zbMATH — the first resource for mathematics

Capillary drops: contact angle hysteresis and sticking drops. (English) Zbl 1168.76007
The authors study an interesting homogenization procedure for the capillary problem which describes a drop resting on a plane. The governing equation is (1) \(2H=\Gamma\rho/\sigma-\lambda\), where \(H\) denotes the mean curvature of the capillary interface \(\mathcal S\), \(\Gamma\) the gravitational potential, \(\rho\) the local density and \(\sigma\) the surface tension. The constant \(\lambda\) is a Lagrange parameter which comes from the volume constraint of the liquid. The contact angle condition (boundary condition) is (2) \(\cos\gamma=\beta\) on \(\partial\mathcal{S}\), where \(\gamma\) is the angle between the capillary surface and the supporting plane, and \(\beta\) is the relative adhesion coefficient.
The paper concerns the case that (3) \(\beta=\beta(x/\varepsilon,y/\varepsilon)\), \(\varepsilon>0\) is small, with a \(\mathbb Z\)-periodic \(\beta(x,y)\). Set \(\langle\beta\rangle=\int_{[0,1]^2}\beta(x,y)\,dx\,dy\). In the case of zero gravity \((\Gamma=0)\), the main result is that there exists a solution of the capillary equation \(2H=-\lambda\) on \(\mathcal S\) which satisfies the boundary condition \(\cos\gamma\leq\langle\beta\rangle\), provided the crucial assumption \(\min_y\max_x\beta(x,y)<\langle\beta\rangle\) is satisfied. The proofs of the results are based on compactness results for viscosity solutions to (1)–(3). Viscosity solutions are minimizers of an energy functional \(J_\varepsilon\) associated to (1)–(3). The existence of the minimizers follows by using appropriate barriers.

76B45 Capillarity (surface tension) for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49Q05 Minimal surfaces and optimization
Full Text: DOI
[1] Caffarelli L.A., C√≥rdoba A. (1993) An elementary regularity theory of minimal surfaces. Differ. Integral Equ. 6(1): 1–13 · Zbl 0783.35008
[2] Caffarelli, L.A., Lee, K.-A., Mellet, A.: Homogenization and flame propagation in periodic excitable media: The asymptotic speed of propagation. Commum. Pure Appl. Math. (2006) (in press) · Zbl 1093.35010
[3] Caffarelli, L.A., Mellet, A.: Capillary drops on an inhomogeneous surface. Preprint (2005) · Zbl 1200.76053
[4] Dussan, V., Chow, R.: On the ability of drops or bubbles to stick to non-horizontal surfaces of solids. J. Fluid Mech. 137, 1–29 · Zbl 0543.76140
[5] Finn R., Shinbrot M. (1988) The capillary contact angle. II. The inclined plane. Math. Methods Appl. Sci. 10(2): 165–196 · Zbl 0657.76080
[6] Finn R. (1986) Equilibrium Capillary Surfaces. Springer, Berlin Heidelberg New York · Zbl 0583.35002
[7] Giusti E. (1984) Minimal Surfaces and Functions of Bounded Variation. Birkhauser, Basel · Zbl 0545.49018
[8] Gonzalez E., Massari U., Tamanini I. (1983) On the regularity of boundaries of sets minimizing perimeter with volume constraint. Indiana Univ. Math. J. 32, 25–37 · Zbl 0504.49026
[9] Hocking L.M. (1995) On contact angles in evaporating liquids. Phys. Fluids 7(12): 2950–2955 · Zbl 1026.76562
[10] Hocking L.M. (1995) The wetting of a plane surface by a fluid. Phys. Fluids 7(6): 1214–1220 · Zbl 1023.76501
[11] Huh, C., Mason, S.G.: Effects of surface roughness on wetting (theoretical). J. Colloid Interface Sci. 60, 11–38
[12] Joanny, J.-F., de Gennes, P.-G.: A model for contact angle hysteresis. J. Chem. Phys. 81 (1984)
[13] Leger L., Joanny J.-F. (1992) Liquid spreading. Rep. Prog. Phys. 55, 431–486
[14] Wente H.C. (1980) The symmetry of sessile and pendent drop. Pac. J. Math. 88, 387–397 · Zbl 0473.76086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.