On the condition of existence of an equilibrium drop in a model that takes into account the elasticity of the intermediate layer

Abstract

In this paper, the 1st variation of the Willmore functional is calculated. Here the Willmore functional is defined as $$\| H \|_2^2=\sigma 2\pi \int\limits_0^L H^{2}y\mathrm{d}s $$

The variational problem for the function $\mathfrak{F}(S)$ is solved. Here $$\mathfrak{F}(S):= F(S)+\mu \| H \| _{2}^{2}(S)$$ $$F(S)=\sigma \Lambda (S)+l_{p}\sigma \Xi^\ast (S)-\beta\sigma S^{\ast}+\lambda \sigma V(S)+V(S)\Gamma \rho$$

It is proved that the extreme surface belongs to an admissible class of surfaces. We consider surfaces that are defined by curves that are graphs of functions of a variable y that lies on an axis perpendicular to the axis of rotation. We prove the existence of a generalized 4th order derivative for functions that define curves that generate an extremal surface. A condition for the existence of an equilibrium drop is derived in the model that takes into account not only the thickness of the intermediate layer, but also the elasticity energy of this layer. This condition is defined by the equation $$\mu {\cdot \Delta }_{S}H+2\mu \cdot H\left( H^{2}-K \right)+\sigma \left( 2H+l_{p}K \right)=\lambda +\frac{1}{\sigma }\Gamma \rho$$

Here $\Delta_{S}$ is the Laplace-Beltrami operator of the surface, $K$ – the Gaussian curvature and $H$ – the mean curvature of it respectively.