Abstract
The planar flow of a steady moving-wall free-surface jet is examined theoretically for moderate inertia and surface tension. The method of matched asymptotic expansion and singular perturbation is used to explore the rich dynamics near the stress singularity. A thin-film approach is also proposed to capture the flow further downstream where the flow becomes of the boundary-layer type. We exploit the similarity character of the flow to circumvent the presence of the singularity. The study is of close relevance to slot and blade coating. The jet is found to always contract near the channel exit, but presents a mild expansion further downstream for a thick coating film. We predict that separation occurs upstream of the exit for slot coating, essentially for any coating thickness near the moving substrate, and for a thin film near the die. For capillary number of order one, the jet profile is not affected by surface tension but the normal stress along the free surface exhibits a maximum that strengthens with surface tension. In contrast to existing numerical findings, we predict the existence of upstream influence as indicated by the nonlinear pressure dependence on upstream distance and the pressure undershoot (overshoot) in blade (slot) coating at the exit.
Highlights
We examine the free-surface flow of a planar moving-wall jet at moderate Reynolds and capillary numbers near and far from the channel exit as encountered in coating flow
The planar laminar free-surface coating flow of a Newtonian fluid is investigated in the current simulation does not indicate the existence of any upstream influence, which is rather inaccurate given study
The second discrepancy concerns the stress singularity numbers, subject to the substrate translation, and an adverse or a favorable constant pressure at the exit, which should inevitably be reflected in the pressure singularity as shown in figure but not gradient applied far upstream of the channel exit as encountered in slot and blade coating flows
Summary
We examine the free-surface flow of a planar moving-wall jet at moderate Reynolds and capillary numbers near and far from the channel exit as encountered in coating flow. Chang et al [8] visualized the slot coating process for a low viscosity fluid, and observed that beyond a critical Reynolds number (Re = 20), both viscous and surface tension effects become negligible, with inertia dominating the flow. One of the earlier numerical work was done by Saito and Scriven [15], who carried out a finite-element analysis coupled to an iterative scheme to examine the capillary number effect on the curved meniscus close to the static contact line in slot coating They showed that the downstream meniscus no longer remains attached to the slot die at higher capillary number and lower flow rate. Ruschak considered a very small capillary number by setting the coating speed close to zero, and determined the film thickness in the slow flow (negligible inertia) limit by carrying out a singular perturbation method. The more complete analysis of the full Navier–Stokes equations and the implementation of a triple-deck approach may be envisageable in a future extension
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