This work investigates the onset of wetting failure for displacement of Newtonian fluids in parallel channels. A hydrodynamic model is developed for planar geometries where an advancing fluid displaces a receding fluid along a moving substrate. The model is evaluated with three distinct approaches: (i) the low-speed asymptotic theory of Cox [J. Fluid Mech.168, 169-194 (1986)], (ii) a one-dimensional (1D) lubrication approach, and (iii) a two-dimensional (2D) flow model solved with the Galerkin finite element method (FEM). Approaches (ii) and (iii) predict the onset of wetting failure at a critical capillary number Cacrit, which coincides with a turning point in the steady-state solution family for a given set of system parameters. The 1D model fails to accurately describe interface shapes near the three-phase contact line when air is the receding fluid, producing large errors in estimates of Cacrit for these systems. Analysis of the 2D flow solution reveals that strong pressure gradients are needed to pump the receding fluid away from the contact line. A mechanism is proposed in which wetting failure results when capillary forces can no longer support the pressure gradients necessary to steadily displace the receding fluid. The effects of viscosity ratio, substrate wettability, and fluid inertia are then investigated through comparisons of Cacrit values and characteristics of the interface shape. Surprisingly, the low-speed asymptotic theory (i) matches trends computed from (iii) throughout the entire investigated parameter space. Furthermore, predictions of Cacrit from the 2D flow model compare favorably to values measured in experimental air-entrainment studies, supporting the proposed wetting-failure mechanism.
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We are grateful to M. L. Pekurovsky and R. C. Moore for many helpful discussions. We acknowledge financial support from the Industrial Partnership for Research in Interfacial and Materials Engineering and the Donors of the American Chemical Society Petroleum Research Fund.