Viscous drag force model for dynamic Wilhelmy plate experiments

In this work we revisit the dynamic Wilhelmy plate method for measuring dynamic contact angles with a specific focus on the viscous drag force which acts along the submerged surface of the plate. Particle image velocimetry shows that the viscous drag force can induce significant contact angle measurement errors if ignored. Preexisting models are found to underestimate the viscous drag force at moderate to high Capillary numbers, in finite-sized fluid domains, and for curved interfaces. Therefore, we propose a new model for the shear stress along a Wilhelmy plate by combining the dominant terms from a moving contact line flow, a Stokes flow past the leading edge of a flat plate, and a Couette-Poiseuille flow. This new model accounts for the shape of the fluid interface, the finite length of the plate, and the finite-size of the fluid container. Good agreement is obtained in comparisons with particle image velocimetry measurements over a Capillary number range from 0.001 to 0.3. This viscous drag model is applied to a dynamic Wilhelmy plate experiment and we obtain dynamic microscopic contact angle measurements and unbalanced Young's forces that are consistent with hydrodynamic theories of the moving contact line.

Figure 1

Schematic of the interface shape near a moving contact line. The microscopic contact angle ${\varphi }_0$ is measured at a distance $\epsilon$ from the contact line and the apparent contact angle ${\varphi }_D$ is measured at a distance ${\ell }_C$ from the contact line where ${\ell }_C$ denotes the capillary length.

Figure 2

Schematic of the control volume and forces acting on (a) static Wilhelmy plate and (b) dynamic Wilhelmy plate. The dashed red line defines the control volume over which the forces are summed. ${\varphi }_{\text{static}}$ denotes the static contact angle and ${\varphi }_0$ denotes the dynamic microscopic contact angle. Schematic is not drawn to scale.

Figure 3

Schematic of the dynamic Wilhelmy plate experiment using an elastic beam and displacement sensor to measure forces. ${\ell }_D$ denotes the submerged depth of the plate, $\delta$ is the deflection the beam tip, $w$ is the distance between the plate and edges of the fluid container, and $h$ is the distance between the plate and the bottom of the fluid container.

Figure 4

(a) Schematic of the vertical stage used to raise and lower the fluid tank. (b) Picture of vertical stage, fluid container, and plate used in the Wilhelmy plate experiment.

Figure 5

Image of the static water-air interface used to validate the static Wilhelmy plate experiment.

Figure 6

Measured force vs. time for a contact line advancing at 5 mm/s. In region I the plate is statically suspended above the fluid. The fluid interface makes contact with the plate at the start of region II, advances up the surface of the plate, and is impulsively stopped at the end of region II. In region III the contact line relaxes to its static shape. Inset shows a magnified view of region II.

Figure 7

(a) Velocity field and plate normal stress for glycerin advancing at 7.5 mm/s. The Wilhelmy plate and interface are denoted by the solid black lines. The contact line lies at $y=0$. Only a quarter of the measured vectors have been plotted for clarity. (b) Comparison of theoretical and experimental shear stress along the plate surface. $y$ denotes the distance from the the moving contact line.

Figure 8

Validation of the current viscous drag force model. Markers denote experimental PIV measurements and solid lines denote the current model. (a) Comparison of the vertical velocity profile in the intermediate region ($y={\ell }_D/2$) for various contact line velocities. Theoretical velocity profiles are determined using Eq. (9). (b) Comparison of the shear stress profile at various plate depths for an interface velocity of 7.5 mm/s. Theoretical shear stress profiles are determined using Eq. (12).

Figure 9

(a) Microscopic dynamic contact angle and (b) force of the moving contact line measured by the dynamic Wilhelmy experiment for glycerin advancing over a glass substrate. Theoretical contact angles and moving contact line forces are modeled by the dynamic Young's equation presented in Eq. (16), which was originally proposed in our previous publication [46].