Dynamic drying transition via free-surface cusps

Abstract

We study air entrainment by a solid plate plunging into a viscous liquid, theoretically and numerically. At dimensionless speeds $Ca=U\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}$ of order unity, a near-cusp forms due to the presence of a moving contact line. The radius of curvature of the cusp’s tip scales with the slip length multiplied by an exponential of $-Ca$. The pressure from the air flow drawn inside the cusp leads to a bifurcation, at which air is entrained, i.e. there is ‘wetting failure’. We develop an analytical theory of the threshold to air entrainment, which predicts the critical capillary number to depend logarithmically on the viscosity ratio, with corrections coming from the slip in the gas phase.

Cox–Voinov theory with slip

“…For the solid line we use the correct value of , the dashed line represents the value . For the smaller capillary number, using the -value as calculated in the present paper, the deviation is negligible, and even for a capillary number , the relative error remains small; when the capillary number becomes of order unity the GL equation fails (Kamal et al 2018), since it is an expansion for small only. On the other hand for the relative error is significant in both cases as one approaches the contact line.…”

“…One of the main aims is to calculate the apparent contact angle as well as the critical capillary number at which fluid 1 is entrained into fluid 2. As shown in Kamal et al (2018), for small values of M (for example when fluid 1 is air), the critical capillary number is so high that it cannot be calculated within an expansion for small Ca. However, here we focus on the regime of smaller Ca up to 0.1, and compute the apparent contact angle, to assess the improvement of our predictions over Chan et al (2013), where the results it was assumed that c = 3.…”

“…We compare full numerical simulations of the dip-coating problem, using the finite elements method (FEM) described in detail in Kamal et al (2018), and based on a framework benchmarked in Sprittles (2015), with results of the GL equation for up to 0.1. We show an example with and one with , with a contact angle of (figure 4(,), respectively).…”

“…We use the GL equation to evaluate the error in predictions of between simulations where the correct value of is implemented, to those where the lubrication value () is used (figure 5). For finite viscosity ratios , above a critical capillary number the stationary meniscus bifurcates to a time-dependent solution, in which the outer fluid is entrained (Snoeijer & Andreotti 2013; Kamal et al 2018). In figure 5( a ), we vary the viscosity ratio as the contact angle is held constant at .…”

Viscous drag force model for dynamic Wilhelmy plate experiments

“…While the measured change in microscopic contact angle appears relatively small, it could have a significant impact on a variety of wetting phenomena such as air entrainment at an advancing contact line [17]. In a recent work by Kamal et al [56], the critical Capillary number marking the onset of air entrainment was correlated to the force acting at the MCL and the microscopic contact angle. The authors noted that the critical Capillary number could change by up to 0.6 over a 20 • range, i.e.…”