{
  "id": "2408.06836",
  "version": "v1",
  "published": "2024-08-13T11:50:02.000Z",
  "updated": "2024-08-13T11:50:02.000Z",
  "title": "Linear stability analysis of a vertical liquid film over a moving substrate",
  "authors": [
    "Fabio Pino",
    "Miguel Alfonso Mendez",
    "Benoit Scheid"
  ],
  "categories": [
    "physics.flu-dyn",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The stability of liquid film flows are important in many industrial applications. In the dip-coating process, a liquid film is formed over a substrate extracted at a constant speed from a liquid bath. We studied the linear stability of this film considering different thicknesses $\\hat{h}$ for four liquids, spanning a large range of Kapitza numbers ($\\rm Ka$). By solving the Orr-Sommerfeld eigenvalue problem with the Chebyshev-Tau spectral method, we calculated the neutral curves, investigated the instability mechanism and computed the absolute/convective threshold. The instability mechanism was studied through the analysis of vorticity distribution and the kinetic energy balance of the perturbations. It was found that liquids with low $\\rm Ka$ (e.g. corn oil, $\\text{Ka}$ = 4) have a smaller area of stability than a liquid at high $\\rm Ka$ (e.g. Liquid Zinc, $\\rm Ka$ = 11525). Surface tension has both a stabilizing and a destabilizing effect, especially for large $\\rm Ka$. For long waves, it curves the vorticity lines near the substrate, reducing the flow under the crests. For short waves, it fosters vorticity production at the interface and creates a region of intense vorticity near the substrate. In addition, we discovered that the surface tension contributes to both the production and dissipation of perturbation's energy depending on the $\\rm Ka$ number. In terms of absolute/convective threshold, we found a window of absolute instability in the $\\text{Re}-\\hat{h}$ space, showing that the Landau-Levich-Derjaguin solution ($\\hat{h}=0.945 \\text{Re}^{1/9}\\text{Ka}^{-1/6}$) is always convectively unstable. Moreover, we show that for $\\text{Ka}<17$, the Derjaguin's solution ($\\hat{h}=1$) is always convectively unstable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-13T11:50:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear stability analysis",
      "vertical liquid film",
      "moving substrate",
      "instability mechanism",
      "fosters vorticity production"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}