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The article delves into the intricate dynamics of bubble departure from surfaces in a liquid cross-flow, focusing on the influence of surface wettability and shear rates. It examines how different substrates, such as plain steel and PDMS pseudo-brush, affect bubble morphology and departure mechanisms. The study identifies distinct regimes of bubble departure, including capillary, transition, and inertial regimes, and analyzes dynamic wetting processes during departure. Key findings include the observation of capillary surface waves on PDMS and the comparison of departure mechanisms to jet breakup at high shear rates. The research provides a comprehensive understanding of bubble departure dynamics, which is essential for optimizing processes in industries like chemical engineering, food and beverage, and pharmaceuticals. The article concludes by highlighting the importance of tailored surface wettability for efficient gas/liquid contact processes and suggests future research directions to further explore the influence of wettability on bubble dynamics.
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Abstract
Bubble departure from walls in cross-flow conditions has been studied. This process plays a fundamental role in numerous applications, from gas-liquid contacting, heterogeneous nucleation in cavitation, to flow boiling. Predicting the size of the detaching bubble is essential for controlling the liquid–gas interfacial area in gas/liquid systems encountered in these applications. However, existing data on bubble departure in cross-flow are often restricted to low wall shear rates where bubble deformation is minor and thus negligible. Moreover, the effect of surface wettability on bubble departure is usually not reported in detail. Therefore, in this study, we analyse bubble deformation and departure from artificial nucleation sites for elevated wall shear rates using high-speed imaging. In addition, the surface wettability was modified and quantities related to dynamic wetting phenomena were examined. Our findings clearly show that wettability of the surface and the dynamic pressure imposed on the bubble by the flow jointly dictate how a bubble deforms prior to departure and how large its size is on departure. In addition, different mechanisms of departure are identified and reported in a regime map based on the critical deformation and the wall shear rate.
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1 Introduction
The formation of bubbles on the interior surface of a drinking glass filled with a carbonated beverage provides a familiar example of heterogeneous nucleation - a process that can be observed by anyone at lunch. In contrast to homogeneous nucleation in the bulk liquid, the energy barrier for nucleation at surface irregularities is considerably lower (Jones et al. 1999a). After formation, bubbles then grow at these nucleation sites, detach once they reach a critical size, and rise to the surface. Even for the relatively simple case of a bubble on a glass surface, it is challenging to predict the precise moment and mechanism of bubble departure. The challenge arises from the presence of a solid surface and the existence of a three-phase contact line between the liquid, gaseous, and solid phases as a boundary. If a contact line is present, wetting phenomena need to be taken into account. Wetting influences a bubble’s dynamics because additional capillary forces arise from it. For the above-described bubble attached to a solid wall, these wetting processes are inevitably coupled to the mass transport of carbon dioxide through the bubble interface.
If a bubble on a wall is additionally exposed to an external flow field, its interface is likely to deform in response to the dynamic pressure distribution around it (Clift et al. 2005). In the following, cross-flow is considered, in which the liquid flows transversely to the bubble’s initial direction of growth. For a hemispherical bubble attached to a wall, this corresponds to liquid motion tangential to the wall and perpendicular to the bubble’s axis of symmetry (Kulkarni and Joshi 2005). As part of the bubble interface, the contact line is forced to move in response to the external forces exerted by the flow (Blake 2006). The external forcing can then also cause the system to undergo a dynamic wetting transition (Snoeijer and Andreotti 2013). One particularly relevant wetting transition in flowing conditions is the bubble leaving the nucleation site in a liquid cross-flow. Once the bubble leaves the nucleation site, it is advected with the bulk flow and becomes a free bubble in the gas-liquid system. Thus, understanding bubble departure is relevant to make predictions about bubble size distributions in disperse gas/liquid systems (Clift et al. 2005).
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Bubble departure in a liquid cross-flow is encountered in problems of bubble injection through orifices or in systems where heterogeneous nucleation on walls occurs (Kulkarni and Joshi 2005). The latter can be driven by a concentration gradient, e.g. a gas desorption process (Jones et al. 1999a; Groß and Pelz 2017), or by a temperature gradient, e.g. flow boiling (James Klausner et al. 2001a). The main goal of the majority of works on bubble departure has been to assess or predict the size of detaching bubbles under various operating conditions. In the following paragraphs key results in these fields regarding bubble departure in cross-flows are summarised to present the current state of the field.
A comprehensive review on bubble formation from submerged orifices was provided by Kulkarni and Joshi (2005). Several authors developed detachment models based on force balance or potential flow theory to derive predictive models for the size of the detaching bubble (Wace et al. 1987; Marshall et al. 1993; Forrester and Rielly 1998; Tan et al. 2000). As Tan et al. (2000) pointed out, bubble shapes for higher shear rates cannot be assumed to be spherical and developed a model for detachment with which they successfully predicted the location of neck formation, the thinning region where the bubble separates. In a more recent work Mirsandi et al. (2020b) showed that for elevated shear rates the bubble interface elongates downstream prior to departure. Drag forces also substantially flatten the bubble and change the location where neck formation occurs.
Heterogeneous bubble nucleation caused by mass transfer in stagnant liquids is extensively reviewed by Jones et al. (1999a, 1999b). The nucleation in liquid cross-flow is relevant for the formation of cavitation nuclei, as shown by (van Wijngaarden 1967; Groß and Pelz 2017; Groß et al. 2018). For diffusively growing air bubbles Al-Hayes and Winterton (1981) conducted experiments to determine the maximum diameter of bubbles in liquid flow on glass substrates before departure. A recent analysis by (Ren et al. 2021) has emphasised that very high wall shear rates \(\sim {1 \times 10^{7}}\,{\text{s}^{-1}}\) cause deformations and tether formation even for very small micro- and nanobubbles.
The detachment of vapour bubbles from heated surfaces alters the heat transfer behaviour and is therefore studied in flow boiling (Zeng et al. 1993; Klausner et al. 1993; Thorncroft et al. 1998; James Klausner et al. 2001a; van Helden et al. 1995). For vapour bubble departure James Klausner et al. (2001b) reviewed force balance models for nucleate and flow boiling. They account for a mildly asymmetric growth of the vapour bubble.
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There is a consensus in all the mentioned fields of research that cross-flow greatly alters the size of the detaching bubbles. Furthermore, the limitation of small deformation in current modelling approaches is acknowledged. Assuming small interface deformation is reasonable in most reviewed studies due to maximum shear rates of \(\approx 10^3\,{\text{s}^{-1}}\). However, shear rates in fluid turbomachinery, industrial pipe networks, or gas-liquid contactors can reach orders of magnitude between \(10^3\) and \(10^6\,{\text{s}^{-1}}\), where it is not justified to assume a minor degree of deformation.
Additionally, we emphasise that departure is closely related to research on disintegration of fluid particles. The splitting of free bubbles or drops was studied in numerous works Hinze (1955); Acrivos (1983); Kang and Leal (1990); Stone (1994). The disintegration is usually categorised with respect to the forces controlling the breakup for various flow patterns. Different mechanisms of bubble detachment were reported by Wace et al. (1987) that included one mechanism similar to the breakup of a gaseous jet (cf. Eggers and Villermaux (2008)) due to hydrodynamic instability, as first by proposed by Silberman (1957).
Despite its obvious important influence on bubble departure from walls, wettability has received relatively little attention in studies. For gas injection through submerged orifices into a shear flow Duhar and Colin (2006) used a force balance approach and derived a departure criterion based on the contact angles for small bubble Reynolds numbers. Wesley et al. (2016) highlighted that the modification of the wettability and topography of the surface strongly affects bubble formation. Al-Hayes and Winterton (1981) in their already mentioned experiments varied the equilibrium contact angles of the glass surfaces used from 22\(^\circ\) to 90\(^\circ\). Their results highlight that the detaching bubble size is almost twice the size for surfaces with higher contact angles. Mirsandi et al. (2020b) experimentally and numerically investigated how wetting conditions affect bubble formation from submerged orifices using hydrophobic surfaces and ethanol-water solutions to vary wettability. They have reported strong effects of wettability on the bubble contact diameter for various liquid surface tensions. Allred et al. (2021) presented a theoretical framework for bubble growth in the case of nucleate boiling. Their results show that the receding contact angle primarily controls bubble size and departure, highlighting the importance of surface characterisation by dynamic rather than static contact angles. To our knowledge, no comprehensive analysis of dynamic wetting during bubble departure for elevated shear rates in a cross-flow has yet been reported. Overall, the literature review indicates that data on bubble departure in liquid cross-flow at high shear rates remain scarce, particularly regarding (i) interface deformation and (ii) the influence of surface wettability. Based on the reviewed literature, we argue that any physically consistent departure criterion for a bubble in flowing liquids must account for these two influencing factors. Reliable criteria could allow one to predict and control bubble size distributions in gas-liquid systems where heterogeneous nucleation takes place. A necessary prerequisite for such criteria are reliable experimental datasets of bubble departure.
This work focusses on the experimental investigation of bubble departure from surfaces in a liquid cross-flow. The bubbles grow due to interfacial mass transfer, as illustrated in Fig. 1 (cf. van Wijngaarden (1967); Groß and Pelz (2017)). The primary objectives are to analyse how altering the surface wettability influences the bubble shape and contact line motion during departure and how the size of the detaching bubble behaves for various shear rates.
Our experimental results show that bubble morphology and mechanisms of departure vary greatly between two substrates of different wettability. The interplay between drag, surface tension, and capillary forces results in distinct regimes of departure that we categorise on the basis of experimental observations. Moreover, owing to high spatial and temporal resolution of the image acquisition system, oscillative phenomena are captured that allow further inspection of the physics of bubble departure in future works.
Fig. 1
Schematic of the diffusion-driven bubble growth at an artificial nucleation site with diameter \(d_{\textrm{p}}\) in a liquid cross-flow (density \(\varrho\), dynamic viscosity \(\mu\)) with velocity field u(z). The bubble grows by interfacial mass transfer \(\dot{m}\), driven by the concentration difference between the bulk and the interface, \(c _{\mathrm {\infty }}- c _{\textrm{I}}\). Bubble departure occurs at the critical time \(t_{\textrm{c}}\), over a much shorter time scale compared to the bubble growth period \(T\). This study focusses on the rapid process of departure
We intend to study the detachment of a single bubble from a nucleation site in a cross-flow subject to wall shear rates \(\dot{\gamma }_{\textrm{w}}\) exceeding \(10^{3}\,{\text{s}^{-1}}\). The level of gas supersaturation \(\zeta \approx 0.3-0.35\) is moderate. Using moderate supersaturation should reflect the conditions in hydraulic circuits and fluid machinery in contact with atmospheric air. This section describes the experimental setup as well as substrate preparation and characterisation.
2.1 Experimental setup
2.1.1 Hydraulic hardware
Experiments are carried out using a hydraulic circuit that features an optically accessible test section for observation; see Fig. 2. A stainless steel pressure vessel with a volume of \({70}\,{\text{L}}\) on the one hand acts as a reservoir to saturate the liquid with gas, while on the other hand it acts as a hydraulic accumulator to dampen the pulsations of the progressing cavity pump that is driving the flow (Netzsch NEMO NM021-BY). All components are interconnected through marine grade 316 L stainless steel pipes and valves. The liquid flow rate is varied by altering the pump’s rotational speed and monitored using an electromagnetic flowmeter (ABB FEH-630).
For all experiments, water purified by a reverse osmosis system was used, whose low total dissolved solids (TDS) concentration was 14–18 ppm and its electrical conductivity ranged from \({10} \, \text{to} \, {15}\,{\upmu \text{S}/\text{m}}\). The water was also filtered using a three-stage submicron filter (\(< 0.3 \, \upmu \text{m}\)) unit.
Inert gas is supplied to the liquid in the pressure vessel as pressurised air through a sinter metal bubble sparger (GKN Powder Metallurgy, Inc.) that is installed on the bottom of the pressure vessel. The sparger produces microbubbles between \({1}\,\upmu \text{m}\) to \({20}\,\upmu \text{m}\) and enhances the efficiency of the gas absorption process. The pressure level during the saturation process is controlled by a pressure regulator (FESTO LRP) and measured in the vessel by a pressure transducer also installed in a bottom flange port. The oxygen level in the liquid phase was monitored using an optical dissolved oxygen (DO) sensor (Hamilton VISIFERM mA 120). To accelerate desorption, the test rig can operate below atmospheric pressure via a vacuum nozzle (FESTO vacuum generator VN).
A plate heat exchanger (HYDAC HEX series) was employed to maintain a nearly constant fluid temperature throughout the experiments by compensating for the dissipative heat input from the pump. When necessary, an auxiliary immersion heater (Vulcanic type 2300) was available to provide additional heating of the working fluid. All measurements reported in this study were conducted at bulk fluid temperatures maintained within the range \({18}\, \text{to}\, {20}\,^{\circ }\,{\text{C}}\). The sensor specifications used in the experimental setup are summarized in Table 1.
Fig. 2
Hydraulic flow diagram showing the main components of the experimental setup
List of used sensor types and respective measurement ranges and accuracies
Sensor
Type
Range
Accuracy
Keller PA-33X
Pressure
0 to 10 bar absolute
0.05 % FS
ABB FEH-630
Flow rate
0.4 to 25 L/min
0.4% relative
Therma GmbH Pt100
Temperature
0 to \(100\,^{\circ }\)C absolute
0.033% FS
Hamilton VisiFerm mA
Oxygen concentration
0 to 25 mg/L
1% relative
The acrylic-glass test section encloses a rectangular conduit with fixed height \(H = {1.86}\,{\text{mm}}\) and width \(W = {12}\,{\text{mm}}\), see Fig. 3. The value of \(H\) was determined from a side-view digital image of the channel with backlight illumination. The aspect ratio of the channel is \(W / H = 6.45\), so a slit flow is established. The velocity field on the centreline of the channel then becomes two-dimensional (unidirectional), and secondary flow effects are avoided. The static pressure within the channel is obtained via a pressure tap, while the fluid temperature is subsequently measured downstream of the test section.
The substrate is mounted on a substrate carrier and fixated with transfer adhesive tape. The substrate carrier is then introduced into the test section through an opening in the acrylic glass from below so that it is flush with the bottom wall of the channel, cf. Fig. 3. To avoid obstruction of view due to undesired nucleation, a clearance of \({1}\,{\text{mm}}\) was maintained between the side walls of the channel and the substrate. The detailed preparation and characterisation routine for the substrate is explained in Sect. 2.4.
Fig. 3
Left: Cross-sectional view of the acrylic-glass test section. The substrate platelet is mounted on a steel carrier using heavy-duty transfer tape. The flow direction is from left to right, entering through a nozzle at the channel entrance. Static pressure is measured upstream via a pressure tap to minimise flow disturbances to the bubble. Right: Enlarged schematic view illustrating a bubble at the nucleation site (diameter \(d_{\textrm{p}}\)) in the cross-flow (velocity component u(z). Up- and downstream contact point locations \(x_{\textrm{u}},\, x_{\textrm{d}}\) and corresponding contact angles \(\theta _{\textrm{u}}, \theta _{\textrm{d}}\) are indicated. The rectangular duct has a height \(H = {1.86}\,{\text{mm}}\)
The experimental procedure consists of preparing a saturated solution of water and dissolved air, the subsequent adjustment of the operating point of the channel flow, and the observation of bubble departure in the test section.
The initial nucleation and growth of bubbles at the nucleation sites is driven by a concentration gradient of dissolved gas between the bulk liquid and the bubble interface \(c _{\mathrm {\infty }}- c _{\textrm{I}}\), cf. Fig. 1. The excess level of dissolved gas is quantified by the dimensionless level of supersaturation \(\zeta\), which is defined as
$$\begin{aligned} \zeta \left( p , T \right) := c \left( p , T \right) / c _{\textrm{s}} \left( p , T \right) - 1, \end{aligned}$$
(1)
where \(c\) is the actual gas concentration and \(c _{\textrm{s}} \left( p , T \right)\) the saturation concentration for a given pressure \(p\) and temperature \(T\). Following Henry’s law (Henry 1803) for a state of ideal dilution, the concentration of dissolved gas in the liquid is proportional to the pressure of the gas phase; hence,
$$\begin{aligned} c _{\textrm{s}} = k_{\textrm{H}}( T ) \, p \, , \end{aligned}$$
(2)
where \(k_{\textrm{H}}\) is the Henry constant.
The gas absorption process to reach supersaturation occurs in the pressure vessel while the liquid is recirculated at low volumetric flow rate. During absorption, air is dissolved into the water at a controlled conditioning pressure \(p _{\textrm{cond}}\) and temperature until equilibrium of the mixture is established. The ratio of \(p _{\textrm{cond}}\) and the desired static pressure in the test section under flowing conditions \(p _{\mathrm {\infty }}\), determines \(\zeta _{\infty }\) for the liquid solution. In all experiments, \(p _{\mathrm {\infty }}\) in the test section was controlled and held constant at \(p _{\mathrm {\infty }}= {1.2}\,{\text{bar}}\), while the flow rate was varied. Using Henry’s law and accounting for a finite vapour pressure of the gas phase of the water, the conditioning pressure \(p _{\textrm{cond}}\) for a bulk supersaturation in the test section \(\zeta _{\infty } = 0.3\) is determined by
$$\begin{aligned} p _{\textrm{cond}} = \left( \zeta + 1 \right) \, \left( p _{\mathrm {\infty }}- p _{\textrm{v}}\right) + p _{\textrm{v}}= {1.55}\,{\text{bar}}. \end{aligned}$$
(3)
For example, a supersaturation \(\zeta _{\infty } = 0.3\) is equivalent to a concentration of dissolved oxygen \(c _{\text{O}_{2}} = {13.65}\,\text{mg}/\text{L}\) at \(T _{\mathrm {\infty }}= {20}\,^{\circ }\,{\text{C}}\) and \(p _{\mathrm {\infty }}= {1.2}\,{\text{bar}}\). Oxygen concentration levels are continuously monitored by the DO sensor in the pressure vessel.
After equilibration to a saturated state, the gas inlet to the bubble sparger is closed, and the intended flow rate for an experiment is set. The imposed flow rate determines the pressure drop along the circuit up to the location of the test section. The static pressure in the circuit is regulated to maintain the target value of \(p _{\mathrm {\infty }}= {1.2}\,{{\text{bar}}}\) in the test section.
Because the relaxation of the dissolved air in the hydraulic circuit is slow compared to the duration of a single experiment, the saturation concentration established at \(p _{\textrm{cond}}\) remains effectively constant. If the oxygen concentration drops below a threshold, the liquid is saturated again. We monitor the level of supersaturation by measuring the dissolved oxygen concentration \(c_{\text{O}_{2}}\). The saturation concentration of air \(c _{ {\textrm{air, s}}}\) for a given temperature and pressure is derived from tabulated solubility data sets (Battino et al. 1984). From additional data on the temperature-dependent composition of dissolved air in water (Winkler 1901) we can derive the corresponding saturation concentration of oxygen \(c _{{{\text{O}_{2}}}, {\textrm{s}}}\).
The value for \(\zeta _{\infty } = 0.3\) is well above the nucleation threshold, i.e. the minimal necessary value \(\zeta _{\infty ,\textrm{min}}\) where initial nuclei form in the pore of the nucleation site (Borkent et al. 2009). In fact, this minimal threshold is negative:
$$\begin{aligned} \zeta _{\infty ,\textrm{min}} = \dfrac{4 S }{ p _{\mathrm {\infty }}\, d_{\textrm{p}}} - \dfrac{ p _{\textrm{v}}}{ p _{\mathrm {\infty }}} \approx - 0.019 . \end{aligned}$$
(4)
In contrast to diffusive bubble growth in a stagnant liquid, the effect of concentration depletion of dissolved gas around the bubble (cf. Moreno Soto et al. (2019)) is negligible, since the supersaturated solution is continuously supplied to the test section from the pressure vessel.
The pump speed was varied to adjust the flow rate in the hydraulic circuit. Changes in flow rate modified the velocity field of the channel flow in the test section, which in turn determined the near-wall flow field where bubble departure was investigated.
2.2 Dimensionless groups characterising bubble departure
The operating point of an experiment is determined by the dynamics of the channel flow by adjusting a volume flow rate \(Q\) and hence a channel Reynolds number \(R\hspace{-0.39993pt}e _{\infty }:= U _{\infty }\, D_{\textrm{h}} / \nu\) based on the mean channel velocity \(U _{\infty }= Q / A\) with flow rate \(Q\) and cross section \(A\), hydraulic diameter \(D_{\textrm{h}} :=2 \, H W / \left( H + W \right) = {3.22}\,{\text{mm}}\) and kinematic viscosity of the liquid \(\nu = 10^{-6}\, \mathrm {m^2/s}\) at the operating temperature \(T_{\infty }\). The amount of dissolved air is given by the previously defined supersaturation \(\zeta _{\infty }\) and the cavitation number \(\sigma :=2\left( p _{\mathrm {\infty }}- p _{\textrm{v}}\right) / \left( \varrho U _{\infty }^2\right)\), where \(p _{\textrm{v}}\) is the vapour pressure of the liquid.
On the length scale of the bubble relevant acting forces are the inertial and capillary forces whose orders of magnitude are \(\varrho \, U^2 \, L^2 \text { and } S \, L\), respectively. Here, \(\varrho\) is the density of the liquid, U the characteristic velocity of the problem, L the characteristic length and \(S\) the surface tension between air and water. The ratio of these two forces is a Weber number \(W\hspace{-1.4pt}e :=\varrho \, U^2 \, L / S\). The characteristic length is set to \(d_{\textrm{p}}\) and the characteristic velocity follows as \(\dot{\gamma }_{\textrm{w}}\, d_{\textrm{p}}\). This way, a shear-based Weber number \(W\hspace{-1.4pt}e :=\varrho \, \dot{\gamma }_{\textrm{w}}^2 \, d_{\textrm{p}}^3 / S\) is identified. By comparing inertial forces to viscous forces in the same manner a shear-based Reynolds number is deduced as \(R\hspace{-0.39993pt}e :=\varrho \, U^2 \, L^2 / (\mu \, U \, L) = \dot{\gamma }_{\textrm{w}} \, d_{\textrm{p}}^2/ \nu\). Both shear-based quantities \(W\hspace{-1.4pt}e\) and \(R\hspace{-0.39993pt}e\) are defined using the wall shear rate \(\dot{\gamma }_{\textrm{w}}\), which is the shear rate \(\dot{\gamma }|_{z = 0}\). For low values of \(\dot{\gamma }_{\textrm{w}}\), inertial forces are comparable in magnitude to surface tension forces, hence \(W\hspace{-1.4pt}e \approx 1\), whereas inertial forces will dominate for large shear rates, hence \(W\hspace{-1.4pt}e \gg 1\). For \(W\hspace{-1.4pt}e> 1\) we expect inertial forces to cause interface deformation for the bubble. Buoyancy forces are not relevant, as the Bond number \(B\hspace{-0.59998pt}o :=\left( \varrho - \varrho _{\textrm{g}}\right) g \, d_{\textrm{p}}^2 / S \approx 5 \times 10^{-3} \ll 1\). Here \(g\) is the gravitational acceleration and \(\varrho _{\textrm{g}}\) the density of the gas phase inside the bubble. By separating \(W\hspace{-1.4pt}e\) into components with time dimensions, namely \(W\hspace{-1.4pt}e = \dot{\gamma }_{\textrm{w}}^2 \, t_\textrm{inert}^2\), a characteristic inertial time scale \(t_\textrm{inert} :=\sqrt{\varrho \, d_{\textrm{p}}^3 / S }\) for bubble departure and breakup is obtained. At \({20}\,^{\circ}{\text{C}}\), this inertial time scale \(t_\textrm{inert} = {0.33}\,{\text{ms}}\).
An artificial nucleation site with \(d_{\textrm{p}}= {200}\,\upmu \text{m}\) can be considered as an idealised and enlarged model of a micro-crevice with an estimated diameter of about \({10}\,{\upmu \text{m}}\). The scaling factor between both nucleation sites is then 20. Upscaling the nucleation site in this way ensures that the optical setup has sufficient spatial resolution to capture bubble departure. However, upscaling leads to incomplete similarity between artificial nucleation sites of different size, because \(W\hspace{-1.4pt}e\) and \(R\hspace{-0.39993pt}e\) cannot be altered independently, cf. Barenblatt (2012). Despite this fact, we expect that findings on the physics of bubble departure are transferable to smaller nucleation sites. Thus, in addition to pores with \(d_{\textrm{p}}= {200}\,{\upmu \text{m}}\), we study this problem of incomplete scaling with few measurements on smaller pores with \(d_{\textrm{p}}= {100}\,{\upmu \text{m}}\).
2.3 Numerical analysis of the channel flow
To obtain detailed information about the flow field, a computational fluid dynamics (CFD) analysis was carried out. For a given channel Reynolds number \(R\hspace{-0.39993pt}e _{\infty }\), the simulation yields a velocity profile u(z) in the wall-normal z-direction. From this, the wall shear rate can be determined as the velocity gradient at the wall, i.e. \(\dot{\gamma }_{\textrm{w}} = \textrm{d}u /\textrm{d}z |_{z = 0}.\)
To obtain detailed information about \(\dot{\gamma }_{\textrm{w}}\), a three-dimensional single-phase RANS (Reynolds-Averaged Navier Stokes) simulation was carried out. The channel height was set to \(H = {1.86}\,{\text{mm}}\), corresponding to an aspect ratio of \(W / H = 6.45\), matching the experimental geometry. Simulations were performed using the simpleFOAM solver in OpenFOAM. To correctly capture the near-wall flow behaviour, the k–\(\omega\) SST turbulence model was used. Within the buffer layer, the universal velocity law proposed by Spalding (Spalding 1961) was applied. A structured O-grid topology is used for meshing to resolve the boundary layer. The height of the first cell adjacent to the wall was set to \({0.5}\,{\upmu \text{m}}\) with a growth factor of 1.2 ensuring that the dimensionless wall distance \(y^+\) remained below 0.5.
The resulting profiles for the velocity component in x-direction u(z) and the shear rate \(\dot{\gamma }(z)\) are shown in Fig. 4. CFD simulations also confirm that for a channel aspect ratio of 6.45 the flow field is two-dimensional on the centreline of the channel. The entrance length in the streamwise x-direction downstream of the nozzle is considered sufficiently long, as the wall shear rate \(\dot{\gamma }_{\textrm{w}}\) does not vary significantly at the substrate location along the flow direction. Under experimental conditions, the flow is classified as transitional, i.e. within the regime between laminar and fully turbulent flow. This computation in the end yields a mapping between \(R\hspace{-0.39993pt}e _{\infty }\) and the previously defined shear-based quantities \(W\hspace{-1.4pt}e , R\hspace{-0.39993pt}e\), see Fig. 5.
Fig. 4
Profiles of the time-averaged streamwise velocity component u(z) and shear rate \(\dot{\gamma } = \textrm{d}u/\textrm{d}z\), extracted from 3D RANS-CFD simulations of a rectangular duct with height \(H = {1.86}\,{\text{mm}}\) and aspect ratio \(W/H = 6.45\). The profiles are shown up to mid-channel height H/2 for four different volumetric flow rates \(Q = 0.5, 1.5, 3.0, 4.5 \,\mathrm {L/min}\) corresponding to \(R\hspace{-0.39993pt}e _{\infty }= 1114,3344,6689,10033\). The inset plot shows that the CFD data follow the law of the wall, which is here represented by the model of Spalding (1961)
Mapping between the channel Reynolds number \(R\hspace{-0.39993pt}e _{\infty }\) and the shear-based dimensionless numbers \(W\hspace{-1.4pt}e\) and \(R\hspace{-0.39993pt}e\) for flow in a rectangular duct with aspect ratio \(W/H = 6.45\). The scatter points represent data at the centre of the substrate obtained from the 3D RANS-CFD simulations
Similar to Borkent et al. (2009) and Groß et al. (2018) cylindrical cavities with diameter \(d_{\textrm{p}}= {200}\,{\upmu \text{m}}\) are used as artificial nucleation sites. In this work, cavities are created by drilling in stainless steel platelets (\({10\, \text{mm} \times 10\, \text{mm}}\)), cf. Fig. 3. We study isolated bubbles emerging from a single, nucleation site in the centre of the substrate. Sharp edges around the cavity’s opening produced by drilling and milling are potential pinning sites for the three-phase contact line (Oliver et al. 1977; Wang et al. 2019). Once the bubble has emerged outside the cavity, its shape only depends on the diameter of the circular cavity opening and not on its geometric form, as shown by Chappell and Payne (2007). Our results should therefore be valid for other axisymmetric cavity geometries. The depth of the blind holes is chosen to be \(2 \, d_{\textrm{p}}\) to ensure that the interface starts deep within the cavity to meet the conditions described by Chappell and Payne (2007).
The steel substrate is a flat, mechanically smooth, and rigid surface, but chemically inhomogeneous. The surface finish for all substrates consists of a series of polishing and lapping steps to achieve a very smooth surface roughness, cf. Table 2 for all processing steps. The final arithmetic mean height of the substrate surface is \(S_\textrm{a} = 17.7 \pm 0.9 \, \text{nm}\) obtained by confocal microscopy (NanoFocus \(\upmu\)surf). The fine polishing of the surface reduces the effect of surface roughness on contact angle hysteresis (Butt et al. 2022). In addition, the majority of potential secondary pinning sites besides the artificial nucleation site is removed by polishing.
Table 2
Fine-polishing routine applied to all stainless steel platelets
Step
Surface
Abrasive
Lubricant
Rot. speed (rpm)
Duration (min)
1
Abrasive paper
Silicon carbide (SiC), 4000 grid
Water
150
3
2
PT Pan
Diamond, \({3}\,{\upmu \text{m}}\)
Lubricant Blue
150
4
3
PT Pan
Diamond, \({1}\,{\upmu \text{m}}\)
Lubricant Blue
150
4
4
PT Skin
Diamond, \({1}\,{\upmu \text{m}}\)
Lubricant Blue
150
2
PT Pan, PT Skin and Lubricant Blue are product of Cloeren Technology GmbH (Germany)
2.4.2 Wettability modifications
We intend to make the steel slightly more hydrophobic by applying a Polydimethylsiloxane (PDMS) coating. Before coating, stainless steel substrates (V2A, material number 1.4301 according to DIN EN 10088) are cleaned with Ethanole and Acetone (Fisher Scientific Co., 95%) and isopropanol (Fisher Scientific Co., 95%) to eliminate potential contaminants adhering to the substrate surface. To coat the substrates with a nanometric PDMS pseudo-brush coating, a thin layer of silicon oil (DMS-T5, Gelest, Inc.) via casting a drop of (\({5}\,{\upmu \text{L}}\) that covers the whole surface of the cleaned substrate. The substrate is then heated on a hotplate for three minutes at \({220}\,^{\circ }{\text{C}}\). Subsequently, the surface is washed with acetone, isopropanol, and ultra-pure water in order to wash free PDMS oligomers away. At the end of this process, a thin, nanoscopic layer of PDMS pseudo-brushes covers the substrate surface (Rostami et al. 2023). As the coating is nanometric in thickness, it does not change the geometric properties of the nucleation site. A detailed procedure for preparing PDMS pseudo-brushes has also been reported in a previous publication by the co-authors (Rostami et al. 2023). Both steel and PDMS are partially wetting substrates, as indicated by a static contact angle \(\ge {90}^{\circ }\). How a liquid spreads on a surface is described by so-called spreading parameter \(\mathcal {S}\), defined as \(\mathcal {S} :=S_{\textrm{sl}} - (S_{\textrm{sg}} + S)\) with \(S_{\textrm{sl}}\) being the surface tension at the solid/liquid, \(S_{\textrm{sg}}\) at the solid/gas, and S at the liquid/gas interface. The spreading parameter can be calculated using \(\mathcal {S} = S \left( \cos \theta _{\textrm{m}}- 1 \right)\). The work of adhesion is connected to \(\mathcal {S}\) via \(W_{\textrm{a}} = -\mathcal {S}\). For partially wetting surfaces \(\mathcal {S} < 0\) (de Gennes et al. 2004).
2.4.3 Wettability characterisation
The substrate material, stainless steel, is a metallic high-energy surface. Although fine polishing produces a mechanically smooth finish, steel is not atomically smooth and therefore represents a non-ideal surface from the perspective of wetting (de Gennes et al. 2004). The liquid’s affinity for the surface is characterised by the mean contact angle, \(\theta _{\textrm{m}}\). For dynamic wetting, the key parameter is the contact angle hysteresis (Butt et al. 2022), defined as \(\Delta \theta :=\theta _{\textrm{a}} - \theta _{\textrm{r}},\) where \(\theta _{\textrm{a}}\) and \(\theta _{\textrm{r}}\) are the advancing and receding contact angles, respectively. \(\Delta \theta\) quantifies the resistance of a drop or bubble to lateral motion on the surface. The corresponding lateral adhesion force can be calculated as follows (Buzágh and Wolfram 1958):
$$\begin{aligned} F_{a}= w S k(\cos (\theta _{r})-\cos (\theta _{a})) \end{aligned}$$
(5)
where \(S\), w, k, \(\theta _{r}\) and \(\theta _{a}\) are surface tension, drop width, geometrical factor, receding contact angle and advancing contact angle, respectively. Consequently, The lower \(\Delta \theta\) the less force in lateral direction is needed to slide the sessile drop over the surface. There are several ways to measure \(\Delta \theta\) (e.g. the Wilhelmy plate (Tretinnikov and Ikada 1994), the drop inflation/deflation (Gaudin et al. 1963; Wong et al. 2020), drops on tilted substrates (Kawasaki 1960; Le Grand et al. 2005) and etc. (Butt et al. 2022)).
In the present study, the drop inflation/deflation method was employed to characterise the wettability of the surfaces. A water drop with a volume of \({3}\,{\upmu \text{L}}\) was first deposited on the substrate. By gradually inflating the drop at a preset volumetric flow rate of \(0.05\,{{\upmu \text{L}} \, \text{s}^{-1}}\) using a syringe pump, the contact angle increased until the contact line advanced on the substrate. The contact angle measured at this point is defined as the advancing contact angle, \(\theta _{\textrm{a}}\). Subsequently, the liquid was withdrawn until the contact angle receded on the surface, resulting in the receding contact angle, \(\theta _{\textrm{r}}\). This process has been followed for five surfaces of steel and PDMS.
The static contact angle \(\theta _{\textrm{m}}\), the advancing angle \(\theta _{\textrm{a}}\) and the receding angle \(\theta _{\textrm{r}}\) are reported in the bar graph in Fig. 6. The static contact angle \(\theta _{\textrm{m}}\) is defined here as the contact angle measured with the droplet at rest after placing it on the substrate and before inflation and deflation. Its value lies between \(\theta _{\textrm{a}}\) and \(\theta _{\textrm{r}}\). When the drop is placed on the substrate, it spreads and the contact line advances, so that \(\theta _{\textrm{m}}\) is closer to \(\theta _{\textrm{a}}\). For steel, \(\theta _{\textrm{m}}= {87}^{\circ }\), while the more hydrophobic surface coated with PDMS exhibits \(\theta _{\textrm{m}}= {106.4}^{\circ }\). The contact angle hysteresis is comparable for both surfaces, with \(\Delta \theta = {32}^{\circ }\) for steel and \({37.9}^{\circ }\) for PDMS.
Fig. 6
Wettability characteristics of the two surfaces, measured using the inflation/deflation method. \(\theta _{\textrm{m}}\): static contact angle; \(\theta _{\textrm{a}}\): advancing contact angle; \(\theta _{\textrm{r}}\): receding contact angle; \(\Delta \theta\): contact angle hysteresis
The higher value \(\theta _{\textrm{m}} = {106.4}^{\circ }\) on PDMS corresponds to a lower work of adhesion \(W_{\textrm{a}}\), which implies that the area of contact between the liquid drop and the modified PDMS surface is smaller compared to steel. Consequently, the spreading parameter \(\mathcal {S}\) on PDMS evaluates to \({-93.4 \times 10^{-3}}\,{\text{N} \text{m}^{-1}}\) while in steel it is only \({-69.0e-3}\,{\text{N} \text{m}^{-1}}\) for \({20}\,^{\circ }{\text{C}}\). In the terminology of de Gennes et al. (2004), both substrates are partially wetting; however, PDMS is considered mostly non-wetting, while steel is classified as mostly wetting.
2.5 Data analysis
2.5.1 Digital image acquisition and processing
Shadowgraphy images of the bubble contour were captured using a CMOS high-speed camera (Photron FASTCAM SA-Z, Japan) at a frame rate of \(100,000\) frames per second (fps) so that the time interval between the images is then \({10}\,{\upmu \text{s}}\). The image resolution at this frame rate is \({640 \times 280}\) px. The long working distance objective (Optem 5x High-Resolution Objectives) with numerical aperture of 0.225 has a working distance of \({35}\,{\text{mm}}\). With the lens system (Optem® FUSION, 12.5:1 zoom objective, 5\(\times\)HiRes) we achieve optical resolutions ranging from \({1.31} \text{to}\, {6.93}\,{\upmu \text{m} / \text{Px}}\) depending on the zoom level. Homogeneous back-illumination was achieved using a fibre optic light source (Schott KL 2500 LED) combined with a diffuser glass sheet (THORLABS \(100 \, \textrm{mm} \times 100 \, \textrm{mm}\) N-BK7 Ground Glass Diffuser, 1500 grit) to homogeneously distribute the light.
Side-view image series were processed using open-source digital image processing libraries scikit image (van der Walt et al. 2014) and OpenCV (Bradski 2000). The different postprocessing steps included conversion to grey scale, background subtraction using a rolling ball algorithm (Sternberg 1983) and rescaling to enhance the subsequent edge detection. To extract the edges, a conventional gradient-based edge detection algorithm (Canny 1986) was applied. Detected edges originating from reflections or small binary objects (blobs) below a size threshold were removed. The remaining bubble edge contour was truncated at the level of the baseline and closed allowing binary hole filling to produce a binarised image of the bubble. All processing steps are illustrated in Fig. 7.
Fig. 7
Subsequent steps in image processing for contour analysis. From left to right: original grey-scale image, background removed, edge detected, and binarised by hole filling
To assess the shape of the bubble before and after departure, the aspect ratio \(\chi :=h/l\) and the projected area \(A_{+} :=4 \, A / (\pi \, d_{\textrm{p}}^2)\) are extracted from high-speed images in side view. The height h and length l of the bubble are measured as indicated by the dimensioning arrows shown in Fig. 8. The projected area \(A_{+}\) is calculated as the sum of all pixels within the closed bubble contour. After departure, the contours obtained from image processing can be attributed to either the portion of the bubble remaining at the nucleation site to the departing bubble advected with the flow, which has the equivalent diameter \(d_{\textrm{e}}\). In this study, we limit the analysis to side-view images. Thus, lateral spreading perpendicular to the direction of flow or flattening cannot be reported.
Contact point positions \(x_{\textrm{u}}, x_{\textrm{d}}\) and dynamic contact angles \(\theta _{\textrm{u}}, \theta _{\textrm{d}}\) are defined with respect to their location upstream (index u) or downstream (index d), see Fig. 8. The contact points \(x_{\textrm{u}}, x_{\textrm{d}}\) in the side view are the most lateral points of the bubble edge at the baseline level on the substrate. To track \(\theta _{\textrm{u}}, \theta _{\textrm{d}}\), we used an algorithm based on the analysis of the local intensity gradient (Biolè and Bertola 2015). The contact point velocities \(u_{\textrm{u}}, u_{\textrm{d}}\) were calculated from the rate of change in contact point positions, \(\Delta x/\Delta t\), with a temporal resolution of \(\Delta t = 10^{-5} \, \textrm{s}\). To reduce pixle-level step noise in \(\Delta x\) and derive a smooth gradient \(\Delta x\), time-series data were smoothed using Whittaker-Henderson smoothing (Schmid et al. 2022).
Fig. 8
Post-processed binary images and tracked quantities of a deformed bubble before (left) and after departure (right). Here, we show the case, in which the bubble splits into two, with one free bubble then being advected by the flow
This section covers two aspects of uncertainty relevant for the experimental data: (i) the image acquisition and processing used to extract geometric quantities, and (ii) the calculation of derived dimensionless quantities based on sensor measurements. The microscope imaging system was calibrated with a calibration target with \({10}\,{\upmu \text{m}}\) divisions (THORLABS R1L3S2P). The imaging is performed through air, an acrylic channel wall, and liquid, introducing refractive interfaces. For side-view image acquisition, the camera was tilted by \({5}^{\circ }\) (rotation about the horizontal axis). By pitching the mirror image of the bubble on the reflective surface can be used to achieve a precise baseline detection. This tilt minimally affects the horizontal measurements by foreshortening, making the horizontal micropore diameter a reliable reference. Vertical measurements experience geometric foreshortening, which we estimated using Snell’s law of refraction. The tilt angle of 5\(^\circ\) in air corresponds to an effective viewing angle of approximately 3.67\(^\circ\) in water, due to the refraction at each interface. This results in a vertical compression factor of \(\cos (3.67^\circ ) \approx 0.998\), which corresponds to an underestimation of less than 0.2% in the vertical direction. An error of one pixel corresponds to the physical uncertainty given by the optical resolution of the system used, i.e. \({1.31} \text{to}\, {6.93}\,{\upmu \text{m}}\). The errors introduced by the edge detection itself are estimated to be 0.5 pixels, hence \({2} \text{to}\, {2.5}\,{\upmu \text{m}}\). The total error is obtained from the root sum of squares of these two contributions, giving approximately \({4.5} \text{to}\, {5.6}\,{\upmu \text{m}}\) for a measured length.
In postprocessing, all dimensionless quantities, e.g. \(R\hspace{-0.39993pt}e _{\infty }\), were determined from sensor readings and tabulated thermophysical data. Their respective uncertainties were evaluated following the approach of GUM Supplement 1 (BIPM et al. 2008), by propagating the input probability distributions with a Monte Carlo method with a sample size of \(10^6\). For illustration, consider
$$\begin{aligned} Re_\infty = \frac{2 \, Q}{( W + H )\,\nu }, \end{aligned}$$
(6)
where Q is the measured volume flow rate. The kinematic viscosity \(\nu\) depends on pressure and temperature. Data for \(\nu\) were sampled from interpolation functions obtained from thermophysical databases (Lemmon et al. 2018). Kernel density estimation (KDE) was used to derive smooth probability distributions from the histograms of Q and \(\nu\). Instead, uniform distributions centred on their nominal values, with width \({0.02\,\mathrm{\text {m}\text {m}}}\) were used for \(H\) and \(W\). Resampling from these distributions leads to the statistical uncertainty (GUM type A) of the time-series data. The systematic (GUM type B) uncertainty components were taken from the manufacturer’s specified tolerances for the sensors. For each of the \(N = 10^6\) Monte Carlo iterations, samples of \(Q, H , W\) and \(\nu\) were drawn and propagated through the equation for \(R\hspace{-0.39993pt}e _{\infty }\) to obtain the resulting distribution. The mean \(\overline{ R\hspace{-0.39993pt}e }_{\infty } = 1/N \, \sum _{i=1}^N R\hspace{-0.39993pt}e _{\infty }^{(i)}\) and sample standard deviation \(\sigma _{ R\hspace{-0.39993pt}e _{\infty }} = \sqrt{ \frac{1}{N - 1} \sum _{i=1}^{N} \bigl ( R\hspace{-0.39993pt}e _{\infty }^{(i)} - \overline{ R\hspace{-0.39993pt}e }_{\infty } \bigr )^2 }\) were then taken to quantify the uncertainty of the channel flow operating point.
3 Results and discussion
In the following section, bubble departure is analysed with regard to bubble deformation, dynamic wetting, and bubble size after the bubble has split into two parts. The range of shear Weber numbers, \(W\hspace{-1.4pt}e\), spans two orders of magnitude, ranging from 0.5 to 150. Adapting the wording of Klausner et al. (1993), in the following we denote departure as the event of the bubble detaching from the nucleation site. The departure can be followed by lift-off from the surface. If these two events occur at the same time, we refer to that process as splitting. To further visualise the departure, supplementary movies 1–8 are provided to support the interpretation of the shown data.
3.1 Bubble shape and size prior departure
We choose a time period of \({5\,\mathrm{\text {m}\text {s}}}\) up to the critical time \(t_\textrm{crit}\), i.e. the last image before the bubble leaves the nucleation site. Using the previously defined inertial time scale, a dimensionless time \(t_{+} :=t/t_\textrm{inert}\) is defined so that the duration of \({5\,\mathrm{\text {m}\text {s}}}\) is equivalent to \(t_{+} = 0...15\). In Figs. 9, 10, 11, 12, time-series data from \(\chi\) and \(A_{+}\) (left side) together with snapshots of the respective image series (right side) for four distinct operating points, given by \(W\hspace{-1.4pt}e , R\hspace{-0.39993pt}e\).
The temporal evolution of a departure process for the lowest values \(W\hspace{-1.4pt}e \approx 0.7\), \(R\hspace{-0.39993pt}e \approx 90\) is shown in Fig. 9. The orientation of the bubble before departure differs markedly between the substrates: On steel, the bubble shape is more slender and orientated more upright, resulting in a higher aspect ratio of approximately \(\chi \approx 1.25\), while on PDMS, the bubble adopts a voluminous, almost hemispherical shape with \(\chi \approx 0.6\). The contact radius (foot of the bubble) is larger on PDMS because of its higher value of \(\mathcal {S} \approx - 52\). As a result, the downstream contact point \(x_{\textrm{d}}\) lies further downstream on PDMS compared to steel. In terms of size, the projected area during departure \(A_{+}\) is approximately twice as large on PDMS as on steel. For both substrates, pinning of the contact line is observed at the upstream edge of the nucleation site. During detachment from the steel, a neck is formed and the bubble pinches off, and lifts off the surface immediately. In contrast, on PDMS, the bubble remains attached to the surface after leaving the nucleation site and slides downstream; no lift-off is observed within the imaging window. Most probably the sliding bubble further grows after departure and lifts off in a similar manner than a large bubble in a quiescent liquid.
Fig. 9
Left: Aspect Ratio \(\chi\) and projected area \(A_+\) against time \(t_+\) for \(W\hspace{-1.4pt}e \approx 0.7, R\hspace{-0.39993pt}e \approx 90\) until departure indicated by the marker symbols. Square symbols \(\blacksquare\) indicate lift-off, triangle symbols \(\blacktriangle\) indicate departure and transition to sliding. Right: Snapshots of the departure process for steel (upper row) and PDMS (lower row). Also see supplementary movies 1 (steel) and 2 (PDMS)
By increasing \(W\hspace{-1.4pt}e \approx 5\), the inertial and surface tension forces become comparable in magnitude. The bubble interface shows only a slight increase in deformation, as reflected in the aspect ratio \(\chi\), which remains similar to that observed at \(W\hspace{-1.4pt}e \approx 0.7\). On PDMS the bubble shape becomes more elongated in the streamwise direction such that \(\chi < 0.5\). The upstream part of the interface adopts a more concave shape, indicating higher dynamic pressure forces acting on it. The projected area \(A_{+}\) decreases considerably at departure by a factor of four on steel and nine on PDMS. For both substrates, the contact line remains pinned at the upstream edge of the nucleation site. On steel, the location of the pinch-off moves farther downstream, resulting in a larger residual bubble on the nucleation site after splitting. During the formation of the neck, the aspect ratio decreases on steel towards departure as the bubble is vertically elongated, corresponding to an increase in height h. On PDMS, the contact line pinning remains upstream while downstream the interface is tapered towards the substrate. After detachment, the bubble slides downstream and subsequently lifts off the substrate.
Fig. 10
Left: Aspect Ratio \(\chi\) and projected area \(A_+\) against time \(t_+\) for \(W\hspace{-1.4pt}e \approx 5, R\hspace{-0.39993pt}e \approx 240\) until departure indicated by the marker symbols. Square symbols \(\blacksquare\) indicate lift-off, triangle symbols \(\blacktriangle\) indicate departure and transition to sliding. Right: Snapshots of the departure process for steel (upper row) and PDMS (lower row). Also see supplementary movies 3 (steel) and 4 (PDMS)
At a higher wall shear rate corresponding to \(W\hspace{-1.4pt}e \approx 40\), the onset of shape oscillations are observed on both substrates, visible in the temporal evolutions of \(\chi\) and \(A_{+}\). We attribute these oscillations to the bubble interface responding to fluctuations in the dynamic pressure field induced by the cross-flow. The aspect ratio \(\chi\) remains higher for steel compared to PDMS. The contact diameter on steel is approximately the pore diameter because of up- and downstream pinning. The angle between the main axis of the bubble and the horizontal substrate decreases. A decreasing angle shows that the bubble tilts more towards the substrate in the downstream region.
On PDMS, the upstream side of the interface becomes more flattened and the aspect ratio drops to \(\chi \approx 0.3\). Despite the increased deformation, the sliding motion after departure persists. The projected area \(A_{+}\) on PDMS exceeds that on steel, reflecting a broader contact region due to stronger hydrophobic adhesion.
The necking point on PDMS shifts further downstream, consistent with the elongated bubble shape and delayed detachment. The bubble still detaches by splitting at the downstream edge of the pore and slides across the substrate. After detachment, the residual bubble at the nucleation site are larger on both substrates, indicating gas retention and potential for subsequent bubble growth cycles.
Fig. 11
Left: Aspect Ratio \(\chi\) and projected area \(A_+\) against time \(t_+\) for \(W\hspace{-1.4pt}e \approx 40, R\hspace{-0.39993pt}e \approx 700\) until departure indicated by the marker symbols. Square symbols \(\blacksquare\) indicate lift-off, triangle symbols \(\blacktriangle\) indicate departure and transition to sliding. Right: Snapshots of the departure process for steel (upper row) and PDMS (lower row). Also see supplementary movies 5 (steel) and 6 (PDMS)
For \(W\hspace{-1.4pt}e \approx 140\) the liquid enters the pore in the case of steel because the adhesive and pinning forces are overcome by the dynamic pressure forces. Consequently, the upstream contact point is no longer pinned to the pore edge. The departure occurs due to a rapid distortion of the bubble as seen by the steep increase of \(\chi\). On the other hand, pinning of the contact line is maintained on PDMS, paired with pronounced high-frequency oscillations visible in the time series of \(\chi\) and \(A_{+}\).
Fig. 12
Left: Aspect Ratio \(\chi\) and projected area \(A_+\) against time \(t_+\) for \(W\hspace{-1.4pt}e \approx 140, R\hspace{-0.39993pt}e \approx 1400\) until departure indicated by the marker symbols. Square symbols \(\blacksquare\) indicate lift-off, triangle symbols \(\blacktriangle\) indicate departure and transition to sliding. Right: Snapshots of the departure process for steel (upper row) and PDMS (lower row). Also see supplementary movies 7 (steel) and 8 (PDMS)
A frequency analysis of \(A_{+}\) for the operating point \(W\hspace{-1.4pt}e \approx 140\) was performed on a series of 150,000 frames by applying the continuous wavelet transform (cf. Torrence and Compo (1998)). A complex Morlet wavelet was used as mother wavelet. The centre frequency of the complex Morlet was 0.5 and its bandwidth 6.0 with a number of scales of 128. In this way, we achieve good frequency resolution and sensitivity for high-frequency features. The analysis led to an observed frequency of \(f = {4029.3\,\mathrm{\text {Hz}}}\). For comparison, we report the frequency of surface waves of a free gas bubble with radius \(R= d_{\textrm{p}}/2 = {100\,\mathrm{\upmu \text {m}}}\) that can be estimated by using Lamb’s formula (Lamb 1953, §275) for mode number \(n=2\) as reference:
The experimentally observed frequency is therefore assigned to surface waves caused by periodic forcing of the interface. The lower value of f compared to \(f_{2}^{\textrm{Lamb}}\) must be due to wall effects, hence wetting. As these oscillations are only observed on PDMS where the bubble still is still pined at the pore edge, this behaviour must be linked to contact line constraints.
Upon evaluation of all experimental data, a regime map based on the critical aspect ratio \(\chi _{\textrm{c}}\) versus \(W\hspace{-1.4pt}e\) is derived and shown in Fig. 13. For increasing \(W\hspace{-1.4pt}e\), \(\chi _{\textrm{c}}\) transitions between plateaus indicating the transition from a capillary to an inertia-dominated regime of departure. The departure regimes can be categorised as follows:
Capillary Regime (\(W\hspace{-1.4pt}e < 1\)): Surface tension dominates over inertia, and the symmetry of the bubble is only slightly disturbed. On the steel substrate, the bubble splits at departure, whereas on PDMS it detaches by sliding away from the nucleation site.
Transition Regime (\(W\hspace{-1.4pt}e \approx 1-10\)): Inertia begins to dominate over surface tension, leading to a pronounced deformation of the bubble shape. This results in a transitional drop in the critical aspect ratio, \(\chi _{\textrm{c}}\).
Inertial Regime (\(W\hspace{-1.4pt}e> 10\)): Departure is dominated by inertia. The bubble is significantly deformed, and oscillatory motions begin as surface tension and adhesive forces act to restore a more compact shape. As a result, the critical aspect ratio decreases and reaches a lower plateau.
For PDMS, two phenomena contribute to the higher standard deviations in \(\chi _{\textrm{c}}\) visible in Fig. 13: the unpinning of the upstream contact point and sliding motion and the oscillatory behaviour for \(W\hspace{-1.4pt}e> 40\). Both effects result in broader distributions of \(\chi _{\textrm{c}}\).
When the two substrates are compared, several differences emerge. For PDMS, \(\chi _{\textrm{c}}\) is generally lower and the mechanism transitions from only sliding, to sliding followed by lift-off, and eventually to a shear induced breakup. This behaviour is likely due to the bubble deforming more easily in the horizontal direction as it moves downstream, allowing for greater lateral stretching. In contrast, on steel the bubble always splits, but changes its orientation for increasing \(W\hspace{-1.4pt}e\). The change of orientation causes a reduction in bubble height h and an increase in its length l, resulting in a more substantial decrease in \(\chi _{\textrm{c}}\) for steel in the region of transition.
On the other hand, the regime transition occurs within the same range of \(W\hspace{-1.4pt}e \approx 4\), which supports the idea that \(W\hspace{-1.4pt}e\) is an appropriate control parameter to predict the transition. However, the bubble shape and the mechanism of departure during transition strongly depend on the wettability of the substrate. The plateaus in the data of \(\chi _{\textrm{c}}\) can be described using logistic (sigmoid) curve fits whose equations are given in the caption of Fig. 13.
Fig. 13
Regime map based on the critical aspect ratio \(h_\textrm{crit}/l_\textrm{crit}\) before departure at various \(W\hspace{-1.4pt}e\) values for steel and PDMS. The regime transition occurs between a capillary regime and an inertial regime. The departure mechanisms are annotated on the plot in the corresponding \(W\hspace{-1.4pt}e\) regions. The trend lines are fitted with logistic functions. For steel, \(\chi _{\textrm{c}} = \frac{0.52}{1 + \left( W\hspace{-1.4pt}e /4.58\right) ^{4.07}} + 0.6\), with a critical Weber number of transition \(W\hspace{-1.4pt}e _{\textrm{c}}=4.58\). For PDMS, \(\chi _{\textrm{c}} = \frac{0.14}{1 + \left( W\hspace{-1.4pt}e /3.99\right) ^{4.42}} + \frac{2.81 \times 10^2}{1 + \left( W\hspace{-1.4pt}e / 7.12 \times 10^4\right) ^{1.2}} - 2.81 \times 10^2\), with critical Weber numbers of transition \(W\hspace{-1.4pt}e _{\textrm{c}}=3.99\) and \(7.12 \times 10^4\)
On steel, a substrate with lower work of adhesion bubble departure occurs by splitting for all values of \(W\hspace{-1.4pt}e\) examined. As \(W\hspace{-1.4pt}e\) increases, the bubble’s orientation becomes increasingly inclined in the direction of the flow, indicating that the drag forces become stronger relative to the adhesive forces. The tilting precedes the necking and the pinch-off. The location where neck formation takes places progressively moves more downstream with increasing \(W\hspace{-1.4pt}e\). In consequence, also the volume of the detaching bubble decreases with increasing \(W\hspace{-1.4pt}e\). The observed phenomena are comparable to those reported by (Mirsandi et al. 2020a).
A higher work of lateral adhesion between the bubble interface and the PDMS surface results in increased interaction strength, promoting sliding of the contact line during bubble growth. This stronger adhesion causes the bubble’s interface to elongate more significantly before it eventually splits or detaches. Additionally, the neck formation-the thinning region where the bubble separates-tends to develop in the direction towards the substrate, influenced by the adhesive forces pulling the interface closer to the solid surface. Unlike studies on free bubbles (Risso 2000), the mechanisms of bubble departure here are clearly influenced by both wettability and shear rate. For high shear the observed shapes are similar to those reported for gaseous jet breakup by (Wace et al. 1987).
While layer formation on PDMS can alter bubble dynamics, we have used PDMS pseudo-brushes with a nanometric thickness in this work; whereas the bubbles are \(\sim {200\,\mathrm{\upmu \text {m}}}\) in size. At this scale, any elastic deformability of the substrate is negligible, and we do not expect the PDMS deformation to influence bubble dynamics. As the contact angle hysteresis is equally high on both steel and PDMS, we also expect the contact line friction to be of equal magnitude on both surfaces.
3.2 Dynamic wetting analysis during departure
When the bubble departs, the liquid de- and rewets the surface. These wetting processes cause motions of the three-phase contact line that can be observed. For a bubble in a cross-flow the contact line motion is caused by shearing and is thus is a forced wetting phenomenon (de Gennes et al. 2004). Although we define up- and downstream contact points \(x_{\textrm{u}},x_{\textrm{d}}\) based on the side-view images, we are aware that these points represent only a projection of the actual two-dimensional contact line. In the analysis of dynamic wetting, the relationship between the contact angles and the velocities of the contact points is of interest. We introduce capillary numbers \(Ca_{\textrm{u}} :=u_{\textrm{u}} \mu / S\) and \(Ca_{\textrm{d}} :=u_{\textrm{d}} \mu / S\) as dimensionless measures for the up- and downstream velocities of the contact points. In the following Figs. 14, 15, 16, 17 representative temporal evolutions of several measurements for \(\theta _{\textrm{u}}, \theta _{\textrm{d}}\) and \(Ca_{\textrm{u}},Ca_{\textrm{d}}\) are shown for steel (left column, black) and PDMS (right column, grey). The upstream position is always indicated by solid lines, whereas the downstream position is always indicated by dotted lines.
For \(W\hspace{-1.4pt}e \approx 0.7\) Fig. 14 (left column) shows that on steel both contact angles \(\theta _{\textrm{u}}, \theta _{\textrm{d}}\) increase on departure. The shape asymmetry of the bubble on departure manifests itself in a difference between \(\theta _{\textrm{u}}\) and \(\theta _{\textrm{d}}\). The upstream contact point is pinned \(Ca_{\textrm{u}} \approx 0\) while downstream the contact point accelerates in the upstream direction, hence \(Ca_{\textrm{d}} < 0\) during departure. Consequently, the liquid rewets the substrate at the downstream location during departure by lift-off.
For the same value of \(W\hspace{-1.4pt}e\) in PDMS the bubble leaves the nucleation site by transitioning to sliding; see Fig. 14 (right column). The sliding is initiated by a rapid acceleration of the upstream contact point \(x_{\textrm{u}}\) visible in the increase in the velocity of the contact point \(Ca_{\textrm{u}}\) for \(t_{+} \approx 11\). This acceleration occurs when the pinning force is overcome, and the contact line leaves the upstream edge of the pore in a rapid motion. After the sliding motion has become steady, \(Ca_{\textrm{u}}\) relaxes to a steady value. In an ideal sliding motion, the equality \(Ca_{\textrm{u}} = Ca_{\textrm{d}}\) should hold.
Fig. 14
Evolution of contact angles \(\theta _{\textrm{u}},\theta _{\textrm{d}}\) and Capillary numbers \(Ca_{\textrm{u}},Ca_{\textrm{d}}\) for steel (left) and PDMS (right) for \(W\hspace{-1.4pt}e \approx 0.7\) from three different measurements are shown, distinguished by line thickness. Upstream values are shown with solid lines, downstream values with dotted lines. Square symbols \(\blacksquare\) mark lift-off from the surface, while triangle symbols \(\blacktriangle\) indicate departure by sliding. Also see supplementary movies 1 (steel) and 2 (PDMS)
For \(W\hspace{-1.4pt}e \approx 5\), see in Fig. 15 (left column), on steel the progression of \(\theta _{\textrm{u}}\) shows a more pronounced increase, while \(\theta _{\textrm{d}}\) shows a more pronounced decrease towards lift-off compared to the case of \(W\hspace{-1.4pt}e \approx 0.7\). Still, the upstream contact point is pinned to the edge of the nucleation site, while the downstream one also travels upstream during departure.
Observing departure on PDMS in Fig. 15 (right column) the beginning of sliding is visible through a rapid decline of \(\theta _{\textrm{u}}\). During this transition, the upper contact point detaches from the upstream pinning site indicated by the triangular marker. In two measurements lift-off also occurs within the observation window as indicated by the square symbol. In these two cases of lift-off, an acceleration of the upstream contact point in the downstream direction is observed. The acceleration leads to \(Ca_{\textrm{u}} \approx 4-6 \times 10^{-2}\).
Fig. 15
Evolution of contact angles \(\theta _{\textrm{u}},\theta _{\textrm{d}}\) and Capillary numbers \(Ca_{\textrm{u}},Ca_{\textrm{d}}\) for steel (left) and PDMS (right) for \(W\hspace{-1.4pt}e \approx 5\) from three different measurements are shown, distinguished by line thickness. Upstream values are shown with solid lines, downstream values with dotted lines. Square symbols \(\blacksquare\) mark lift-off from the surface, while triangle symbols \(\blacktriangle\) indicate departure by sliding. Also see supplementary movies 3 (steel) and 4 (PDMS)
For clarity, only two representative measurements are shown for \(W\hspace{-1.4pt}e \approx 40\) and one for \(W\hspace{-1.4pt}e \approx 140\), as including more would compromise the readability of the plot.
For \(W\hspace{-1.4pt}e \approx 40\) the time evolution of the contact angles and capillary numbers shows oscillative behaviour on both substrates. On steel in Fig. 16 (left column), the spread of the contact angles is observed from \(t_{+} = 10-15\) visible in an increase of \(\theta _{\textrm{u}}\), while simultaneously \(\theta _{\textrm{d}}\) decreases. This behaviour can be traced back to the visible inclination of the bubble in the downstream direction in Fig. 11 (upper row, middle frame) for the steel surface. With neck formation (upper row, right frame), the downstream contact angle increases sharply again.
On PDMS (Fig. 16, right column), the bubble elongates in the downstream direction. The unpinning from the upper edge of the nucleation site is again visible as a sudden drop in \(\theta _{\textrm{u}}\). The discontinuity of \(\theta _{\textrm{u}}\) results from the change of the algorithmically tracked object - from the bubble at the nucleation site to the sliding bubble (see also Fig. 16 for comparison). Following departure from the nucleation site, \(\theta _{\textrm{u}}\) decreases linearly until the bubble lifts off the substrate.
Fig. 16
Evolution of contact angles \(\theta _{\textrm{u}},\theta _{\textrm{d}}\) and Capillary numbers \(Ca_{\textrm{u}},Ca_{\textrm{d}}\) for steel (left) and PDMS (right) for \(W\hspace{-1.4pt}e \approx 40\) from two different measurements are shown, distinguished by line thickness. Upstream values are shown with solid lines, downstream values with dotted lines. Square symbols \(\blacksquare\) mark lift-off from the surface, while triangle symbols \(\blacktriangle\) indicate departure by sliding. Also see supplementary movies 5 (steel) and 6 (PDMS)
For the highest value of \(W\hspace{-1.4pt}e \approx 140\), the liquid begins to penetrate the upstream region of the nucleation site on steel, as shown in Fig. 12 and supplementary movie 7. The penetrating water displaces the upstream interface of the bubble and eliminates the upstream pinning. Consequently, the time signal in (Fig. 17, left column) rather shows interface oscillations at the baseline level. However, pinning on the downstream region of the pore edge persists. The bubble is then rapidly ejected from the nucleation site at departure.
Both contact points remain pinned for PDMS for \(W\hspace{-1.4pt}e \approx 140\). As the bubble at departure takes on a thread-like gaseous shape, high-frequency oscillations also become visible in \(\theta _{\textrm{u}}, \theta _{\textrm{d}}\) and \(Ca_{\textrm{u}}, Ca_{\textrm{d}}\) (Fig. 17, right column). These oscillations are a result of the capillary waves propagating towards the contact points. For clarification, see supplementary movie 8.
Fig. 17
Evolution of contact angles \(\theta _{\textrm{u}},\theta _{\textrm{d}}\) and Capillary numbers \(Ca_{\textrm{u}},Ca_{\textrm{d}}\) for steel (left) and PDMS (right) for \(W\hspace{-1.4pt}e \approx 140\) from one representative measurement are shown. Upstream values are shown with solid lines, downstream values with dotted lines. Square symbols \(\blacksquare\) mark lift-off from the surface. Also see supplementary movies 7 (steel) and 8 (PDMS)
Previous observations show that the bubble departs from the nucleation site either by splitting or sliding. In the following, we analyse the critical equivalent diameter, \(d_{\textrm{c}}\), of the bubble right before splitting and the equivalent diameter, \(d_{\textrm{e}}\), of the detached bubble for both substrates.
Both diameters are expected to be a function of the \(W\hspace{-1.4pt}e\), and wettability, here represented by the mean static contact angle \(\theta _{\textrm{m}}\) as a dimensionless measure for the work of adhesion. Taking into account the fact that \(R\hspace{-0.39993pt}e \gg W\hspace{-1.4pt}e\) and \(R\hspace{-0.39993pt}e \gg 1\) for all experiments, we can omit the dependence on \(R\hspace{-0.39993pt}e\) in the dimensional analysis. Then it follows:
The mean equivalent diameters \(d_{\textrm{c}} / d_{\textrm{p}}\) and \(d_{\textrm{e}} / d_{\textrm{p}}\) for various \(W\hspace{-1.4pt}e\) are shown in Figs. 18 and 19. Both dimensionless diameters \(d_{\textrm{c}} / d_{\textrm{p}}\) and \(d_{\textrm{e}} / d_{\textrm{p}}\) decrease significantly by factors of \(3-4\) in the range covered by \(W\hspace{-1.4pt}e\). Additional data on smaller micropores of \(d_{\textrm{p}}= {100\,\mathrm{\upmu \text {m}}}\) indicate that although incomplete similarity is present, it does not lead to significant deviations between the two pore sizes examined. Further reducing the pore size may require applying the Barenblatt method (Barenblatt 2012), using the ansatz \(d_{\textrm{c}} / d_{\textrm{p}}\sim W\hspace{-1.4pt}e ^{\alpha } R\hspace{-0.39993pt}e ^{\beta }\) to account for the incomplete similarity.
The data of \(d_{\textrm{c}} / d_{\textrm{p}}\) in Fig. 18 show that the critical size of a bubble is larger on PDMS for \(W\hspace{-1.4pt}e < 4\), where the mechanism is sliding on PDMS and splitting on steel. Between \(W\hspace{-1.4pt}e \approx 4-20\) the values between both substrates coincide until there is a plateau visible for PDMS for \(W\hspace{-1.4pt}e> 40\). From \(W\hspace{-1.4pt}e \approx 4-20\) on PDMS the regime changes to sliding and lift-off, whereas on steel the bubble’s orientation changes and the neck forms farther downstream. The transition between regimes results in a noticeable change in the slope of the fitted trendlines for both substrates. For comparison of regimes depending on \(W\hspace{-1.4pt}e\) see Fig. 13 and supplementary movies 1–8.
Fig. 18
Critical equivalent diameter \(d_{\textrm{c}} / d_{\textrm{p}}\) of the bubble before departure for various \(W\hspace{-1.4pt}e\) on steel and PDMS. The circular symbols \(\bullet\) mark experimental data for \(d_{\textrm{p}}= {200\,\mathrm{\upmu \text {m}}}\), while triangle symbols \(\blacktriangle\) data on \(d_{\textrm{p}}= {100\,\mathrm{\upmu \text {m}}}\). The insert numbers indicate the slope m of the trendline, given by \(d_{\textrm{c}} / d_{\textrm{p}}= m \, \log _{10}( W\hspace{-1.4pt}e ) + b.\)
The size of the detaching bubble \(d_{\textrm{e}} /d_{\textrm{p}}\) is generally larger on PDMS, see Fig. 19. Instead of reaching a plateau for \(W\hspace{-1.4pt}e> 40\) on PDMS, the values for \(d_{\textrm{e}} /d_{\textrm{p}}\) continue to decrease and approach those observed on steel. We conclude that for the gas jet breakup regime, the volume of the detaching bubble further decreases with increasing \(W\hspace{-1.4pt}e\). For each substrate a power law of the form \(d_{\textrm{e}} /d_{\textrm{p}}= a \, W\hspace{-1.4pt}e ^b\) is fitted to the data using nonlinear least squares regression. The exponent \(b \approx -0.30\) to \(-0.34\) has a similar value for both substrates, while the prefactor a differs by 0.83. The shift to higher values is attributed to the difference between the spreading parameters \(\mathcal {S}\) of the substrates.
Fig. 19
Equivalent diameter \(d_{\textrm{e}} / d_{\textrm{p}}\) of detaching bubbles after splitting at various \(W\hspace{-1.4pt}e\) for steel and PDMS. The corresponding power law fits are displayed in the legend. The circular symbols \(\bullet\) mark experimental data for \(d_{\textrm{p}}= {200\,\mathrm{\upmu \text {m}}}\), while triangle symbols \(\blacktriangle\) data on \(d_{\textrm{p}}= {100\,\mathrm{\upmu \text {m}}}\)
The dynamic viscosity ratio between air and water, \(\mu _{\textrm{air}} / \mu _{\textrm{water}} \approx 0.018\) at 20 \(^{\circ }\text {C}\), is relatively small. Our results are therefore expected to remain valid for systems with higher liquid viscosities, since in that case the viscosity ratio becomes even smaller while the shear Reynolds number \(R\hspace{-0.39993pt}e\) remains very high. This trend is consistent with the findings of Mirsandi et al. (2020a), who reported similar behaviour at lower \(W\hspace{-1.4pt}e\).
4 Conclusions and outlook
This study aimed to experimentally study the deformation and departure of bubbles from artificial nucleation sites in a cross-flow. The findings for the low viscosity system air/water demonstrate that the mechanism of bubble departure is governed by both the wall shear rate and the wettability of the substrate. Different departure mechanisms and regimes were identified and dynamic wetting was analysed on two substrates, plain steel and a PDMS pseudo-brush anchored on steel. The range of shear Weber numbers \(W\hspace{-1.4pt}e\) in experiments covers two orders of magnitude ranging from 0.4...140. The main contribution of this paper is the observation and analysis of bubble departure on short inertial time scales of \(t_\textrm{inert} = {0.33}\,{\text{ms}}\), capturing the complete dynamics of departure and splitting with high spatial and temporal resolution. To date, such detailed data for interface deformation and dynamic wetting for bubble departure from walls have not been reported for high values of \(W\hspace{-1.4pt}e\).
On the more hydrophobic PDMS pseudo-brush surface, larger contact diameters, lower bubble aspect ratios \(\chi\) and larger projected areas \(A_+\) were observed compared to plain stainless steel. On steel, the bubble leaves the site mainly by splitting and lifting off. The higher the wall shear rate, the more the bubble tilts towards the surface, and the further downstream the location moves where necking occurs. On PDMS, bubble departure occurs via transition to sliding and subsequent lift-off. Neck formation primarily occurs towards the surface. Pinning of the contact line on the sharp edges of the micropore was observed for most of the operating points on both substrates. At \(W\hspace{-1.4pt}e \approx 140\), capillary surface waves were observed on PDMS and analysed by using a wavelet transform. The observed departure mechanism for high shear rates is comparable to those encountered in jet breakup.
The results have important implications for the design and optimisation of processes where bubble dispersion from walls affects performance, such as membrane distillation, electrolysis, boiling heat transfer, and chemical reactors. By showing how surface properties and hydrodynamic conditions jointly govern bubble departure regimes, this work provides a first glance at how a tailored surface wettability is detrimental to more efficient processes in gas/liquid contact.
Future work should further explore the influence of wettability on the frequency of bubble oscillatory motion. Analogies between bubble and jet breakup could offer insight into characteristic frequencies. To capture additional information on the deformation of departing bubbles, top-view visualisation is needed. Quantifying the flattening of the bubble and its spreading using top-view imaging could be used to estimate the evolution of the interfacial area as a function of aspect ratio. Lastly, altering the microstructure of nucleation sites, here, for example, the edge radius of the micropore could offer a way to study the influence of contact line pinning forces on departure.
Acknowledgements
We kindly acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)-Project-ID 265191195-Collaborative Research Center 1194 (CRC 1194) "Interaction between Transport and Wetting Processes", projects A02 and C06. The authors also thank Mr A. Schuler for manufacturing of the specimens as well as Mr K. Habermann and Mr U. Trometer for the technical support. We thank the anonymous reviewers for their valuable feedback and suggestions to improve the quality of this work.
Declarations
Conflict of interest
The authors declare no Conflict of interest.
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Battino R, Rettich TR, Tominaga T (1984) The solubility of nitrogen and air in liquids. J Phys Chem Ref Data 13(2):563–600. https://doi.org/10.1063/1.555713CrossRef
BIPM, IEC, IFCC, et al. (2008) Evaluation of measurement data - supplement 1 to the “guide to the expression of uncertainty in measurement” - propagation of distributions using a monte carlo method. Joint Committee for Guides in Metrology JCGM 101:2008
Borkent BM, Gekle S, Prosperetti A et al. (2009) Nucleation threshold and deactivation mechanisms of nanoscopic cavitation nuclei. Phys Fluids doi 10(1063/1):3249602
Bradski G (2000) The opencv library. Dr Dobb’s J Softw Tools
Buzágh A, Wolfram E (1958) Bestimmung der haftfähigkeit von flüssigkeiten an festen körpern mit der abreißwinkelmethode. ii. Kolloid-Zeitschrift 157(1):50–53. https://doi.org/10.1007/BF01734033CrossRef
Chappell MA, Payne SJ (2007) The effect of cavity geometry on the nucleation of bubbles from cavities. J Acoust Soc Am 121(2):853–862. https://doi.org/10.1121/1.2404629CrossRef
Clift R, Grace JR, Weber ME (2005) Bubbles, drops, and particles, unabridged republication of the work first published by Academic Press, Inc., New York, in 1978 edn. Dover Publications Inc, Mineola, New York
Forrester SE, Rielly CD (1998) Bubble formation from cylindrical, flat and concave sections exposed to a strong liquid cross-flow. Chem Eng Sci 53(8):1517–1527. https://doi.org/10.1016/S0009-2509(98)00019-0CrossRef
Gaudin A, Witt A, Decker T (1963) Contact angle hysteresis–principles and application of measurement methods. Trans 107–112
Henry W (1803) Iii. experiments on the quantity of gases absorbed by water, at different temperatures, and under different pressures. Philos Trans R Soc Lond 93:29–274. https://doi.org/10.1098/rstl.1803.0004CrossRef
Kulkarni AA, Joshi JB (2005) Bubble formation and bubble rise velocity in gas\(-\)liquid systems: a review. Ind Eng Chem Res 44(16):5873–5931. https://doi.org/10.1021/ie049131pCrossRef
Lamb H (1953) Hydrodynamics, 6th edn. Cambridge University Press, Cambridge
Le Grand N, Daerr A, Limat L (2005) Shape and motion of drops sliding down an inclined plane. J Fluid Mech 541:293–315CrossRef
Lemmon EW, Bell IH, Huber ML, et al. (2018) Nist standard reference database 23: Reference fluid thermodynamic and transport properties database (refprop), version 10.0
Moreno Soto Á, Enríquez OR, Prosperetti A et al. (2019) Transition to convection in single bubble diffusive growth. J Fluid Mech 871:332–349. https://doi.org/10.1017/jfm.2019.276CrossRef
Rostami P, Hormozi MA, Soltwedel O et al. (2023) Dynamic wetting properties of pdms pseudo-brushes: Four-phase contact point dynamics case. J Chem Phys doi 10(1063/5):0142821
Silberman E (1957) Production of bubbles by the disintegration of gas jets in liquid. In: Proceedings of the fifth midwestern conference on fluid mechanics, Michigan, pp 263–285
Thorncroft GE, Klausner JF (2001) Bubble forces and detachment models. Multiph Sci Technol 13(3 and 4):35–76
Thorncroft GE, Klausner JF, Mei R (1998) An experimental investigation of bubble growth and detachment in vertical upflow and downflow boiling. Int J Heat Mass Transf 41(23):3857–3871. https://doi.org/10.1016/S0017-9310(98)00092-1CrossRef
Tretinnikov ON, Ikada Y (1994) Dynamic wetting and contact angle hysteresis of polymer surfaces studied with the modified Wilhelmy balance method. Langmuir 10(5):1606–1614CrossRef
van Helden W, van der Geld C, Boot P (1995) Forces on bubbles growing and detaching in flow along a vertical wall. Int J Heat Mass Transf 38(11):2075–2088. https://doi.org/10.1016/0017-9310(94)00319-QCrossRef
van Wijngaarden L (1967) On the growth of small cavitation bubbles by convective diffusion. Int J Heat Mass Transf 10:127–134CrossRef
Winkler LW (1901) Die löslichkeit der gase in wasser. [dritte abhandlung]. Berichte der deutschen chemischen Gesellschaft 34(2):1408–1422. https://doi.org/10.1002/cber.19010340210
Wong WS, Hauer L, Naga A et al. (2020) Adaptive wetting of polydimethylsiloxane. Langmuir 36(26):7236–7245CrossRef
Zeng LZ, Klausner JF, Bernhard DM et al. (1993) A unified model for the prediction of bubble detachment diameters in boiling systems–II. Flow boiling. Int J Heat Mass Transf 36(9):2271–2279. https://doi.org/10.1016/S0017-9310(05)80112-7CrossRef
