Asymptotic solutions for the relaxation of the contact line in the Wilhelmy-plate geometry: The contact line dissipation approach

Abstract

The relaxation of straight contact lines is considered in the context of the Wilhelmy-plate experiment: a homogeneous solid plate is moving vertically at constant velocity in a tank of liquid in the partial wetting regime. We apply the contact line dissipation approach to describe the quasistatic relaxation of the contact line toward the stationary state (below the entrainment transition). Asymptotic solutions are derived from the differential equations describing the capillary rise height and the contact angle relaxation for small initial deviations of the height from the final stationary value in the model considering the friction dissipation at the moving contact line, in the model considering the viscous flow dissipation in the wedge, and in the combined model taking into account both channels of dissipation. We find that for all models the time relaxation of the height and the cosine of the contact angle are given by sums of exponential functions up to a second order in the expansion of the small parameter. We analyze the implications which follow when only one dissipation channel is taken into account and compare them to the case when both dissipation channels are included. The asymptotic solutions are compared with experimental results and with numerically obtained solutions which are based on hydrodynamic approach in lubrication approximation with and without a correction factor for finite contact angles. The best description of the experimental data, based on multicriteria testing, is obtained with the combined contact line dissipation model which takes into account both channels of dissipation.