^{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.

^{1,b)}

### Abstract

Many microfluidics devices, coating processes, or diphasic flows involve the motion of a liquid meniscus on a wet wall. This motion induces a specific viscous force, that exhibits a nonlinear dependency in the meniscus velocity. We propose a review of the theoretical and experimental work made on this viscous force, for simple interfacial properties. The interface is indeed assumed either perfectly compressible (mobile interface) or perfectly incompressible (rigid interface). We show that, in the second case, the viscous force exerted by the wall on the meniscus is a combination of two power laws, scaling such as Ca 1/3 and Ca 2/3, with Ca the capillary number. We provide a prediction for the stress exerted on a foam sliding on a wet solid and compare it with experimental data, for the incompressible case.

We thank N. Denkov and S. Tcholakova for their experimental data and A. Saint-jalmes, B. Dollet, J. Seiwert, S. Jones, and H. A. Stone for their constructive remarks.

I. DYNAMICAL MENISCUS SHAPE

A. Resolution of the lubrication problem

B. Asymptotic parabolic shapes

C. Comparison with some experimental data

1. Wetting film thickness in the LLD geometry

2. Film profile for bubbles with incompressible interfaces

3. Flux around a bubble for pure water

II. VISCOUS FORCES IN A MENISCUS

A. Theoretical predictions

B. Validity range of the models

1. Velocity range

2. Interfacial rheology

3. Meniscus size and length of the wetting film

III. PRESSURE DROP FOR AN ISOLATED BUBBLE OR A LAMELLA

A. Force balance

1. Single bubble

2. Lamella

B. Bretherton's pressure derivation for a single bubble

IV. BOUNDARY CONDITIONS FOR FOAM FLOWS

A. Film orientation

B. Influence of the liquid fraction

1. 3D dry foams

2. Wet foams

C. Comparison with some experimental results

1. 3D foams

2. 2D foams

V. CONCLUSION

##### B81B

## Figures

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 13 ) or in a Hele-Shaw cell; a lamella in a tube; a plate or a fiber 8 pulled out of a liquid bath (Landau-Levich-Derjaguin (LLD) problem 11,40 ); a droplet spreading on a wet plate (Tanner's problem 41 ). U 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.

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 13 ) or in a Hele-Shaw cell; a lamella in a tube; a plate or a fiber 8 pulled out of a liquid bath (Landau-Levich-Derjaguin (LLD) problem 11,40 ); a droplet spreading on a wet plate (Tanner's problem 41 ). U 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.

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

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

Family of rear meniscus profiles, obtained by solving Eq. (6) . The boundary conditions at X = 0 are derived from Eq. (9) using ɛ0 = 10−6 and different values of Φ. The dashed line (red online) is the r* solution, having the same asymptotic curvature as the front meniscus. Inset: solution of Eq. (6) for the front meniscus.

Family of rear meniscus profiles, obtained by solving Eq. (6) . The boundary conditions at X = 0 are derived from Eq. (9) using ɛ0 = 10−6 and different values of Φ. The dashed line (red online) is the r* solution, having the same asymptotic curvature as the front meniscus. Inset: solution of Eq. (6) for the front meniscus.

Asymptotic parabolic shapes of the meniscus. The solid lines are the numerical solutions of Eq. (6) for the rear meniscus r* (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 X p . The apparent contact point of the rear meniscus is X c and the apparent contact angle at this point is .

Asymptotic parabolic shapes of the meniscus. The solid lines are the numerical solutions of Eq. (6) for the rear meniscus r* (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 X p . The apparent contact point of the rear meniscus is X c and the apparent contact angle at this point is .

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. 51 , for Ca = 1.15 10−4, 2.3 10−4, and 4.6 10−4, 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.

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. 51 , for Ca = 1.15 10−4, 2.3 10−4, and 4.6 10−4, 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.

Fractional speed difference W of the liquid phase around an isolated bubble in a tube. •: experimental data from Fig. 4 , 18 with Ca = 3 10−5 and r t = 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.

Normal and tangential force distribution in the front and r* meniscus, in dimensionless units. (a) Upper graph: Front meniscus profile. Lower graph: solid line: pressure P = 0.643 − H ″(X); dashed line: meniscus contribution to the viscous stress (H − 1)/H 2; dotted-dashed line: velocity divergence −H ′/H 2. (b) Same functions for the rear meniscus r*. Full line: P = 0.643 − H ″(X); dashed line: (1 − H)/H 2; dotted-dashed line: H ′/H 2.

Normal and tangential force distribution in the front and r* meniscus, in dimensionless units. (a) Upper graph: Front meniscus profile. Lower graph: solid line: pressure P = 0.643 − H ″(X); dashed line: meniscus contribution to the viscous stress (H − 1)/H 2; dotted-dashed line: velocity divergence −H ′/H 2. (b) Same functions for the rear meniscus r*. Full line: P = 0.643 − H ″(X); dashed line: (1 − H)/H 2; dotted-dashed line: H ′/H 2.

as a function of the asymptotic curvature H ″(−∞). F r 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 r* solution.

as a function of the asymptotic curvature H ″(−∞). F r 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 r* solution.

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

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

Sketch of a lamella.

Sketch of a lamella.

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

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

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.

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.

Stress at the wall rescaled by γ/r b as a function of the capillary number Ca for a 3D foam. Symbols: Experimental data obtained by Denkov et al. in Ref. 29 , Fig. 8(c) . The solid line is obtained from Eq. (49) with Z = 0.7 and ℓ/r m = 0.318, and the dotted-dashed line with Z = 0.5 and ℓ/r m = 0.5. The dashed lines are, respectively, power laws of exponent 1/3, 1/2, and 2/3.

Stress at the wall rescaled by γ/r b as a function of the capillary number Ca for a 3D foam. Symbols: Experimental data obtained by Denkov et al. in Ref. 29 , Fig. 8(c) . The solid line is obtained from Eq. (49) with Z = 0.7 and ℓ/r m = 0.318, and the dotted-dashed line with Z = 0.5 and ℓ/r m = 0.5. The dashed lines are, respectively, power laws of exponent 1/3, 1/2, and 2/3.

Stress at the wall, rescaled by γ/r b , plotted as a function of the capillary number for a 3D foam of Amilite GCK-12. △, ▲, ○, ●: Experimental data from Fig. 8 in Ref. 30 , 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⟩/r m = 0.5; 0.25; 0.09, and 0.004, as obtained from Eqs. (45) and (47) and Z fit = 2 Z theo = 0.6; 0.44; 0.32, and 0.18. The dashed lines are power laws of exponent 1/3 and 2/3.

Stress at the wall, rescaled by γ/r b , plotted as a function of the capillary number for a 3D foam of Amilite GCK-12. △, ▲, ○, ●: Experimental data from Fig. 8 in Ref. 30 , 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⟩/r m = 0.5; 0.25; 0.09, and 0.004, as obtained from Eqs. (45) and (47) and Z fit = 2 Z theo = 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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