2. Experimental set-up and data analysis

The experimental set-up used in the current study is shown schematically in figure 3. A thin glass plate with dimensions $75\,{\rm mm}\times 25\,{\rm mm} \times 1\,{\rm mm}$ was dipped into an acrylic tank of dimensions $100\,{\rm mm}\times 100\,{\rm mm} \times 27\,{\rm mm}$ at a constant speed. A motorized traverse with a stepper motor was used to vary the speed of the plate from $100\,\mathrm {\mu }{\rm m}\,{\rm s}^{-1}$ to $2\,{\rm cm}\,{\rm s}^{-1}$. A DM542 digital microstepper driver was connected to a computer through a data acquisition system from National Instruments. To ensure that a contact line is present in all the experiments, we only allowed the plate to dip into the liquid bath. This restricts our experiments only to advancing cases. To prevent contamination, the glass plate and the tank were thoroughly cleaned with isopropyl alcohol followed by distilled water, and dried before each experiment. The meniscus was illuminated using a thin laser sheet (thickness of approximately 0.5 mm) that was created by a combination of biconvex and cylindrical plano-convex lenses from a 532 nm 2W diode laser. We placed a spherical biconvex lens and two cylindrical plano-convex lenses at distances of 25 mm, 235 mm and 335 mm from the laser. The focal lengths of the lens are 150 mm, 75 mm and 100 mm. The flow was seeded with polyamide particles of an average diameter of $5\,\mathrm {\mu }{\rm m}$ for all the experiments. To ensure that the particles faithfully follow the flow, we require the Stokes number to be very small. The Stokes number ($St$), defined as the ratio of particle relaxation time scale ($t_p$) to flow time scale, i.e. $St=t_p/(l_c/U)$, ranged from $2\times 10^{-10}$ to $2.2\times 10^{-6}$ over the entire range of velocities. The particle relaxation time based on the density and diameter of the particle, $t_p = \rho _p d_p^2/(18 \mu _{oil})$, was always less than $0.2\,\mathrm {\mu }{\rm s}$. All PIV images were captured using a Photron Fastcam Nova S9 high-speed camera connected to a macro lens with the camera being placed perpendicular to the laser sheet. The images were captured at frame rates ranging from 10 fps to 1000 fps depending on the speed of the plate. To ensure that the surface level of the fluid did not increase when the plate was dipped into the liquid bath, a programmable syringe pump was employed to withdraw fluid at a prescribed flow rate from the bottom of the tank. In all the experiments the interface shape and the flow settled into a steady state after a short transient. Only the PIV images after this initial transient were processed. The steady nature of the flow allowed us to also generate streakline images, and a sample image obtained from the experiment is shown in figure 4. The absence of crossings in the streaklines indicates that the flow is indeed steady (also see supplementary material available at https://doi.org/10.1017/jfm.2024.486 for a sample transient image).

Figure 3.

A schematic of the PIV experimental set-up with the following key components: (i) rectangular tank, (ii) glass slide, (iii) motorized traverse mechanism with DC power source and controller, (iv) laser with controller and power source along with associated optics to produce a thin laser sheet, (v) camera with a macro lens, (vi) programmable syringe pump used to maintain a constant liquid level.

Figure 4.

A streakline image for 500 cSt silicone oil at $Re = 3.92 \times 10^{-4}$ and $Ca = 4.19 \times 10^{-3}$. The glass plate, represented by a grey vertical slab is immersed in a liquid bath at constant speed. The blue solid curve represents the interface between the liquid and air, whereas the red arrow represents the direction of the flow in the liquid phase.

A digital delay generator was used to synchronize the laser and the high-speed camera. The field of view of the high-speed camera with the macro lens varied from $4.2\,{\rm mm} \times 4.2\,{\rm mm}$ to nearly $8\,{\rm mm} \times 8\,{\rm mm}$ with a spatial resolution in the range of $4\,\mathrm {\mu }{\rm m}\,{\rm pixel}^{-1}$ to nearly $8\,\mathrm {\mu }{\rm m}\,{\rm pixel}^{-1}$. Preprocessing of the images was performed prior to the PIV analysis, including average background subtraction, image equalization and masking. The particle images were analysed using a multigrid, window-deforming PIV algorithm. In addition, since the flow is steady, an ensemble PIV correlation was employed to improve the signal-to-noise ratio in a small interrogation window (typically $8\text { pixels} \times 8\text { pixels}$).

As shown in figure 2, the primary goal of this paper is to investigate advancing contact angles with dynamic contact angle $\theta _d < 90^{\circ }$. This was most easily achieved with the immersion of a glass plate into a bath of silicone oil at low speeds. The properties of the silicone oils used in the experiments are given in table 1. The wetting properties of the glass substrates were also characterized by measuring the static advancing and receding angles with the help of a contact angle meter (Kruss DSA25S). As clearly evident in table 2, the static hysteresis in all the experiments was found to be very small.

Table 1.

Properties of the fluids used in the experiments.

Table 2.

Characterizing the hysteresis for silicone oils on glass substrates.

$^{a}$Since there is an overlap in the error bars in the receding and advancing angles, only the mean value of the hysteresis is reported.

To determine the interface location and shape, several particle images were combined and the locus of particle streaks at the interface are considered to be representative of the interface itself. The interface angle was determined by fitting a two-term exponential function to the interface data points in the form

(2.1)\begin{equation} p(x) = c_1 e^{c_2 x} + c_3 e^{c_4 x}, \end{equation}

which resulted in an excellent fit for all experiments with an $R$-square value always above $0.99$. The local angle along the interface was then determined from this fit. Dynamic contact angles for various capillary numbers shown in figure 5 were compared with the classical study of Hoffman (Reference Hoffman1975). Despite the differences in the geometry between the present study and that of Hoffman, the collapse of all contact angle data in the two experiments shows that the relation between dynamic contact angle and capillary number is universal. Fitting the Cox–Voinov model of the form $\theta _d^3-\theta _e^3 = A\,Ca$, the data of Hoffman (Reference Hoffman1975) results in the value of the slope as $A\approx 81.1$, which is very close to the value 78.7 in the present experiments. More information about comparison with dynamic contact angle models is available in the supplementary material.

Figure 5.

Variation of dynamic contact angle, $\theta _d$, with an adjusted capillary number, $Ca + F(\theta _e)$. Here $F(\theta _e)$ represents the shift factor that compensates for the effect of static contact angle in terms of $Ca$. The shift factor is calculated considering $\theta _d = \theta _e$ and the corresponding value of $Ca$ is assigned to $F(\theta _e)$. All the markers in black are taken from Hoffman (Reference Hoffman1975) and those in red (, red) represent present data obtained using different grades of silicone oil.

In the next section we discuss the theoretical basis for the comparison of flow fields from experiments with models.

3. Theoretical background

The earliest theoretical models for moving contact lines were aimed at developing a local model valid only in the vicinity of the contact line. A schematic of such a ‘local’ framework is shown in figure 6(a) where the dynamics is governed by the Stokes equations for $r \ll L$ with $L$ being the macroscopic length scale in the problem. Huh & Scriven (Reference Huh and Scriven1971) were among the first to develop a simple model of a moving contact line assuming the interface to be flat and employing the no-slip condition on the moving wall. Their study showed the presence of a singularity in the shear stress at the contact line. Nevertheless, their model predicts flow fields in both phases that have regular and smooth behaviour away from the contact line. A key prediction in the HS71 model is that the higher viscosity fluid exhibits a rolling motion and the lower viscosity fluid exhibits a split-streamline motion. This means that the fluid particles at the interface move towards the moving contact line in an advancing contact line problem as shown in figure 6(b).

Figure 6.

(a) Cylindrical polar coordinate system $(r,\theta )$ used in HS71 for flow in a flat wedge separating two immiscible fluids, A and B, with a constant wedge angle $\phi$, caused by a plate moving at constant speed $U$. (b) Typical flow pattern predicted by HS71's theory when $\lambda \ll 1$. The arrow at the interface represents the direction of motion of fluid particles along the interface. When fluid B undergoes a ‘rolling’ motion, fluid A exhibits a ‘split-streamline’ motion with the splitting streamline shown by a dashed line. (c) Coordinate system for a wedge with a curved interface with angle $\beta$ varying along the interface.

To relieve the HS71 singularity, slip models (Cox Reference Cox1986; Kirkinis & Davis Reference Kirkinis and Davis2013, Reference Kirkinis and Davis2014; Febres & Legendre Reference Febres and Legendre2017) were subsequently developed. Cox (Reference Cox1986), in a seminal paper, developed an asymptotic model by dividing the flow into three regions: an outer region where the interface shape is affected by the geometry of the problem, a slip-dominated inner region in the vicinity of the moving contact line and an intermediate region merging these two regions as shown in figure 1. The model of Cox (Reference Cox1986) is valid in the limit of $Re \ll 1$ and $Ca \ll 1$ where $Re = \rho U L/\mu$ and $\mu$ is typically taken to the viscosity of the more viscous phase. This restriction on $Re$ ensures that the flow in the intermediate and inner regions is always in the viscous regime. By ‘intermediate’ region we refer to regions of interest that are smaller than the macroscopic length scale, $L$: $l_s \ll l_i < L$. Sibley, Nold & Kalliadasis (Reference Sibley, Nold and Kalliadasis2015) noted that distinct inner and outer regions exist with an intermediate region sandwiched between the two, consistent with figure 1, if the following conditions hold:

(3.1)\begin{equation} r_{outer} \gg L \mathrm{e}^{{-}1/|Ca|} \quad \text{and} \quad r_{inner} \ll l_s \mathrm{e}^{1/|Ca|} \quad \implies \quad \left|Ca \ln \left(\frac{l_s}{L}\right)\right| \ll 2. \end{equation}

Here, $r_{inner}$ and $r_{outer}$ define the radial extent of the intermediate region between the inner and outer regions as shown in figure 1. The slip length, $l_s$, is typically in nanometres, thus, the above condition will always be satisfied if the capillary number is kept small, typically less than or equal to $10^{-2}$. This ensures that there is a distinct and wide intermediate region in the experiments.

3.1. Models for flow fields

In our experiments the geometry resembles that of a wedge formed between a vertically moving plate and a curved interface. Our aim is to compare flow fields from experiments in this geometry with well-known models in the literature. Following the seminal work of Moffatt (Reference Moffatt1964) who determined the flow in a wedge formed between two flat plates, HS71 determined the flow in a wedge formed between a moving plate and a flat interface, and was subsequently amended by Chen et al. (Reference Chen, Ramé and Garoff1997) to incorporate the effects of a curved interface.

3.1.1. Fixed wedge solution

The ‘local’ solution developed by HS71 assumes the interface to be flat as shown in figure 6(a). The flow in the vicinity of the moving contact line is governed by the biharmonic equation for the streamfunction Moffatt (Reference Moffatt1964),

(3.2)\begin{equation} \nabla^4 \psi = 0, \end{equation}

where the velocity components are defined as

(3.3a,b)\begin{equation} v_r = \frac{1}{r}{\frac{\partial\psi}{\partial\theta}}, \quad v_{\theta} ={-}{\frac{\partial\psi}{\partial r}}. \end{equation}

The solution of the biharmonic equation can be assumed to be of the form (Moffatt Reference Moffatt1964; Huh & Scriven Reference Huh and Scriven1971)

(3.4)\begin{equation} \psi(r,\theta) = r(a \sin \theta + b \cos \theta + c \theta \sin \theta + d \theta \cos \theta). \end{equation}

Using conditions of no-slip, no-penetration on the moving solid surface, continuity of tangential velocity and tangential stress at the interface (no Marangoni effects), the streamfunction reduces to

(3.5)\begin{equation} \psi(r,\theta;\phi) = rf(\theta,\phi) = rU\left(\frac{\phi\sin\theta - \theta\sin\phi \cos(\theta -\phi) }{\phi - \sin\phi \cos\phi}\right). \end{equation}

In the above expression the gas above the liquid is assumed to be passive, i.e. $\lambda = 0$. It can be easily shown that the shear stress obtained from the above expression diverges like $1/r$ as $r\rightarrow 0$. In the HS71 framework, the interfacial velocity, $v_i^{HS}$, is identically equal to the radial velocity and is independent of the radial location along the interface and is given by

(3.6)\begin{equation} v_i^{HS} = v_r(r,\phi) = U \left(\frac{\phi \cos \phi - \sin \phi}{\phi - \sin \phi \cos \phi}\right). \end{equation}

In experiments the interface is always curved due to the presence of external forces such as gravity that induces a mean curvature to the interface. To facilitate a comparison of the streamfunction with experiments, we modify (3.5) for a curved interface. The recipe for doing this was first given by Chen et al. (Reference Chen, Ramé and Garoff1997).

3.1.2. Modulated wedge solution

If the interface is curved, shown schematically in figure 6(c), with the interface angle at any radial position given by $\beta (r)$, then the streamfunction in (3.5) can be modified by replacing the constant angle $\phi$ with a variable angle $\beta (r)$:

(3.7)\begin{equation} \psi(r,\theta;\beta(r)) = rf(\theta,\beta) = rU\left(\frac{\beta\sin\theta - \theta\sin\beta \cos(\theta -\beta) }{\beta - \sin\beta \cos\beta}\right). \end{equation}

This solution, also used by Chen et al. (Reference Chen, Ramé and Garoff1997), is referred to as the MWS. The above solution is identical to the leading-order solution in the intermediate region given by Cox (Reference Cox1986). As one approaches the contact line, the modulated wedge angle $\beta$ approaches the fixed wedge angle $\phi$. Therefore, as $r \rightarrow 0$, the expressions (3.5) and (3.7) become identical. This allows us to make a three-way comparison between the HS71 solution, the MWS and experiments.

The function $\beta (r)$ can be obtained from experiments and inserted into (3.7) to obtain the streamfunction everywhere in the fluid domain. The radial and tangential velocities can also be easily computed as

(3.8)\begin{gather} v_r(r,\theta;\beta) = \frac{\partial f}{\partial\theta}, \end{gather}

(3.9)\begin{gather}v_{\theta}(r,\theta;\beta) ={-}f - r\frac{\partial f}{\partial \beta}\frac{\partial \beta}{\partial r}. \end{gather}

Using the expression for $f(\theta,\beta )$ from (3.7), we have

(3.10)\begin{gather} \frac{v_r(r,\theta;\beta)}{U} = \frac{\beta \cos\theta + \theta \sin \beta \sin (\theta - \beta) - \sin \beta \cos (\theta-\beta)}{\beta - \sin\beta \cos\beta}, \end{gather}

(3.11)\begin{gather} \frac{v_{\theta}(r,\theta;\beta)}{U} = \frac{-\beta \sin\theta + \theta\sin \beta \cos (\theta-\beta)}{\beta - \sin\beta \cos\beta} \nonumber\\ + r\frac{{\rm d}\beta}{{\rm d}r} \left(\frac{2\sin^2\beta(\beta \sin\theta - \theta\sin\beta \cos(\theta-\beta))}{(\beta - \sin\beta \cos\beta)^2} - \frac{\sin\theta -\theta \cos(2\beta - \theta)}{\beta - \sin\beta \cos\beta}\right). \end{gather}

The velocity at the interface can be obtained in terms of radial and angular velocity components by setting $\theta = \beta$ in (3.10) and (3.11),

(3.12)\begin{equation} v_i^{MWS} = v_r(r,\beta) \cos(\alpha-\beta) + v_{\theta}(r,\beta) \sin(\alpha-\beta), \end{equation}

where $\alpha (r)$, shown in figure 6(c), is a measure of the slope of the interface and is related to $\beta (r)$ by the expression

(3.13)\begin{equation} \alpha(r) = \beta(r) + \tan^{{-}1}\left(r{\frac{{\rm d}\beta}{{\rm d}r}}\right). \end{equation}

After simplification, the expression (3.12) reduces to

(3.14)\begin{equation} v_i^{MWS} = \frac{U}{\cos(\alpha - \beta)}\left(\frac{\beta \cos \beta - \sin \beta}{\beta - \sin \beta \cos \beta}\right). \end{equation}

At the contact line, $\alpha =\beta =\phi$. Equation (3.14) reduces to the HS71 solution in (3.6) as $r\rightarrow 0$. But for other radial positions, the interfacial speed $v_i^{MWS}$ will no longer be the same as $v_i^{HS}$ but will increase in magnitude while moving away from the contact line and then becomes constant as the change in the difference ($\alpha -\beta$) becomes small.

The expression for interface shape, expressed in terms of $\beta (r)$, can fully describe the flow field using the streamfunction (3.7). In the next section we apply analytical models and show how interface shapes in the intermediate and outer regions can be compared against experiments.

3.2. Models for complete interface shape

In the present problem there are three main forces at play: viscous forces, surface tension and gravity. Inertia is negligible if the Reynolds number is kept small as is the case in the present study. Far away from the plate, surface tension and gravity are in balance with each other leading to the formation of a static meniscus. However, near the plate, viscous forces created due to the moving plate cannot be ignored. In steady state, viscous forces balance surface tension in the vicinity of the moving contact line. The subtle balance of the three forces and the extent of the static effects is dictated by the value of $Ca$. In the low $Ca$ limit, viscous effects remain confined to very small regions near the moving wall while the interface shape away from the wall is dictated by the balance of surface tension to gravity. The deviation of the dynamic meniscus from the static meniscus can be referred to as viscous deformation. Hence, in the low $Ca$ limit, viscous deformation is small while it is large in the moderate to high $Ca$ limit.

In most theoretical models the apparent contact angle is related to the equilibrium contact angle and the hydrodynamics of the flow. In the case of the Cox–Voinov relationship (see the review of Snoeijer & Andreotti (Reference Snoeijer and Andreotti2013) for more details), we have

(3.15)\begin{equation} \theta_{ap}^3 = \theta_e^3 + 9 Ca \ln\left(\frac{L}{l_s}\right), \end{equation}

where $L$ is the outer (macroscopic length) that is equal to the capillary length in the present study and $l_s$ represents an inner (microscopic) length over which the singularity is regularized (also see figure 1). According to such theoretical models, as one approaches the contact line from the intermediate to the inner region, the interface rapidly bends to meet the equilibrium contact angle at the wall. This rapid change in shape of the interface to satisfy the equilibrium contact angle is referred to as viscous bending. Unlike viscous deformation that can be observed in experiments, viscous bending is beyond the reach of experiments. Even if viscous bending exists in real flows, it exists only over very small length scales, typically in the range of nanometres to fractions of micrometres from the wall. A simple estimate for the scale of the inner region can be obtained by letting the equilibrium contact angle be very small, i.e. $\theta _e \rightarrow 0$. In this case, the Cox–Voinov relationship can be used to estimate the length of the inner region as

(3.16)\begin{equation} l_s \sim L \exp{\left(\frac{-\theta_{ap}^3}{9 Ca}\right)}. \end{equation}

The above length scale is also a simple estimate for the length scale of viscous bending. For $\theta _{ap} = {\rm \pi}/3$, we obtain $l_s/L \approx 2.8 \times 10^{-6}$ when $Ca = 0.01$, while it is 0.88 when $Ca=1$. Using $L = l_c = 1500\,\mathrm {\mu }{\rm m}$, it is evident that viscous bending persists over a length of $4.3\,{\rm nm}$ to $1300\,\mathrm {\mu }{\rm m}$ for the above range of $Ca$.

The above discussion clearly shows that there is a large variation in the length scale between the inner and outer regions in the low $Ca$ limit. Distinct models have been developed to account for the variation of interface shape. The first model relies on developing a composite solution for the interface shape by using the Cox model. This composite solution is parameterized by a single scalar parameter that is determined by matching the solution to experiments. The second approach uses a generalization of lubrication equations and reduces the problem to a system of coupled differential equations.

Recall that in the three-region framework of Cox (Reference Cox1986), the inner slip-dominated region is matched to an outer region via an intermediate region. Since the inner region is often beyond the resolution of any experiment, a direct comparison of interface shape with the Cox model with experiments is likely to result in poor agreement. To improve the prediction of interface shape over all length scales, it is necessary to incorporate the effect of the outer region that may be dominated by other forces such as gravity. Dussan V. et al. (Reference Dussan V., Ramé and Garoff1991) precisely carried out such an extension, referred to as the DRG model hereafter.

In the DRG model the problem of finding the full interface shape is reduced to a search for a single algebraic parameter, termed $\omega _0$. In the absence of motion, an interface deforms near the plate forming a static meniscus of characteristic length $l_c$, the capillary length. According to the classical three-layer model of Cox (Reference Cox1986), the interface shape in the intermediate region is given by

(3.17)\begin{equation} g(\alpha)\sim d_0 + Ca\ln(r/l_c), \end{equation}

where the function $g(x)$ is given by

(3.18)\begin{equation} g(x) = \int^x_0 \frac{t - \cos t \sin t}{2\sin t} \,{\rm d}t. \end{equation}

The viscosity ratio, $\lambda$, has been assumed to be negligible in the above expression for simplicity, though it can be included as per the full Cox model without any difficulty. Following Cox (Reference Cox1986), matching the solutions between the intermediate region and the inner region (see figure 1), we obtain

(3.19)\begin{equation} d_0 = g(\theta_{e}) + Ca\ln(l_c/l_s), \end{equation}

where $l_s$ is the slip length and $\theta _{e}$ is the microscopic contact angle. Combining (3.17) and (3.19), we obtain

(3.20)\begin{equation} g(\alpha) = g(\theta_{e}) + Ca\ln(r/l_s). \end{equation}

Similarly, by matching the outer to the inner solution (see Dussan V. et al. Reference Dussan V., Ramé and Garoff1991), we obtain

(3.21)\begin{equation} d_0 = g(\omega_0), \end{equation}

where $\omega _0$ plays the same role as the apparent contact angle. Dussan V. et al. (Reference Dussan V., Ramé and Garoff1991) showed that $\omega _0$ can be determined by matching the interface shape from the analytical model with experimental data. In the outer region the interface shape is dictated by the shape of the static interface such that it matches the solution in the intermediate region. For example, the shape of a static meniscus would be determined not only by the far-field boundary conditions where the slope of the interface vanishes but also by a boundary condition on the plate. In the same way, the outer solution will have to be parameterized by an ‘effective’ contact angle boundary condition that is equal to $\omega _0$. This is shown schematically in figure 7(a). Let us assume that the interface shape in the outer ‘static’ region assumes the form

(3.22)\begin{equation} \theta_s(r) = f_0\left(\frac{r}{l_c};\omega_0\right), \end{equation}

where $\theta _s(r)$, shown in figure 7(a), is a local slope estimated along the static interface. The exact form of $f_0$ for a two-dimensional static meniscus is given in § 4.1. Matching the intermediate solution (3.17) to (3.21), we obtain an expression for the interface angle given by

(3.23)\begin{equation} g(\alpha)=g(\omega_0) + Ca\ln\left(\frac{r}{l_c}\right). \end{equation}

The composite DRG solution is obtained by adding the static and Cox solutions, i.e. (3.22) and (3.23), to give

(3.24)\begin{equation} \alpha(r) = g^{{-}1}\left[g(\omega_0)+Ca\ln\left(\frac{r}{l_c}\right)\right] + \left({\theta_s(r)} - \omega_0\right). \end{equation}

The parameter, $\omega _0$, can be interpreted as an empirical parameter that can be evaluated by matching (3.24) with an experimentally obtained interface shape.

Figure 7.

(a) The geometrical set-up for the DRG model with the outer solution, $h_s(x)$ making an angle $\omega _0$ at the moving wall and $\theta _e$ is the microscopic contact angle in the inner region as per Cox (Reference Cox1986). (b) The dynamic meniscus in the GLM framework with $\alpha (s)$ being the local angle along the curved interface, $s$ is the arc length measured from the contact line, $h(x)$ is the position of the interface measured from the moving plate.

An alternative approach to obtain the full interface shape was proposed by Snoeijer and coworkers. This is a simple and direct approach involving the generalized lubrication equations used in free-surface flows even for problems with large interface slopes (Snoeijer Reference Snoeijer2006; Chan et al. Reference Chan, Srivastava, Marchand, Andreotti, Biferale, Toschi and Snoeijer2013, Reference Chan, Kamal, Snoeijer, Sprittles and Eggers2020). This is achieved by expanding the Stokes equations about flow in a wedge with a constant wedge angle (Huh & Scriven Reference Huh and Scriven1971), perturbed around $Ca\ll 1$, i.e. variations of the interface slope are assumed to be slow. The model is described in terms of two coupled differential equations for the interface shape, $h(s)$, and local interface angle, $\alpha (s)$, as shown schematically in figure 7(b), and given by

(3.25a)\begin{gather} {\frac{{\rm d}^2\alpha}{{\rm d}s^2}} = \frac{3 Ca}{h(h+cl_s)} f(\alpha,\lambda) - \frac{1}{l_c^2}\cos(\alpha), \end{gather}

(3.25b)\begin{gather}{\frac{{\rm d}h}{{\rm d}s}} = \sin(\alpha), \end{gather}

where $s$ is the arc length along the interface and $c$ is a constant chosen to match with the slip region. For small contact angles ($\theta _e\ll 1$) and free-surface flows ($\lambda =0$), one can take $c=3$ (Chan et al. Reference Chan, Kamal, Snoeijer, Sprittles and Eggers2020). In the limit of $\lambda =0$, the function $f(\alpha,\lambda )$ is given by

(3.26)\begin{equation} f(\alpha,0)={-}\frac{2\sin^3(\alpha)}{3(\alpha-\sin(\alpha)\cos(\alpha))}. \end{equation}

Integrating (3.25) and using $Ca\ll 1$, Snoeijer (Reference Snoeijer2006) showed that the structure of (3.20) can be recovered. In contrast to the method of matched asymptotics involving matching solutions from distinct regions (Cox Reference Cox1986; Dussan V. et al. Reference Dussan V., Ramé and Garoff1991), the generalized lubrication model (GLM) in (3.25) gives a full description of the interfacial profile in a more convenient way. For the present experiments, the GLM equations are solved using the following boundary conditions on the moving plate and the far field:

(3.27)\begin{equation} \alpha(s=0)=\theta_e;\quad \alpha(s\to\infty)=\frac{\rm \pi}{2}, \quad h(s=0)=0. \end{equation}

Detailed comparisons of the interface shape from the above two models with experiments are discussed in § 4.1.

5. Summary and discussions

In this study the velocity field in the vicinity of a moving contact line is determined experimentally and compared against well-known theoretical models. The nature of the flow field crucially depends on the static and dynamic contact angles, viscosity ratio and capillary number. All these parameters are systematically varied in the current study and compared with predictions of theoretical models, though viscosity ratio did not appear to play a very large role due to the small value of $\lambda$ in our experiments. Experiments are performed by vertically immersing a plate into a liquid bath at controlled speeds. Moving contact line studies can also be carried out in a number of other configurations. However, the reason behind choosing the above-mentioned configuration was to ensure that we have precise control over the speed of the contact line as well as the dynamic contact angle. Furthermore, the two-dimensional nature of the flow in the middle of the plate allowed accurate measurement of the flow field using PIV techniques. Another important reason for choosing this configuration was to ensure that the ‘local’ nature of the flow near the moving contact line is not affected by confinement effects, which are likely to be present in the flow fields inside drops or capillary tubes. Since care was taken to ensure that no extraneous flows were present before the start of the experiment, we believe that the local flow generated due to a moving contact line can be observed at considerably larger length scales than in other flow configurations. This also allows for accurate measurement of flow fields using conventional PIV techniques. The variation of dynamic contact angle with the capillary number, shown in figure 5, was found to be in excellent agreement with the classical work of Hoffman (Reference Hoffman1975), which also suggests that results are configuration/geometry independent. Therefore, the obtained flow fields should be similar to flow near a moving contact line in other flow geometries.

The results are grouped into three distinct categories based on the nature of the theoretical models. In § 4.1, interface shapes are compared against the DRG model of Dussan V. et al. (Reference Dussan V., Ramé and Garoff1991) and the GLM of Chan et al. (Reference Chan, Kamal, Snoeijer, Sprittles and Eggers2020). In § 4.2, flow fields from the experiments are compared against the MWS that in turn is based on the classical work of HS71. In § 4.3, interfacial speeds are compared against predictions from HS71's theory and the MWS.

All the experiments in this study involve silicone oils of varying viscosities. Since surface tension and density are nearly constant in these fluids, the capillary length, which is used as a measure of the scale of the ‘outer’ region, is the same across all experiments. The Reynolds number varied from $O(10^{-4})$ to $O(1)$ and the capillary number ranged from $O(10^{-5})$ to $O(10^{-2})$.

The low $Re$ and $Ca$ in all the experiments allow us to test various types of theoretical models. The DRG model was found to be capable of determining the interface shape over a wide range of length scales except in a narrow region near the contact line where viscous bending from the inner solution causes deviation of the interface shape. At the lowest $Ca$ values, the agreement between the DRG model and experiments was found to be nearly perfect over the entire range of spatial measurement. A test of the more recent model based on the generalization of lubrication theory for high interface slopes yielded satisfactory results. Since the value of slip length is not known, we fitted this model for a range of slip lengths. A slip length of 14 nm was found to give the best results for all the cases. The value of this slip length is consistent with published values of slip length reported in the literature. In the case of the DRG model, the static interface shape has a large influence on the overall shape of the full dynamic interface. This is especially evident when viscous effects become small such as in the 10 cSt oil experiments. However in the case of GLM, the static interface shape is not explicitly used in the model, but the model mainly relies on viscous effects to deform the dynamic interface. It is therefore not surprising that GLM best agrees with experiments in viscous-dominated cases such as when using 500 cSt oil. A similar effect of viscosity is also observed when comparing flow fields from experiments with theory.

An ambitious test of theoretical models can be made by directly comparing the predicted velocity field in the bulk against experiments. We test this by directly comparing the streamfunction values between experiments and the MWS, shown in figure 12. Excellent agreement is found for both the shape of the streamlines and the values of the streamfunction. The prediction from the MWS is found to be within 6 $\%$ deviation from the experiment, as shown in figure 13. It is worth reiterating that the HS71 or the MWS are identical to the leading-order solution in the intermediate region of Cox (Reference Cox1986); hence, a test of HS71 is equivalent to a direct test of the intermediate solution of Cox's model. Understandably, the agreement between theory and experiment deteriorates as one moves away from the contact line since the theoretical model of HS71 is only a local theory valid in the vicinity of the contact line. In the present study the MWS model required inputs of the local interface angle, $\beta (r)$, from experiments. We suggest that it is possible to predict flow fields at all length scales without any inputs from experiments, but this requires determining $\beta (r)$ from the generalized lubrication theory of Chan et al. (Reference Chan, Kamal, Snoeijer, Sprittles and Eggers2020).

One of the most interesting findings of the present study is the measurement of the interfacial velocity. Since the fluid particles approach the contact line, the interfacial velocity is negative. Away from the contact line, the interfacial velocity is largely similar to that predicted by HS71's theory. This is similar to what was also observed by Fuentes & Cerro (Reference Fuentes and Cerro2005), but their measurements were at much larger length scales than the present experiments. There is a small deviation in the experiments from the value predicted by HS71's theory. This deviation can be easily accounted for by recognizing that the interface is curved. The inset of figure 14(a) shows that accounting for the curvature of the interface in the MWS increases the velocity marginally and is in excellent agreement with experiments. Recall that in HS71's theory, the interfacial velocity is independent of the radial location from the contact line. This means that fluid particles just below the interface approach the contact line at a constant speed and are required to make an instantaneous turn at the contact line, and subsequently move along the plate. These fluid particles will be required to possess an infinite acceleration as $r \rightarrow 0$. This is yet another manifestation of HS71's singularity. Contrary to theoretical predictions, what we find in the experiments is that the interfacial velocity rapidly decreases to very small values as one approaches the contact line. This rapid reduction in speed to a near-zero velocity is non-monotonic. There is a small increase in the speed at a distance of approximately $150\,\mathrm {\mu }{\rm m}$ from the contact line and then a rapid decrease to a very small speed as shown in figure 14(a,b). The singularity is thus prevented by allowing the fluid to approach the contact line at smaller speeds, shown schematically in figure 15. These fluid particles are then expected to slip along the moving plate, but then soon adjust to the no-slip condition along the surface of the plate. To the best of our knowledge, such a behaviour of fluid particles near the contact line has not been reported in experiments before, and is completely consistent with the maxim that ‘nature abhors singularities’.

Figure 15.

Typical flow pattern observed in all experiments showing rolling motion in the bulk fluid (black dashed curves) consistent with figure 12. Fluid particles at the interface rapidly slow down as the contact line is approached consistent with figure 14, before eventually turning down to move along the plate.

We hope that these experimental findings will spur new theoretical developments by incorporating necessary physics in the inner and intermediate regions. Furthermore, quantitative data about interface shape and flow fields in the viscous phase is available for download as supplementary material and we hope new computational models will be validated against the PIV data provided in this paper.