Capillary rise with velocity-dependent dynamic contact angle

The classic description of the rate of capillary rise given by the Washburn equation, which assumes that the contact angle preserves the equilibrium value at all times, has been recently questioned in the light of the known experimental dependence of the dynamic contact angle on the velocity of the contact line. For a number of such proposed functions of velocity for the dynamic contact angle, we analyze the resulting dependences of the contact angle and of the time of rise, respectively, on the height of the capillary rise. By applying our results to the particular cases of a high-viscosity silicone oil and water, respectively, in a glass capillary, we show that, in general, strong similarities arise between the various approaches and the classic theory in what concerns the time dependence of the capillary rise, which explains the lack of consistent experimental evidence for deviations in the rate of capillary rise from the Washburn equation. However, for a strong dependency of the contact angle on the velocity in the range of small velocities, as in the case of water on glass, one of the models predicts significant deviations even for the time dependence of the capillary rise. Moreover, our results show that the time or height dependence of the contact angle during the capillary rise can clearly discriminate between the various models.