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Liquid meniscus friction on a wet plate: Bubbles, lamellae, and foamsa)
a)This paper is based on an invited lecture, which was presented by Isabelle Cantat at the 64th Annual Meeting of the Division of Fluid Dynamics of the American Physical Society, held 20–22 November 2011 in Baltimore, MD.
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View: Figures


Image of FIG. 1.
FIG. 1.

Different diphasic flows involving the motion of a meniscus on a wet wall. From left to right: a single bubble in a tube (Bretherton's problem ) or in a Hele-Shaw cell; a lamella in a tube; a plate or a fiber pulled out of a liquid bath (Landau-Levich-Derjaguin (LLD) problem ); a droplet spreading on a wet plate (Tanner's problem ). is the velocity of the solid in the frame of the meniscus. The front and rear menisci are denoted respectively by (f) and (r). Each meniscus is connected to a wetting film of length ℓ.

Image of FIG. 2.
FIG. 2.

Schematic view of a liquid meniscus on a solid that moves in the -direction at velocity . The thickness profile is (), the pressure is (), and the x-component of the velocity in the meniscus frame is (, ). The radius of curvature of the meniscus tends to for → −∞ (meniscus side) and the film thickness tends to for → ∞ (wetting film side). The plate velocity > 0 corresponds to a front meniscus (f) and < 0 to a rear meniscus (r). The problem is invariant in the direction.

Image of FIG. 3.
FIG. 3.

Family of rear meniscus profiles, obtained by solving Eq. (6) . The boundary conditions at = 0 are derived from Eq. (9) using ɛ = 10 and different values of Φ. The dashed line (red online) is the * solution, having the same asymptotic curvature as the front meniscus. Inset: solution of Eq. (6) for the front meniscus.

Image of FIG. 4.
FIG. 4.

Asymptotic parabolic shapes of the meniscus. The solid lines are the numerical solutions of Eq. (6) for the rear meniscus * (left) and for the front meniscus (right). The dashed lines are the best parabolic fits. The horizontal dotted lines show the height of the parabola minimum and , for the rear and front cases, at the position . The apparent contact point of the rear meniscus is and the apparent contact angle at this point is .

Image of FIG. 5.
FIG. 5.

Wetting film profiles for a 2D bubble in contact with a wall moving to the right, for incompressible interfaces. +, ×, and •: experimental data from Fig. 12(a) in Ref. , for = 1.15 10, 2.3 10, and 4.6 10, respectively. Full lines: solutions of Eq. (6) (shown in Fig. 3 ) rescaled using β = 2 (sliding case). Inset: Zoom on the rear meniscus of the first profile. Dashed, full, and dashed-dotted lines: solutions of Eq. (6) rescaled using β = 1 (stress free case, red online), β = 2 (sliding case), and β = 4 (rolling case, blue online), respectively.

Image of FIG. 6.
FIG. 6.

Fractional speed difference of the liquid phase around an isolated bubble in a tube. •: experimental data from Fig. 4 , with = 3 10 and = 0.5 mm. The solid lines are the predictions, from top to bottom, for the rolling case, the stress-free case, and the sliding case.

Image of FIG. 7.
FIG. 7.

Normal and tangential force distribution in the front and * meniscus, in dimensionless units. (a) Upper graph: Front meniscus profile. Lower graph: solid line: pressure = 0.643 − (); dashed line: meniscus contribution to the viscous stress ( − 1)/ ; dotted-dashed line: velocity divergence − / . (b) Same functions for the rear meniscus *. Full line: = 0.643 − (); dashed line: (1 − )/ ; dotted-dashed line: / .

Image of FIG. 8.
FIG. 8.

as a function of the asymptotic curvature (−∞). is the dynamical meniscus contribution to the viscous force per unit length exerted on the rear meniscus. It is related to the corresponding physical quantity by the Eq. (13) . The circle represents the * solution.

Image of FIG. 9.
FIG. 9.

Sketch of a single bubble in a tube, with the two subsystems Ω and used for the force balance. The two subsystems are defined by the liquid bounded by the tube wall, the median plane and, respectively, the plane and .

Image of FIG. 10.
FIG. 10.

Sketch of a lamella.

Image of FIG. 11.
FIG. 11.

Network of menisci in contact with a wall, for a 2D or 3D dry foam. Each meniscus makes an angle θ with the direction , normal to the foam velocity. The contact area between a bubble and the wall is .

Image of FIG. 12.
FIG. 12.

Wet foam in contact with a wall. Each bubble occupies a hexagonal domain of area . The contact area between the bubble and the wall is a disc of area represented in grey.

Image of FIG. 13.
FIG. 13.

Stress at the wall rescaled by γ/ as a function of the capillary number for a 3D foam. Symbols: Experimental data obtained by Denkov in Ref. , Fig. 8(c) . The solid line is obtained from Eq. (49) with = 0.7 and ℓ/ = 0.318, and the dotted-dashed line with = 0.5 and ℓ/ = 0.5. The dashed lines are, respectively, power laws of exponent 1/3, 1/2, and 2/3.

Image of FIG. 14.
FIG. 14.

Stress at the wall, rescaled by γ/ , plotted as a function of the capillary number for a 3D foam of Amilite GCK-12. △, ▲, ○, ●: Experimental data from Fig. 8 in Ref. , with liquid fractions ϕ=0.10, 0.15, 0.20, and 0.25, respectively. The solid lines are predictions of Eq. (49) using the theoretical values of ⟨ℓ⟩/ = 0.5; 0.25; 0.09, and 0.004, as obtained from Eqs. (45) and (47) and = 2  = 0.6; 0.44; 0.32, and 0.18. The dashed lines are power laws of exponent 1/3 and 2/3.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Liquid meniscus friction on a wet plate: Bubbles, lamellae, and foamsa)