects the effect of kinetic energy of the fluid motion within the drop, it significantly under-predicts the frequency, by up to 21 %. In contrast, the B–S model, which neglects the effect of $Bo$, over-predicts the frequency, by up to 17 %. As shown in figure 4, the present model yields very similar results to the B–S model for $Bo =0$ in the range of $\theta _0$ considered in the present experiments. By incorporating the contribution of $Bo$, the present model yields more accurate predictions than the other two models for all cases. The relative errors are less than 12 % for all cases. Excluding the cases for copper, the errors are less than 9 %.

Figure 6.

Comparison between experimental results of the normalized frequency $\omega /\omega _c$ and model predictions with the present, B–S and C–K models.

Table 3.

Comparison between experimental results of the normalized frequency $\omega /\omega _c$ and model predictions with the present, B–S and C–K models.

The present model over-predicts the frequency for all cases, compared with the experimental results. There are a couple of potential reasons that contribute to the discrepancy between the model and experimental results. Similar to the B–S and C–K models, the contact area is taken to be a circle with radius as $R_c=R_0\sin {\theta _0}$ in the present model. This ideal contact area corresponds to the spherical-cap shape of the drop when gravity is absent. The contact area is then fixed during oscillation, since the contact line is pinned. In the experiment, the contact area may be slightly smaller than the ideal contact area considered in the model, since gravity is present when the drop is deposited on the surface. The smaller contact area will result in a weaker constraint on the drop and a lower oscillation frequency. Furthermore, although in the experiment the oscillation amplitude is controlled to be very small, the contact line may still not be fully pinned and may have oscillated with a very small amplitude near the equilibrium position. In contrast, the contact line is perfectly pinned in the model and simulation. The mobility of the contact line will result in a reduction of the oscillation frequency for the same $\theta _0$ and $Bo$, according to our previous study (Sakakeeny & Ling Reference Sakakeeny and Ling2021).

In the experiment, $Bo$ is varied only in a very small range ($0.12< Bo < 0.15$). Drops with larger $Bo$ are very unstable and may slide even when the surface vibration amplitude is very small. To better pin the contact line and to enable experiments for larger $Bo$, a more sophisticated surface treatment, such as that by Chang et al. (Reference Chang, Bostwick, Daniel and Steen2015), is required, which will be relegated to future work. In order to validate the model for higher $Bo$, the model predictions are compared with the fully resolved simulation results.

The model and simulation results of the normalized frequency $\omega /\omega _c$ as a function of $Bo$ for $\theta _0=100^{\circ }$ are shown in figure 7(a). It is seen that $\omega /\omega _c$ decreases with $Bo$ and excellent agreement between the present model and the simulation results is observed. It is conventionally considered that the effect of gravity on the the supported drop oscillation is negligible if $Bo <1$, namely when the drop radius is smaller than the capillary length (Chang et al. Reference Chang, Bostwick, Daniel and Steen2015). However, it can be seen that the frequency decreases for more than 25 % when $Bo$ increases from 0 to 0.62.

Figure 7.

Comparison between the model and simulation results for oscillation frequency as a function of $Bo$ and $\theta _0$.

The variation of $\omega /\omega _c$ over $Bo$ can be better depicted if $\omega$ is normalized by its $Bo =0$ counterpart $\omega _{ Bo =0}$. From (3.10), it can be shown that

(3.13)\begin{equation} {\omega^2}/{\omega_{ Bo =0}^2} = 1+ ({3\xi}/{2\eta^3}) Bo _1^2. \end{equation}

The equilibrium drop results show that $\xi <0$ for all $\theta _0$ (see figure 5a), indicating that that the curvature of the $S$–$y_c$ curve (3.1) decreases as $y_c$ increases. As a result, $\omega /\omega _c$ decreases with $Bo$, see figure 7(a). Furthermore, the leading correction term for ${\omega ^2/\omega _{ Bo =0}^2}$ scales with $Bo ^2$. Therefore, for finite and small $Bo$, $\omega /\omega _c$ decreases quadratically with $Bo$. It is difficult to vary $Bo$ over a wide range in experiments to examine the scaling relation, but that can be done in the simulations. We have change $g$ from 0 to 0.9 $\textrm {m }\,\textrm {s}^{-2}$, yielding $0\le Bo \le 0.78$. The model and simulation results for $\theta _0={70}^{\circ }, 100^{\circ }$ and ${130}^{\circ }$ are plotted in figure 7(b). Since drops with large $\theta _0$ become unstable for large $Bo$, the range of $Bo$ for $\theta _0={130}^{\circ }$ is smaller than other cases. The model predictions (3.13) agree very well with the simulation results, validating the quadratic scaling law revealed in the model.

In the present study, $Bo$ is varied by changing the gravity $g$. Therefore, the results imply that the natural frequency of the lateral oscillation mode will increase if the drops oscillations occur in a reduced gravity environment. This knowledge is important to the surface vibration dropwise condensation applications in the space station.

The present model can also be used to estimate the frequencies for the drops of different volume. For a given gravity, $Bo$ increases with $V_d$. As a result, $\omega /\omega _c$ decreases as $V_d$ increases. Since the model is based on an inviscid assumption and the viscous effect on the oscillation frequency (Lamb Reference Lamb1932; Prosperetti Reference Prosperetti1980) is not accounted, it is valid only for $Oh\ll 1$. For low-viscosity liquids like water, $Oh$ for millimetre drops is generally small. Yet, as $Oh$ increases as the drop size decreases, caution is required if the model is applied to very small drops.