微通道气泡中堵塞现象检测方案(流量计)

Open Access Articles The Role of Contact Line (Pinning) Forces on Bubble Blockage inMicrochannels The Faculty of Oregon State University has made this article openly available.Please share how this access benefits you. Your story matters. Citation Mohammadi， M.， & Sharp， K.V. (2015). The Role of Contact Line (Pinning) Forceson Bubble Blockage in Microchannels. Journal of Fluids Engineering-Transactions of the ASME， 137(3)， 031208. doi：10.1115/1.4029033. DOI 10.1115/1.4029033 Publisher The American Society of Mechanical Engineers (ASME) Version Accepted Manuscript Terms of Use http：//cdss.library.oregonstate.edu/sa-termsofuse The RIoleof ContactLine(Pinning)) ForcesonBubbleBlockage inMicrochannels Mahshid Mohammadi E-mail： mahshid@lifetime.oregonstate.edu Kendra V. Sharp E-mail： kendra.sharp@oregonstate.edu 204 Rogers Hall， Oregon State University， Corvallis， OR 97331 Abstract This paper highlights the influence of contact line (pinning) forces on the mobility of dry bubblesin microchannels. Bubbles moving at velocities less than the dewetting velocity of liquid on thesurface are essentially dry， meaning that there is no thin liquid film around the bubbles. For these“dry”bubbles， contact line forces and a possible capillary pressure gradient induced by pinningact on the bubbles and resist motion. Without sufficient driving force (e.g. external pressure) adry bubble is brought to stagnation. For the first time， a bipartite theoretical model that estimatesthe required pressure difference across the length of stagnant bubbles with concave and convexback interfaces to overcome the contact line forces and stimulate motion is proposed. To validateour theory， the pressure required to move a single dry bubble in square microchannels exhibitingcontact angle hysteresis has been measured. The working fluid was deionized water. Theexperiments havebeenconductedon coatedglasschannels with differentSsurfacehydrophilicities that resulted in concave and convex back interfaces for the bubbles. Theexperimental results were in agreement with the model’s predictions for square channels. Thepredictions of the concave and convex btack models were within 19% and 27% of theexperimental measurements， respectively. 1 Introduction Bubble clogging in microchannels is a well-known problem in microfluidics. Immobile bubblesinduce strong resistance effects on the flow in the channels and may disturb the performance andreduce the efficiency of the device [1]. Therefore， management of gas bubbles is an importantmatter for microfluidic devices. The emergence of the bubbles can be caused by various factors，e.g.， simply by diffusion of air into the device or by the degassing of the liquid due totemperature or pressure variations. Vibration can stimulate the formation of the bubbles as well[2]. Bubbles can also get introduced into the system while connecting or disconnecting tubing tothe valves， pumps， etc. Overall， in practice， the chances of the appearance of unwanted bubblesin microfluidic devices are relatively high. Furthermore， bubbles may be generated as a reactionproduct in the device， e.g. the production of carbon dioxide during the oxidation of methanol onthe anode side of a micro direct methanol fuel cell (uDMFC) [3]. Two-phase micro-heatexchangers may also experience bubble removal problems [4-6]. In particular， dryout iss acommon problem in two-phase microchannel heat sinks used for electronic cooling. Theformation of sufficiently large bubbles that block the channels leads to the diversion of the fluidinto other channels which greatly reduces heat removal in the blocked channels [5， 7， 8]. Some of the methods suggested for facilitating bubble removal in microchannels include the useof T-shaped non clogging microchannels [9]， tapered structures [1]， hydrophilic-hydrophobicpatterning [10]， and hydrophobic porous membranes [11， 12]. Most of the previous studies attributed the lack of motion of stagnant bubbles to resistantcapillary forces related to pressure differences across the front and back interfaces [2， 13-17].Herein we investigate pinning forces along triple contact lines as another factor hindering the motion of dry bubbles. Note that lubricated bubbles have no triple contact line and do notexperience pinning. To the best of authors’knowledge， the only studies to account for the pinning forces acting on abubble in the force balance are studies by Blackmore et al. [18] and Metz et al. [1]. Blackmore etal. [18] modeled the forces acting on a small stagnant bubble confined between two parallelplates at the moment of detachment. Their model included viscous drag and pinning forces butoverlooked capillary forces. The pinning force was derived as an integration of the projection ofsurface tension on the solid surface plane over the bubble’s contact perimeter. Metz et al. [1]modeled the forces acting on a moving bubble in a microchannel as the superposition of thecapillary force， viscous drag， and the drag induced due to thin film deposition， pinning， andcontact angle dynamics. Their simple model for the pinning forces linearly correlates this forceto a constant pinning coefficient， the length of the gas bubble contact line， and the surfacetension. For their experiments， the pinning coefficient and subsequently the pinning force turnedout to be negligible which was attributed to the high surface quality of the glass channels and thedisappearance of contact lines in lubricated bubbles. Because of dewetting， elongated bubbles moving in microfluidic systems are not alwayslubricated. Dewetting refers to the spontaneous withdrawal of a liquid film from a partiallywetting surface [19]. On a totally wetting substrate， a film is always stable while on a partiallywetting surface， thin films below a critical thickness are metastable and may dewet by nucleationand the growth of a dry zone. The critical thickness for an ideal surface is given by where k is the capillary length defined by √ol pg [19]. Non-ideal surfaces which are marredby chemical or physical surface irregularities exhibit contact angle hysteresis. For those surfaces，eu must be replaced with e，ec in Eq. (1). The nucleation of a dry zone may start with a local surface defect or by a perturbation； the zoneexpands if its radius exceeds a critical value. For an air-water system with a receding contactangle as low as 1， the critical thickness becomes about 45 um， much thicker than a possibleliquid film around an elongated bubble in microchannels. Because microfluidic systems are notusually free of surface defects， the chances for dewetting of the thin films around bubbles arerelatively high. The dewetting velocity with which the dry zone opens up is constant， where k is an empirical coefficient that depends on the nature and molecular weight of the fluidand also the solid surface quality [20]. The existence of lubricating films around bubbles in a partially wetting microchannel depends onthe bubble velocity， the dewetting velocity， the length of the bubbles， and the microchannelcross-sectional dimensions [21]. Experimental results from Cubaud and Ho [21] have shownthat， in a partially wetting square channel， bubbles moving slower than the dewetting velocity arecompletely dry. For velocities between the dewetting velocity and a critical velocity， the rear ofthe bubble is dried out while the front of the bubble remains lubricated (hybrid bubbles). Atvelocities higher than the critical velocity， bubbles are lubricated. The critical velocity wasestimated by where ， is the length of the bubble and h is the width of the square microchannel [21]. Furtherexperimental data is required to develop such a correlation for rectangular channels. Figure 1presents images of dry， hybrid， and lubricated bubbles. The presence of droplets on the channelwalls around a bubble is indicative of dewetting. Figure 1： Bubbles with different wetting conditions： a) dry bubble， b) consecutive images of ahybrid bubble， and c) lubricated bubble， adapted from Cubaud and Ho [21] We reviewed the conditions leading to the presence of dry bubble in microchannels. Because ofthe contact angle hysteresis on a non-ideal surface， a dry moving bubble forms different dynamiccontact angles at the front and back. The pinning forces applied on the contact lines and thepossible capillary pressure induced by pinning resist the motion of a dry bubble. At low flowvelocities， the pressure difference across the length of the bubble may not be enough toovercome these forces. In such a situation， the bubble becomes stationary. If there is no bypass for the flow， the bubble eventually starts crawling due to the liquid accumulation and pressurebuildup behind it. When a bypass (e.g. a parallel channel or corner flow) exists， the bubble mayremain stationary indefinitely and an increase of flow rate would be required to move the bubbleforward [18]. Dry stationary bubbles present in our microchannel array device， as shown inFigure 2， further motivate this theoretical and experimental study of the pressure required tostimulate motion of dry bubbles and effectively clear the channel. Figure 2： Dry stationary bubbles in a polycarbonate microchannel array where clear channels actas a bypass for the flow. Upon stagnation small droplets start to condense and grow on thechannel walls inside the saturated bubbles. a and b were taken upon stagnation and 17 minutesafter， respectively. 2 Theoretical model In this section a theoretical model that describes the roles of contact line forces on the bubblepinning phenomenon is proposed. de Gennes et al. [19] have described the forces acting on adynamic triple contact line on non-ideal surfaces. Herein， we develop the force balance equationsfor the situation where an external force on the contact line is not enough to initiate movement.Then we expand our model for a clogging bubble having two contact lines at its front and back. Young's equation presents the balance of forces for a triple contact line at equilibrium as When an external force is applied on a contact line， the contact angle varies from the localequilibrium contact angle. For the liquid segment in the circular tube described in Figure 3 theforce equilibrium yields where s refers to the arc length of the contact line. The magnitude ofthis force is equal to themagnitude of the pinning force which acts in the opposite direction and resists motion.Substituting the term - from Young’s equation we get Figure 3： Pinning forces resisting motion under the action of a piston in a circular tube， adaptedfrom de Gennes et al. [19] Similar forces acts on the back and front (left and right) triple contact lines (S and sp) of aclogging bubble in a rectangular channel as shown in Figure 4a. Depending on the hydrophilicity and geometry of the channel liquid regions may or may not bepresent in the corners of the channel， as illustrated in Figure 4b and Figure 4c. Liquids with equilibrium contact angles below 45° most likely occupy the corner regions in rectangularchannels[221. In the presence of corner liquid regions， we can assume slip boundary conditions on the gas-liquid interface since p<

90°)， as shown in Figure 4d， therequired condition for motion of the back interface might be different. Pinning forces cannoteffectively contribute to the force balance written for a fluid element at the bulged interface. Ifthe pressure difference across the interface exceeds what interfacial forces can compensate foraccording to the Young-Laplace equation， the interface will break up and move forward. In thiscase This condition is met before the pinning forces are overcome.Capillary valves often work basedon this principle [11， 24， 25]. Note that at the front interface， pressure differences up to4o cos egu /h can be easily held by surface tension. However， for higher pressure differencesand subsequently front contact angles below eou， the pinning forces are responsible for the integrity of the interface. The motion of the front interface is always triggered by overcoming thepinning forces. Adding Eq. (17) and Eq. (14) written for a square microchannel results in the necessarycondition for the motion of both the front and the back contact lines Therefore， our theoretical model consists of two parts： Eq.(16) for concave back bubbles and Eq.(18) for convex back bubbles with some exceptions. Our experimental results suggest that formoving bubbles with a slightly bulged back interface (90°≤0aiv≤93°)， Eq. (16) can predict thepressure difference across the bubble much better than Eq. (18). For slightly bulged interfaces，the pinning forces may be still able to assist the surface tension to withstand the pressuredifference across the meniscus and the motion of the back contact line happens by overcomingthe pinning forces. It is important to note that in the absence of liquid regions in the corners of the channel， pinningforces are the only resistive forces acting on the bubbles. Moreover， the length of the bubble doesnot influence the pressure difference across the bubble. However， when liquid regions exist，capillary forces may act on the bubble as well [13， 26] and the pressure drop across the bubblebecomes a function of the bubble length. 2.1 Differentiation between capillary and pinning forces It is easy to confuse capillary and pinning forces because the mathematical form of the right handofEq. (15) looks similar to capillary pressure if we replace s and A with 2 (w+h) and wxh，respectively. It is important to distinguish between the concepts of capillary and pinning forcesFE-14-1213 Sharp 11 and to understand that both of these forces may act on a dry bubble. In our example， the capillarypressure gradient was caused by the difference in the back and front curvatures due to pinningand contact angle hysteresis. An integration of capillary pressure over the interface results in thecapillary force [1]. It is also possible to have capillary forces in the absence of pinning； for example， lubricatedbubbles passing through contractions or expansions experience capillary forces [1]. The capillarypressure gradient and subsequently the capillary force can also be produced by a surface tensiongradient due to variations in the temperature， concentration，or electric field [22]. Unlike pinning forces which always resist the motion， capillary forces may stimulate it. Aninteresting example presented by Paust et al. [27] demonstrates a conflict between the capillaryand pinning forces： A growing bubble in a tapered geometry is pinned at the back side. Thebubble growth toward a bigger cross section reduces the front curvature and increases thecapillary force which eventually overcomes the pinning force at the back and detaches the bubblefrom the pinning point. 2.2 Capillary Pressure In the presence of liquid regions in the corner of a channel， the capillary pressure gradientinduces an interfacial liquid flow on the gas-liquid menisci (represented by dotted line in Figure4b) from the back to front that wants to move the bubble backward [22， 27]. Indeed， the cornerliquids act as a medium for implementing the capillary pressure. The maximum capillarypressure gradient occurs when both front and back contact angles hold their limiting values. Fora rectangular channel with a width of w and depth of h the Young-Laplace equation yields The possibility of the capillary pressure to be completely transferred through the small gas-liquidmenisci is an open question. If it does AP should be added to the right hand side of Eq. (15) toresult in the necessary condition for the onset of bubble motion The pressure drop along the length of the bubble is maintained by the bypassing corner flow.Treating the corner flow as Poiseuille flow， Wong et al. [13] approximated the pressure dropalong a stationary bubble by where g is the volumetric flow rate and der is the effective dynamic diameter of the liquidflow. Longer bubbles apply more resistant to the corner flow and experience a larger pressuredifference across their length. Equation (21) suggests that for any bubble length there is a criticalflow rate that provides a pressure drop sufficient for bubble motion. The dynamic contact anglesof the front and back of the dry moving bubble vary between receding and advancing contactangles as the triple contact line still exists. If the velocity of a dry bubble becomes high enough athin film around the bubble will start to form and eventually the bubble will become lubricatedagain [28]， as shown in Figure 5. Asthe velocity increases the bubble caps deformasymmetrically resulting in the so-called“bullet-shaped”appearance [22， 29]. Figure 5： Progression of bubble shape with increasing velocity： a) dry dynamic bubble， b) thinfilm forming， c) lubricated bubble， and d) bullet-shaped lubricated bubble， adapted from Jensen[28] 3 Theoretical model validation The objective of the experiments is to measure the pressure required behind a dry bubble (withno corner flow and hence no resistive capillary pressure) to overcome the pinning forces andmaintain its crawling motion in a square channel. At low velocities (Re<<1)， inertial forces arenegligible and we can assume that the pinning forces are the only forces resisting the motion.Furthermore， at low velocities (Ca<10=)， dynamic contact angles are independent of velocityand are equal to the static receding and advancing contact angles [30]. Equations (16) and (18)will be validated for bubbles with concave or convex back interfaces， respectively. 3.1 Experimental setup and procedure A schematic view of the experimental setup is presented in Figure 6. Square borosilicate glasschannels (VitroCom PN 8100) with a height and width of 976±8 um and a length of 10 cmhave been used as the test channels. In order to decrease the hydrophilicity of the glass and havecontact angle readings with high accuracy， four brand new channels were washed with“Rain XFE-14-1213 Sharp 14 Original” glass treatment solution for 1-3 seconds and then rinsed with water and dried withcompressed air. Depending on the duration of the treatment and the velocity of solution injection，each glass channel show different receding and advancing contact angles in the experiments. Thefluid used in the experiments was deionized water. The treatment increased the equilibriumcontact angle of water on the channel surface from below 10° to 60°-85. Figure 6： Experimental setup for pressure measurements A syringe pump (Harvard Apparatus PHD 2000 Programmable) was used for flow delivery. A100-ul gastight syringe (Hamilton PN 81020) was used for bubble injection. The imagingsystem consisted of a LaVision sCMOS camera coupled with a Navitar 12X Zoom lens. Anincandescent 100-Watt globe lamp was used for backlighting. A pressure transducer (Omega PX409-10WG5V) with a range of 0-2490 Pa and a 0.08%accuracy (±2 Pa) was used for pressure measurement. The pressure signal was recorded with anNI-USB 6211 DAQ system and LabVIEW. Several precautions were taken to reduce thepressure fluctuations in the room and test loop including limiting changes in the lab environment.The pressure transducer was connected to its fittings under water to avoid air trapping in thefittings and the transducer branch. The flow rate was set at 2 pl/min which corresponds to an average velocity of 2 mm/min in thechannels (Ca<10-). A single bubble with a volume between 5 to 10 ul was injected into theflow stream for each experiment. To verify the repeatability， on two of the channels， two sets ofexperiments were performed. The required pressure to maintain a flow rate of 2 ul/min as measured prior to bubbleinjection. The periodically averaged pressure values for every 0.2 s interval prior to bubbleinjection were relatively steady and the variations were within the ±2 Pa accuracy range of thetransducer. The change in the water reservoir level between this reading and the beginning ofimage and pressure acquisition is carefully calculated for each experiment (<120um) andaccounted for. We are interested in the pressure difference across the bubble length； therefore，the pressure prior to bubble injection was subtracted from the measured pressure during bubblemotion. The change in the reservoir water level during image and pressure acquisition was about29 um (0.28 Pa hydrostatic pressure) and was neglected. Image and pressure data acquisition started almost simultaneously at rates of 5 fps and 1000 Hzrespectively and continued for 12 minutes. The pressure data has been condensed to an averagepressure value for every 0.2 s interval. By matching a pressure peak in the pressure signal to itscorresponding closely-timed image with an apparent maximum pinning， the lag between theimaging and pressure measurements was revealed. Using this information， each image has beenmapped to a single pressure reading. 3.2 Results In order to calculate the required pressure to keep the bubbles moving from Eq. (16) and Eq. (18)the contact angles must be measured from the images. Accurate measurement of the contactangles is only possible when the image of the bubble interface is a uniform line as shown inFigure 7a and Figure 7c， in contrast to Figure 7b and Figure 7d. For the selected images， accuratemeasurement of the contact angles was possible. Furthermore， within one second before andafter the image was captured the motion of the bubble was steady and there was no observedrapid change in the shape of the front and back interfaces. For those cases， the variation in themeasured pressure during the two second period was always within and often much lower than±2 Pa accuracy range of the transducer. Figure 7： Variation in the contact angles during the motion， a and b are images of one bubble andc and d are images of another. a) concave back interface， c) convex back interface， b and d)nearly flat back interfaces and unequal front contact angles at the side walls The top and bottom contact angles of an interface (see Figure 4d) are often different and theaverage of their cosine values should be used instead in the equations. To use Eq. (18) theequilibrium contact angle must be known. We have estimated the cosine of the equilibriumcontact angle by averaging the cosine of the advancing and receding contact angles of all the FE-14-1213 selected occurrences in each dataset [31，32]. In reality， the equilibrium contact angle varieslocally but we have assumed it to be a single value in each data set. Table 1 presents the length of the bubbles in each experiment and the range of the advancing andreceding contact angles for the selected occurrences. It is possible to either have a concave or aconvex back interface when one of the back contact angles (top or bottom contact angles) isbelow 90° and the other one is above 90°. Table 1： Details of the six sets of experiments Experiment Channel Length of the injected o(deg)of Bady(deg) of No. No. bubble (mm) occurrences occurrences 1 1 8.1 70-83 87-97 2 1 10.2 69-84 86-94 3 2 9.3 56-70 77-90 4 3 5.3 44-60 93-98 5 4 5.6 68-81 89-97 6 4 5.4 66-78 86-96 Figure 8 presents a sample measured pressure difference across the bubble length over time. Thevalues of receding and advancing contact angles vary locally and the peak pressure valuescorrespond to severe pinning cases. Figure 9 presents an overview of the measured pressures in the six sets of experiments and themodel’s predictions. For bubbles witha aconcave back interface， the convex back modelunderestimates the required pressure to move the bubbles while for bubbles with a convex backinterface， the concave back model overestimates the required pressure. Time (s) Figure 8： Variation in pressure difference across the length of a crawling bubble over time whichis due to pinning and variation in the local surface conditions and contact angles Measurement No. Figure 9： Comparison between the model’s predictions and the measured pressures for the 6 setsof experiments. There could be a large difference between the concave and convex back modelpredictions. Measurement No. Figure 10： Comparison between the measured pressure and the matching models’ prediction.There are four irregular occurrences that have a slightly convex back interface but match theconcave back model's prediction better. The measured pressure is compared with the model’s predictions for bubbles in the six sets ofexperiments in Figure 10. Four of the occurrences have advancing contact angles slightly largerthan 90°(90°≤0adv≤93°) but they matched the concave back model’s prediction better. Wepresume for those four consequences the motion of the back contact line was achieved byovercoming pinning forces and not by failure of the meniscus in withstanding the pressuredifference across it. Figure 11 gives an overview of the normalized model’s predictions withrespect to measured pressures. The model successfully predicts the required pressure to maintainthe crawling motion ofthe bubbles. Measurement No. Figure 11： Normalized models’predictions based on the measured pressure The errors were calculated using the Kline and McClintock method [33]. The quantified errorsassociated with pressure measurements prior to bubble injection and during bubble motion aredue to the accuracy limits of the transducer (±2 Pa).The variations of the periodically averagedpressure values prior to bubble injection were within the accuracy range. Therefore， pressurefluctuations sourced from the syringe pump or room pressure were not quantifiable and havebeen neglected in error estimations. The errors associated with the concave model's prediction include uncertainty in the size of thechannel (±8 um) and uncertainty in the measurement of the four contact angles for a bubble dueto image resolution and thresholding (±2.25°). An additional error is involved in the predictionof convex back model due to the estimation of the equilibrium contact angle which results inlarger error bars for the convex back model's predictions. The cosine of equilibrium contactangle was estimated by averaging the cosine of advancing and receding contact angles of studiedoccurrences in each data set. A standard error of ±2.25°， equal to the uncertainty in contact anglemeasurements， was assigned to this estimation of the equilibrium contact angle. One of the most important factors contributing to the differences observed between the measuredpressure and the model’s predictions is the contact angle measurement. With a two dimensionalimage from the bubble caps it is not possible to measure the contact angles on all 4 sides of thechannel. We only measure the contact angles on the top and bottom sides of the bubble caps (asshown in Figure 4d). We assume that the cosine of contact angles on the other two sides are anaverage of the cosine of contact angles on the top and bottom sides. For a uniform cap thisassumption is reasonable but may not be accurate. Some assumptions were made in thedevelopment of our theoretical model such as assuming equal cross sectional areas， contact linelengths， and equilibrium contact angles at the front and the back of the bubbles. There is adefinite deviation from these assumptions in the experiments which also contributes to thedifferences between the measured pressure and the model's predictions. For example in thedevelopment of the model， Eq. (15) was derived by adding Eq. (13) and (14). If it turns out thats，/4 =0.99 sp/4g=0.99(4/h) the model prediction by Eq. (16) will overestimate therequired pressure by 0.04og(cosBegu-cosFadv)/h 4 Conclusions A bipartite theoretical model that describes the required pressure to initiate the motion of a drybubble in a partially hydrophilic channel is proposed. It has been shown that the motion of thefront interface of the bubble always occur by overcoming pinning forces. However， the motionof the back interface may occur either by overcoming the pinning forces or by failure of themeniscus in withstanding the pressure difference across it. The theoretical model has beenvalidated by experimental results. This work differentiates the resistive pinning forces from the capillary pressure which may get transmitted by the gas-liquid interface in the presence of cornerflow. The use of highly hydrophilic surfaces is an effective solution for reducing bubble clogging atlow velocities. 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[32] Della Volpe， C.， Maniglio， D.， Siboni， S.， and Morra， M.， 2001， "An ExperimentalProcedure to Obtain the Equilibrium Contact Angle from the Wilhelmy Method，" Oil & GasScience and Technology Rev IFP， 56(1)， pp. 9-22. [33] Kline， S. J.， and McClintock， F. A.， 1953， "Describing Uncertainties in Single-SampleExperiments，" Mechanical Engineering， 75， pp. 3-8. List of Table Captions Table 1： Details of the six sets of experiments List of Figure Captions Figure 1： Bubbles with different wetting conditions： a) dry bubble， b) consecutive images of ahybrid bubble， and c) lubricated bubble， adapted from Cubaud and Ho [21] Figure 2： Dry stationary bubbles in a polycarbonate microchannel array where clear channels actas a bypass for the flow. Upon stagnation small droplets start to condense and grow on thechannel walls inside the saturated bubbles. a and b were taken upon stagnation and 17 minutesafter， respectively. Figure 3： Pinning forces resisting motion under the action of a piston in a circular tube， adaptedfrom de Gennes et al.[19] Figure 4： a) Side view of a stationary bubble and forces applied on it by the pressure field andthe channel walls， b and c) channel cross section at the left contact line with and without liquid inthe corner regions， respectively， note that the dotted border is not a part of the triple contact linebut rather a gas liquid interface， d) a moving bubble with a convex back interface Figure 5： Progression of bubble shape with increasing velocity： a) dry dynamic bubble， b) thinfilm forming， c) lubricated bubble， and d) bullet-shaped lubricated bubble， adapted from Jensen[281 Figure 6： Experimental setup for pressure measurements Figure 7： Variation in the contact angles during the motion， a and b are images of one bubble andc and d are images of another. a) concave back interface， c) convex back interface， b and d)nearly flat back interfaces and unequal front contact angles at the side walls Figure 8： Variation in pressure difference across the length of a crawling bubble over time whichis due to pinning and variation in the local surface conditions and contact angles Figure 9： Comparison between the model's predictions and the measured pressures for the 6 setsof experiments. There could be a large difference between the concave and convex back modelpredictions. Figure 10： Comparison between the measured pressure and the matching models’ prediction.There are four irregular occurrences that have a slightly convex back interface but match theconcave back model's prediction better. Figure 11： Normalized models’predictions based on the measured pressure FE- FE-harp