Homogenization of contact line dynamics
A fluid droplet on a macroscopically flat surface whose height is assumed to be a graph \(h(x,y)\) with bounded support \(\Omega,\;\partial\Omega\) is the contact line \(\Gamma\). In the small angle limit the linearized contribution of surface energy is given by \(\gamma\int_{\Omega}(1+|\nabla h|^2)\,d {\mathbf{x}}\) where \(\gamma\) is the liquid-vapor surface energy, which is constant here. The minimizer of this expression subjected to the constraint of constant volume is the quasi-steady droplet shape. The relevant Euler-Lagrange problem is \[ \triangle h=\lambda,\quad h|_{\Gamma},\quad\int_{\Omega}h\,dx=M\equiv \text{volume}, \] where \(\lambda<0\) is a Lagrange multiplier, essentially just the negative hydrostatic pressure. The ``apparent'' contact angle \(\theta\) is defined on the boundary \(\partial\Omega\), in the small angle limit, as \(\theta=-\frac{\partial h}{\partial \mathbf{n}},\;\mathbf{n}\) is the outward normal of \(\Gamma\). The motion of \(\Gamma\) is then specified by prescribing the normal velocity \(\frac{d\Gamma}{dt}\) as a function of this angle and the spatial location of the contact line \(\frac{d\Gamma}{dt}=F(\theta;\mathbf{x})\). This can be recast in terms of \(h\) as the free boundary condition \[ h_t=F(|\nabla h|;\mathbf{x})|\nabla h|,\qquad \text{when }\;h=0. \] The function \(F\) is a constitutive law relating the contact line velocity to the apparent contact angle \(\theta\), that must be a continuous and increasing function of \(\theta\). There is a local equilibrium angle \(\theta_l(\mathbf{x})\) where \(F(\theta_l(\mathbf{x});\mathbf{{x}})=0.\) The spatial dependence of \(F\) arises as a consequence of substrate inhomogeneities. Considering two-dimensional periodicity with some small period \(\varepsilon\) \(F(\theta;\mathbf{x}+\varepsilon \mathbf{k})=F(\theta;\mathbf{x})\) for all \(\mathbf{k}\in \mathbb{Z}^2\) the author examines the homogenized limit \(\varepsilon \to 0\) by deriving an effective velocity law that is independent on \(\varepsilon.\) Formal multiple scales analysis yields a local, periodic problem whose time-averaged dynamics corresponds to the homogenized front velocity.
scientific article; zbMATH DE number 5081446