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.
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 .
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.
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.
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.
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.
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.
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