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

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.

Figure 1

(Color online) Schematic drawing of the system.

Figure 2

(Color online) Critical value of the stationary contact angle ${\theta }_{st}^{\ast }$ as a function of the equilibrium contact angle ${\theta }_{eq}$ (both given in degrees). For the HDM the solid line; for the HDM-L short dotted line. For the combined CLDM and the combined CLDM-L the dashed-dotted and the dotted lines, respectively, for several values of the parameters $\xi /{\psi }_1,\xi /{\psi }_2$. The de Gennes result [Eq. (49)] is shown with the dashed line.

Figure 3

(Color online) Dimensionless height of the contact line as a function of the plate velocity (expressed in terms of the dimensionless capillary number $\text{Ca}=u\eta /\gamma$): experimental results—solid squares, squares with error bars; the MKM—solid line; the HDM—dotted line; the HDM-L—dashed line; the combined CLDM $\left(\xi /{\psi }_1=9.091\right)$ and the combined CLDM-L $\left(\xi /{\psi }_2=6.66\right)$—dashed-dotted lines; HA—short dotted line; and HA-L—short dashed line.

Figure 4

(Color online) Dimensionless relaxation rate $\sigma$ as a function of the capillary number Ca: experimental results—solid squares with error bars; the MKM results—solid line; the HDM results—dotted line; the HDM-L results—dashed line; combined CLDM $\left(\xi /{\psi }_1=9.091\right)$ and the combined CLDM-L $\left(\xi /{\psi }_2=6.66\right)$—dashed-dotted lines; HA—short dotted line, and HA-L—short dashed line within the experimental error interval. This criterion determines uniquely $h_{st}\left(\text{Ca}\right)$ and $\sigma \left(\text{Ca}\right)$ in the MKM and in the HDM. By extrapolating the experimental results in [31] for $\sigma \left(\text{Ca}\right)$ to zero plate velocity we require that it lies in the interval $\left[0.029-0.0025,0.029+0.0025\right]$.