A theory of capillary rise of a liquid in a vertical cylindrical tube and in a parallel-plate channel: Washburn equation modified to account for the meniscus with slippage at the contact line

Abstract

The classic equation of Washburn and Rideal for the rate of displacement due to surface tension forces of air by a liquid in a vertical capillary tube is based on the steady-flow parabolic velocity profile of Poiseuille across a section of the capillary. However, such a model of incompressible liquid flow cannot apply at the two ends of the liquid column in the capillary. The departure from Poiseuille flow in the vicinity of the advancing meniscus in a vertical cylindrical capillary is examined under the following assumptions. (i) The meniscus retains a fixed shape, which is the sector of a sphere making an angle of contact α with the capillary wall. (ii) The liquid has risen up the tube to reach a sufficiently slow quasi-steady flow, when both the acceleration and nonlinear inertia terms may be ignored. (iii) The usual nonslip flow condition is assumed at the tube wall except in the immediate region of the three-phase contact line, where the slip velocity is proportional to the shear stress exerted on the solid tube. The two regions are matched by the method of “inner” and “outer” expansions. (iv) At the meniscus, the tangential shear stress is zero and continuity of normal stress is approximated by assuming a fixed spherical shape. A generalization of Washburn's formula for the rate of rise of the meniscus as a function of height in a cylindrical tube is obtained. The meniscus exerts an extra drag which increases the effective height by an amount that increases with decrease in contact angle and the dimensionless ratio (slip coefficient/tube radius). A similar treatment of the flow near the meniscus moving up a parallel-plate channel is presented and a corresponding increase in drag due to the meniscus is obtained.