Capillary drops: contact angle hysteresis and sticking drops
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.
scientific article; zbMATH DE number 5146200
scientific article