3.1. Rheological properties of CMCell and Carbopol solutions

Rheological measurements of the CMCell and Carbopol solutions are performed with a controlled stress rheometer (DHR-3, TA Instrument) using a parallel-plate geometry (with a diameter of 25 mm) at a controlled temperature of 25 $^{\circ }$C. Under a simple shear, the rheological properties of shear-thinning fluids are classically modelled by the power-law model $\mu =\kappa \dot {\gamma }^{n_{p}-1}$ (Kamişli & Ryan Reference Kamişli and Ryan2001; de Sousa et al. Reference de Sousa, Soares, de Queiroz and Thompson2007). However, the power-law model cannot predict the viscosity at the low-shear-rate region (Picchi et al. Reference Picchi, Poesio, Ullmann and Brauner2017, Reference Picchi, Ullmann, Brauner and Poesio2021) where the viscosity approaches to a constant value, known as the zero-shear-rate viscosity. Instead, the Ellis model (Reiner & Leaderman Reference Reiner and Leaderman1960) was proposed to capture such a viscosity plateau with a constitutive equation as

(3.1)\begin{equation} \frac{\mu}{\mu_0}=\frac{1}{1+(\tau/\tau_{1/2})^{\alpha-1}}, \end{equation}

where $\tau _{1/2}$ is the shear stress at which the viscosity is half of the Newtonian limit, while $\mu _0$ and $\alpha$ are the zero-shear-rate viscosity and the degree of shear-thinning, respectively.

Here, we consider the C-Y model, which has been used to describe emulsions, protein solutions and polymer melts (Myers Reference Myers2005; Picchi et al. Reference Picchi, Poesio, Ullmann and Brauner2017). The C-Y model is more convenient for experimental analysis since it expresses the viscosity as an explicit function of the shear rate (Carreau Reference Carreau1972; Pipe, Majmudar & McKinley Reference Pipe, Majmudar and McKinley2008; Morozov & Spagnolie Reference Morozov and Spagnolie2015). The constitutive equation of the C-Y model is $\mu =(\mu _0-\mu _\infty )(1+(\lambda \dot {\gamma })^a)^{(n_c-1)/a}+\mu _\infty$, where $\mu _0$, $\mu _\infty$, $n_c$ and $a$ are the zero-shear-rate viscosity, infinite-shear-rate viscosity, the power-law index and dimensionless parameter, respectively. Here, $\lambda$ is the inverse of a characteristic shear rate at which shear-thinning becomes apparent. Figures 2(a) and 2(b) demonstrate that the C-Y model can well capture the rheological behaviours for both the CMCell and Carbopol solutions in the range of shear rates over five orders of magnitude. We note that $\mu _\infty$ is neglected in the following discussion since $\mu _\infty /\mu _0< O(10^{-2})$ and therefore the constitutive equation for the C-Y model can be written as

(3.2)\begin{equation} \mu\approx\mu_0(1+(\lambda\dot{\gamma})^a)^{(n_c-1)/a}.\end{equation}

Figure 2.

(a) Rheogram of the glycerin and carboxymethyl cellulose (CMCell) solutions with different mass fractions: viscosity $\mu$ versus shear rate $\dot {\gamma }$. (b) Rheogram of the Carbopol solutions with different mass fractions: $\mu$ versus $\dot {\gamma }$. The dashed lines represent a fitting with the Carreau–Yasuda (C-Y) model. (c) Dimensionless parameter $a$ versus the power-law index $n_c$ in the C-Y model or the degree of shear-thinning $\alpha$ in the Ellis model. (d) Ellis number $El$ versus Carreau number $Cu$ in the current experiments. Error bars are smaller than the symbols.

A very recent theoretical work by Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021) derived the film thickness as a function of the Ellis number $El$ and the degree of shear-thinning $\alpha$ for an Ellis fluid, where $El$ is the ratio between the characteristic shear rate of the fluid and the characteristic shear rate in the liquid film as

(3.3)\begin{equation} El=\frac{\tau_{1/2}h}{U\mu_0}. \end{equation}

To compare our experimental measurements with the lubrication theory of Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021), we rewrite the C-Y model with a similar form to the Ellis model as

(3.4)\begin{equation} \frac{\mu}{\mu_0}=(1+{({Cu\tilde{\dot{\gamma}}})}^a)^{(n_c-1)/a} \quad \text{with} \ Cu = \frac{\lambda{U}}{h} \ \text{and} \ \tilde{\dot{\gamma}}={\frac{\dot{\gamma}}{U/h}},\end{equation}

where the Carreau number $Cu$ is the ratio between the effective shear rate $U/h$ in the film and the cross-over strain rate $1/\lambda$ (Datt et al. Reference Datt, Zhu, Elfring and Pak2015), and $\tilde {\dot {\gamma }}$ is the dimensionless shear rate. With the rheological data in figure 2(a,b), we used the method of least squares fitting to obtain all the parameters for the Ellis and C-Y models. Notably, the value of $a$ in the C-Y model is chosen to impose $n_c=1/\alpha$ for an analogy between the Ellis and C-Y models, and we find $a\approx 0.45 n_c ^{-1}$, as shown in figure 2(c). Next, for all the experimental cases, once the bubble speed and the film thickness are determined, $El$ and $Cu$ are calculated as shown in figure 2(d). In the current work, $Cu$ is found to be inversely proportional to $El$ with an experimentally fitted relation of $Cu\approx 1.3El^{-1}$. The rheological parameters of the CMCell and Carbopol solutions are reported in table 2, and we will focus on the effects of $Cu$ and $n_c$ on the deposition dynamics of the working fluids.

Table 2.

Rheological properties of the CMCell and Carbopol solutions with different mass fractions computed by the C-Y model.

We note that Carbopol solutions can exhibit both yield stress and shear-thinning behaviours. However, we use a low mass fraction of Carbopol to diminish the yield stress effect (Spiers, Subbaraman & Wilkinson Reference Spiers, Subbaraman and Wilkinson1975; Ma et al. Reference Ma, Bai, Chang, Yi, Jiao and Du2015), so that only the shear-thinning behaviour dominates. We further justify this consideration by fitting our rheological data of Carbopol with the Herschel–Bulkley model $\tau =\tau _y+k_{HB}\dot {\gamma }^{n_{HB}}$, where $\tau _y$ is the yield stress and $k_{HB}$ is the consistency factor. A dimensionless number $B=\tau _y R/\sigma$ is suggested to compare the yield stress to the capillary pressure (Deryagin & Levi Reference Deryagin and Levi1964; Laborie et al. Reference Laborie, Rouyer, Angelescu and Lorenceau2017). In our experiments, $B$ is $O (10^{-3})$ while $B$ is $O (1)$ in Laborie et al. (Reference Laborie, Rouyer, Angelescu and Lorenceau2017), thus we neglect the yield stress effect in the following discussion.

3.2. Scaling of the film thickness with the modified capillary number

The film thickness measured in the experiments with the CMCell and Carbopol solutions is shown in figures 3(a) and 3(b), respectively, as a function of the zero-shear-rate capillary number $Ca_0$, as well as the modified capillary number $\widehat {Ca}$. As shown in figure 3(a), compared to a Newtonian fluid at the same $Ca_0$, the bubble forms a thinner liquid film in both the CMCell and Carbopol solutions because of the shear-thinning effect. The thinner liquid film formed in the Carbopol solutions compared with that in the CMCell solutions at the same $Ca_0$ results from that the stronger shear-thinning effect in the Carbopol solutions than that in the CMCell solutions, as indicated by lower power-law indices of the Carbopol solutions compared with those of the CMCell solutions in table 2.

Figure 3.

(a) Non-dimensional liquid film thickness h/R as a function of $Ca_0$. The black line represents prediction of (1.1). (b) $h/R$ versus $\widehat {Ca}^{2/{2n_p+1}}$, where $\widehat {Ca}=\kappa (U/R)^{n_p}/(\sigma /R)$, with $\kappa$ and $n_p$ ranging from 0.4 to 5.4 $\textrm {Pa s}^{n_p}$ and 0.16 to 0.51, respectively. The black line represents the prediction of (1.2) with a prefactor of 1. The experimental measurements are shown as open symbols, and error bars are smaller than the symbols.

With $\widehat {Ca}$ to compare the shear stress from the power-law model and the capillary pressure, (1.2) has been used to predict the film thickness (Gutfinger & Tallmadge Reference Gutfinger and Tallmadge1965; Hewson et al. Reference Hewson, Kapur and Gaskell2009). We note that the values of $\kappa$ and $n_p$ for $\widehat {Ca}$ are obtained by fitting the rheological data with a power-law model considering the range of shear rates exhibiting a shear-thinning behaviour, i.e. $\dot {\gamma } = O(10\unicode{x2013}10^3$ s$^{-1})$ for CMCell and $O(10^{-1}\unicode{x2013}10^2$ s$^{-1}$) for Carbopol. As shown in figure 3(b), we report that the data do not precisely follow the power law of $2/(2n_p+1)$, as predicted by (1.2). The results imply that all the dynamics of the coating process cannot be captured by using $\widehat {Ca}$, since the power-law model alone is not sufficient to describe the rheological behaviours of the working fluids around the bubble. In particular, the deviation of the experimental data for the CMCell solutions is larger than those for the Carbopol solutions when comparing to the prediction of (1.2), which can be attributed to the range of the effective shear rates. In the experiments, considering the effective shear rates in the film $\dot {\gamma }=U/h$, we obtain $Cu=O(10\unicode{x2013}10^2)$ for the CMCell solutions while $Cu=O(10^5)$ for the Carbopol solutions. The experiments for the CMCell solutions include low- to intermediate-shear-rate regions, while the experiments for the Carbopol solutions are performed at intermediate- to comparably high-shear-rate regions. Therefore, the viscosity plateau at low-shear-rate is required to be considered to obtain the better prediction for the case of the CMCell solutions, in addition to the power-law dependence at intermediate-shear-rates. Furthermore, in the vicinity of the uniform film thickness region, the fluid is at rest. However, the shear-thinning effect plays an important role at the bubble front meniscus. Approaching the meniscus, the film starts growing rapidly in the thickness and the shear rate also increases. However, the local shear rate will decrease again in the region of the re-circulating flow ahead of the bubble. Such a change of the local shear rate at different regions requires an accurate viscosity model for a correct representation of the flow physics, which also highlights the importance of a more realistic rheological model in free surface flow analyses.

3.3. Scaling of the film thickness with the effective capillary number

To explore the effect of the shear-thinning rheology on bubble characteristics, the following ordinary differential equation for the bubble profile has been obtained in the theoretical work by Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021) considering an Ellis fluid

(3.5)\begin{equation} \frac{\mathrm{d}^3\eta}{\mathrm{d}\xi^3}+\frac{3^\alpha}{(\alpha+2){El}^{\alpha-1}} \frac{\mathrm{d}^3\eta}{\mathrm{d}\xi^3} \left|\frac{\mathrm{d}^3\eta}{\mathrm{d}\xi^3}\right|^{\alpha-1}\eta^{\alpha-1}= \frac{\eta-1}{\eta^3}, \end{equation}

where $\xi =x/[h(3Ca_0)^{-1/3}]$ and $\eta =y/h$. Different from the Newtonian case, the bubble profile $\eta$ becomes a function of $\xi$, $\alpha$ and $El$. The two terms in the left-hand side of (3.5) represent the Newtonian and shear-thinning contributions, respectively. The curvature of the parabolic region needs to be matched to the curvature of the bubble spherical cap, $1/R$, and once that is done, the thickness of the uniform film region can be deduced.

By introducing the effective capillary number $Ca_e$ that considers both the zero-shear-rate and the shear-thinning effects, we obtain

(3.6)\begin{equation} \frac{h}{R}=P(3Ca_0)^{2/3}=0.643(3Ca_e)^{2/3} \quad \text{with}\ Ca_e=\frac{\mu_eU}{\sigma}, \end{equation}

where $\mu _e$ is the effective viscosity defined as $\mu _e=\mu _0(P/0.643)^{3/2}$, and $P$ is the dimensionless curvature related to the capillary pressure as the second derivative of $\eta$ with respect to $\xi$, i.e. $P=\mathrm {d}^2\eta /\mathrm {d}\xi ^2$ at the limit of $\eta \gg 1$. Here, $P$ is determined by numerically solving (3.5) using the fully implicit solver $ode15i$ of Matlab and the exact initial conditions given by Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021), and then taking a limit when $\eta \gg 1$ toward the front meniscus. Furthermore, 0.643 is the numerical factor for the Newtonian limit (Bretherton Reference Bretherton1961). In addition, Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021) used the numerical results of $P$ from (3.5) to obtain a master fitting curve of $\mu _e$ as a function of $El$ and $\alpha$. Here, we revise the fitting curve with $Cu$ and $n_c$ considering the experimental rheological data ($Cu\approx 1.3{El}^{-1}$ and $n_c=1/\alpha$, § 3.1) as follows:

(3.7)\begin{equation} \frac{\mu_e}{\mu_0}=\left(\frac{P}{0.643}\right)^{3/2}= \begin{cases} 1 & \text{if} \ Cu \rightarrow 0, \\ \dfrac{10-7n_c}{4}\left(\dfrac{Cu}{1.3}\right)^{n_c-1} \equiv \varTheta & \text{if} \ Cu \gg 1. \end{cases}\end{equation}

The prefactor $(10-7{n_c})/4$ for $Cu \gg 1$ comes from the fitting curve obtained by Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021) at $El\rightarrow 0$. When $Cu \rightarrow 0$, (3.7) reduces to a Newtonian case. When $Cu \gg 1$, the shear-thinning effect dominates, and thus the viscosity depends on $n_c$ and $Cu$ given that $\mu _\infty$ is neglected in the current experiments. For other practical non-Newtonian fluids with non-negligible $\mu _\infty$, the limit of $Cu \rightarrow \infty$ corresponds to the high-shear-rate viscosity plateau with constant viscosity $\mu _\infty$.

For each experiment, we compute the effective shear rate in the film $U/h$, using the measured bubble speed and film thickness, and thus acquire the effective viscosity $\mu _e/\mu _0$ of the CMCell and Carbopol solutions by using the C-Y model (figure 2). We find the experimental values of $\mu _e/\mu _0$ agree well with $(P/0.634)^{3/2}$, which is numerically calculated from (3.5). Indeed, figure 4(a) provides a plot of the effective viscosity as a function of $Cu$ and $n_c$ and all the viscosity data collapse around the fitting curve of the master (3.7), as shown in figure 4(b), which suggests a universal scaling for the effective viscosity to define the effective capillary number. Therefore, the comparison of the effective capillary number $Ca_e$ obtained from (3.6) against the experimental data show good agreement over the entire range of $Ca_e$ (figure 4c). The smaller values of $\mu _e$/$\mu _0$ for the Carbopol solution demonstrate the greater extent of shear-thinning compared to the CMCell solution, which are also indicated by the higher values of $Cu$ for the Carbopol solutions, as shown in table 2. We further recast (3.6) in the following expression proposed by Aussillous & Quéré (Reference Aussillous and Quéré2000) for the range of $Ca_e$ ($10^{-3}< Ca_e<0.6$) in the current experiments as

(3.8)\begin{equation} \frac{h}{R}=\frac{1.34Ca_e^{2/3}}{1+2.5\times1.34Ca_e^{2/3}}.\end{equation}

Figure 4.

(a) Non-dimensional effective viscosity $\mu _e/\mu _0$ as a function of the Carreau number $Cu$ and the power-law index $n_c$. (b) $\mu _e/\mu _0$ of the experimental data as a function of $\varTheta$. All the experimental data from the present work show good agreement with the master curve of (3.7). (c) Comparison of the effective capillary number, $Ca_e$, between the theoretical and experimental results. All the data lie along the solid line with a slope of a unity. (d) h/R as a function of $Ca_e$. The black line represents prediction of (3.8) with $Ca_e$ (Aussillous & Quéré Reference Aussillous and Quéré2000; Picchi et al. Reference Picchi, Ullmann, Brauner and Poesio2021). Error bars are smaller than the symbols.

Figure 4(d) shows a plot of the non-dimensional liquid film thickness $h/R$ as a function of $Ca_e$. The values of $h/R$ increase consistently with $Ca_e$ for the non-Newtonian fluids, as the viscous effect is increasingly important. The experimental data for both the CMCell and Carbopol solutions agree well with (3.8), including the trend in low $Ca_e$ and the saturation behaviour at relatively large $Ca_e$. Thus, we demonstrate that the liquid film thickness of the shear-thinning fluids can be estimated with the scaling law proposed by Aussillous & Quéré (Reference Aussillous and Quéré2000) using $Ca_e$. The better prediction accuracy compared with (1.2) also highlights the importance of a practical rheological model. To the best of our knowledge, our work serves as the first experimental validation of (3.8) with realistic shear-thinning fluids based on $Ca_e$, which is helpful to assess the true range of applicability for different scaling laws.

3.4. Scaling of the bubble speed with the effective capillary number

In addition, we experimentally measure the ratio of the bubble speed, $U$, to the average velocity of the fluid flowing far from the bubble, $U_\infty$. The scaling law for the ratio $U/U_\infty$ can be derived by applying the mass balance near the thin film region in a reference frame moving with the bubble (Picchi et al. Reference Picchi, Ullmann, Brauner and Poesio2021). Using $Ca_e$, we obtain

(3.9)\begin{equation} \frac{U}{U_\infty}=\frac{1}{(1-h/R)^2}=\frac{1}{\left(1- \dfrac{1.34Ca_{e}^{2/3}}{1+2.5\times1.34Ca_{e}^{2/3}}\right)^2}.\end{equation}

The experimentally obtained $U/U_\infty$ collapse well with (3.9) (figure 5), showing that $Ca_e$, as a function of $Cu$ and $n_c$, can be used to describe the evolution of the bubble speed with the rheological parameters of the shear-thinning fluids.

Figure 5.

Ratio of the bubble speed to the average velocity of the fluid far from the bubble $U/U_\infty$, as a function of $Ca_e$ obtained from (3.6). Error bars are smaller than the symbols.

3.5. Characteristics of the bubble front and rear menisci

We further investigate the shape variations of the bubble translating in the CMCell solution, in which the front and rear menisci of the bubble could be identified clearly compared to those in the Carbopol solution. In this section, the bubble shape profiles near the front and rear menisci are computed by solving (3.5) as a function of $El$ and $\alpha$. Then we plot in figure 6 the numerically obtained profiles considering $Cu$ and $n_c$ to be given by the correlations $Cu \approx 1.3 El^{-1}$ and $n_c=1/\alpha$, and these correlations having already been introduced back in § 3.1. Using the experimentally obtained bubble profiles, we first identify the points at which the local film thickness increase by one pixel ($\approx 1.3\ \mathrm {\mu }$m) compared with the uniform film thickness. Such a thickness increase corresponds to ${\approx }1.7$–$8.5\,\%$ of the uniform thickness of the deposited film. Then, these points are overlapped with those corresponding to the same thickness increase in the numerically obtained bubble profiles, as shown in figure 6. We note that the bubble profiles at the front and the rear menisci are solved separately by integrating (3.5) with a different set of boundary conditions. At the bubble front, we assume that the thin film region extends to $\xi \rightarrow -\infty$ (i.e. $\eta (-\infty )=1$), and the front meniscus is obtained by integrating (3.5) towards positive $\xi$. However, at the rear meniscus, the thin film region is at $\xi \rightarrow +\infty$, and the profile at the rear meniscus is obtained by integrating (3.5) towards negative $\xi$ starting from the boundary condition $\eta (+\infty )=1$ (Bretherton Reference Bretherton1961; Picchi et al. Reference Picchi, Ullmann, Brauner and Poesio2021).

Figure 6.

Bubble front meniscus as a function of (a) the Carreau number $Cu$ with the power-law index $n_c = 0.48$ and (b) $n_c$ with $Cu = 13.8$. Bubble rear meniscus as a function of (c) the Carreau number $Cu$ with the power-law index $n_c = 0.48$ and (d) $n_c$ with $Cu = 13.8$.

For the bubble front meniscus, figure 6(a) shows good agreement between the experimental results and the numerical predictions (Picchi et al. Reference Picchi, Ullmann, Brauner and Poesio2021) of the shape changes when $Cu$ increases at fixed $n_c=0.48$. Although the bubble maintains the rounded shape similar to that in the Newtonian fluids, the delayed transition from the uniform film to the parabolic region characterized by a constant dimensionless curvature becomes significant due to the higher effective shear rate in the film. Considering the viscosity field obtained from the numerical simulations by Moreira et al. (Reference Moreira, Rocha, Carneiro, Araújo, Campos and Miranda2020), the appearance of high viscosity in the film is due to the almost stagnant liquid, while in the axis of the channel, it is due to a low velocity gradient. In between, as $Cu$ can be interpreted as the ratio between the representative shear rate in the film to the onset of the shear-thinning effects, larger values of $Cu$ (i.e. lower $El$) indicate stronger shear-thinning effects and thus the weight of the second term on the left-hand side of (3.5) increases, resulting in the delayed transition to the parabolic region for the bubble shape. A similar trend is observed when decreasing $n_c$ at fixed $Cu=13.8$, as shown in figure 6(b). The decrease of $n_c$ indicates stronger shear-thinning effects and thus the transition to the parabolic region is also expected to be delayed. We note that the numerical solution of (3.5) starts to deviate from the experimental data as $Ca_0 > 0.3$ (figure 6b) since the lubrication approximation will no longer strictly hold for the relatively large $Ca_0$. We note that such a delayed transition from the thin film to the parabolic profile is also theoretically observed in the case where a charged oil droplet moves through a charged capillary. When the electrostatic interaction between the capillary wall and the droplet surface is attractive, the visco-electro-osmotic balance might not only reduce the film thickness, but also delay the transition because of the cooperation of the electro-osmotic and capillary pressure (Grassia Reference Grassia2020, Reference Grassia2022).

For the bubble rear meniscus, (3.5) is solved following the initial and boundary conditions given by Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021). The general observation of the rear meniscus is similar to that of the Newtonian case, where the bubble profile exhibits one main crest and one main valley (Magnini et al. Reference Magnini, Beisel, Ferrari and Thome2017) with a high degree of undulations, as shown in figure 6(c,d). Although the experimental results do not agree with the numerical solutions very well, the general trends are consistent with the results of Picchi et al. (Reference Picchi, Ullmann, Brauner and Poesio2021), considering the effect of the shear-thinning rheology. As the shear-thinning effect becomes more important with increasing $Cu$ or decreasing $n_c$, the bubble profile stretches along $\xi$. Unlike the front meniscus, where the viscosity profile is regular, the axial velocity gradient near the rear meniscus should be considered when computing the shear rate due to the undulations (Picchi et al. Reference Picchi, Ullmann, Brauner and Poesio2021). Therefore, an accurate description of the viscosity field at the rear meniscus of the bubble requires a further correction for the axial derivative of the velocity in future work.