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A Study on Capillary Flow Under the Effect of Dynamic Wetting Journal of JSEM, Vol.10, Special Issue (2010) 62-66 Copyright Ⓒ 2010 JSEM A Study on Capillary Flow under the Effect of Dynamic Wetting Kenji KATOH1, Tatsuro WAKIMOTO1 and Sinichiro NITTA1 1 Department of Mechanical Engineering, Osaka City University, Osaka 558-8585, Japan (Received 29 January 2010; received in revised form 17 April 2010; accepted 12 June 2010) Abstract those theories have succeeded in some degree to explain the A theoretical model was proposed to consider the dynamic experimental results for the velocity dependence of contact contact angle from a macroscopic viewpoint. A slight angles, it is still not clear whether they represent the correct convex profile of liquid surface is assumed to avoid the mechanism of dynamic wetting or not. divergence of viscous stress near the three-phase contact line If we consider the wetting phenomenon on a real solid on the wall. The theoretical results of the dynamic contact surface with roughness or defects with much larger scale angle are dependent on the parameter ε, i.e., the ratio of than molecules, it seems natural that the behavior of surface area occupied by the defects on the solid surface. apparent contact angle should be controlled mainly by the The movement of liquid column in a capillary is investigated macroscopic characteristics on the solid surface. In this experimentally. The modified Lucas-Washburn equation report, first a theoretical model is proposed to estimate the including the effect of dynamic wetting represents well the dynamic contact angle from a macroscopic viewpoint. In measured results. The theoretical model can approximate the order to discuss the velocity dependence of contact angles, measured dynamic contact angles within the experimental the viscous stress is estimated during the stick-slip motion uncertainty. of the contact line between the defects on the solid surface. Next we discuss the behavior of liquid column in a Key words capillary tube as the real system influenced by the dynamic Capillary Tube, Dynamic Wetting, Contact Angle, Surface wetting. The capillary phenomenon is related to many Tension, Viscous Stress applications in science, industry and daily life, e.g., the movement of liquid in a small gap in heat pipes, the rise of 1. Introduction blood in the blood sugar level sensor, and so on. However The wetting behavior between solid and liquid phases has very few reports have investigated the liquid motion in the an important topic in various engineering fields treating capillary based on a systematic experiment to measure small amount of liquid on a solid surface. When the three- precisely the behavior of the liquid column or the dynamic phase contact line at the tip of the liquid surface on the solid contact angle [2,3,4]. In this report, the liquid height and the wall moves with a finite velocity, one can observe that the dynamic contact angle are measured simultaneously in a contact angle changes with the velocity. When the capillary glass capillary. The results are compared with the modified number, i.e., the ratio between viscous force and surface Lucas-Washburn equation for the movement of liquid tension is larger than roughly 10-4 at which the velocity may column including the effect of dynamic wetting. The be typically 0.1 mm/s, the effect of dynamic wetting should theoretical model for the dynamic contact angle proposed be taken into consideration to discuss the liquid motion on here is validated through comparison with the measured the solid surface. Many theoretical or experimental studies results in this experiment. have discussed the velocity dependence on the contact angle [1,2]. There are two kinds of theoretical models to express 2. Theoretical Consideration the dynamic contact angle, one of which is called 2.1 Dynamic contact angle hydrodynamic model to consider the viscous dissipation in Based on the hydrodynamic model stated in section 1, Cox the wedge of liquid near the contact line, and the other is discussed the dynamic wetting and obtained the following called molecular kinetic theory (MKT) to consider the relation for the dynamic contact angle [5]. molecular movement at the contact line [1,2]. In the former 3 3/1 model, the liquid motion is analyzed from the theory of d ()θθ 9 ⋅+= CaD (1) fluid dynamics under the assumption of slip condition at the contact line. In order to avoid stress divergence at the where θd and θ indicate the dynamic contact angle and the contact line where the thickness of liquid layer becomes static contact angle, respectively. D is an experimental constant defined as: zero, the no-slip condition is discarded and the slip length is introduced so that the integral of viscous stress on the wall ⎛ L ⎞ can be determined. The dynamic contact angle is then D = ln⎜ ⎟ (2) S related to the viscous force. In the latter, back or forward ⎝ ⎠ jump of liquid molecules at the contact line is considered where L is the typical length scale of the system and S is the based on the Frenkel-Eyring theory of liquid transport slip length. In Eq. (1), Ca is the capillary number defined which is applied to the determination of viscosity and so on. as: The excess surface tension caused by the contact angle μU change was related to the statistical average of molecular Ca = (3) motion, i.e., the macroscopic velocity. Both models include σ some properties relating to molecular length scale. Although -- -- Journal of JSEM, Vol.10, Special Issue (2010) μ, U and σ indicate viscosity, velocity of contact line and the defect. Since very large wall shear stress concentrates surface tension, respectively. Cox discussed the liquid flow mainly near the contact line, the resultant of frictional force near the contact line separately in three regions, i.e., inner may be estimated in the region of λ, the characteristic region where the slip condition is applied, intermediate distance between defects as shown in Fig. 1(b). de Gennes region and outer region of macroscopic scale. Equation (1) calculated the viscous dissipation in the straight wedge was derived from the solution obtained by the method of region near the contact line, which has often been treated as matching asymptotic expansions. Here, we try to propose an the lubrication problem [8]. When the wall shear stress is approximate model to discuss the dynamic wetting on the integrated, the slip length should be introduced to avoid the solid surface having roughness or defect whose scale is stress divergence at the contact line. If we consider the much larger than the microscopic slip length irreversible slip motion, the contact angle may be different Figure 1 shows the schematic of contact line from the static equilibrium angle at the edge of the contact movement on the solid surface with defects such as line. In the present model, a slight convex profile of liquid roughness or patch of adsorbed foreign molecules. Although surface is assumed instead of the straight line as shown in we consider the receding of the contact line as shown in Fig. Fig. 1(b). The origin of the coordinate system is taken at the 1, the mechanism discussed below can be directly applied to edge of the defect. x and y coordinates are taken in the the advancing case. In the practical wetting behavior, it is horizontal and the wall-normal directions, respectively. The usually observed that the contact line movement is not liquid surface profile b(x) is approximated by the following continuous but shows a stick-slip type behavior even in the expression as: macroscopically quasi-static movement, i.e., the contact line 1−ε is once trapped at the defect as shown in Fig. 1, then it b ⎛ x − ε ⎞ = tanθ ⎜ ⎟ (4) moves irreversibly with a finite velocity to the next defect d ⎜ ⎟ λ ⎝ 1− ε ⎠ [6]. When the contact line is trapped at the defect, the liquid surface is distorted and some excess work is necessary to where x≡ x / λ and ε is the ratio occupied by the defect. move over the defect. This energy is dissipated during the The above profile satisfies the following geometrical slip motion and is considered as the cause of the contact conditions. angle hysteresis (i.e., the static contact angle is dependent on the moving direction of the contact line) [7]. When the b db x=ε : b = 0, x =1 : = tanθ , = tanθ , (5) contact line moves with a macroscopically finite velocity U, λ d dx d U is just added to the slip velocity during the irreversible motion, which leads to the increases of viscous drag on the Equation (4) means that the liquid surface is assumed to be wall τW, compared with the quasi-static case. In the present distorted by ελ, i.e., the scale of defect. The velocity model, this drag increase is related to the difference distribution u(y) in the liquid is simply approximated by the between the static and the dynamic contact angles. following equation to satisfy the boundary conditions at the As shown in Fig. 1(b), the viscous drag is estimated wall and the liquid surface. during the slip motion after the contact line is released from u 3 2 =⎜⎛2y − y ⎟⎞ (6) U 2 ⎝ ⎠ where y≡ y/ b . The force balance on the area ABC shown in Fig. 1(b) is now considered. The solid-gas and solid-liquid interfacial tensions, σ S and σ SL act on the contact line A while the liquid-gas surface tension σ acts on the liquid surface at C with the dynamic contact angle θ d as shown in the figure. (a) Movement of contact line Since the Reynolds number should be quite small, the influence of inertial force can be neglected.
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