{
  "id": "2405.02874",
  "version": "v1",
  "published": "2024-05-05T10:08:23.000Z",
  "updated": "2024-05-05T10:08:23.000Z",
  "title": "Weakening the effect of boundaries: `diffusion-free' boundary conditions as a `do least harm' alternative to Neumann",
  "authors": [
    "Yufeng Lin",
    "Rich Kerswell"
  ],
  "comment": "17 pages, 9 figures and comments welcome",
  "categories": [
    "physics.flu-dyn"
  ],
  "abstract": "In this note, we discuss a poorly known alternative boundary condition to the usual Neumann or `stress-free' boundary condition typically used to weaken boundary layers when diffusion is present but very small. These `diffusion-free' boundary conditions were first developed (as far as the authors know) in 1995 (Sureshkumar & Beris, J. Non-Newtonian Fluid Mech., vol 60, 53-80, 1995) in viscoelastic flow modelling but are worthy of general consideration in other research areas. To illustrate their use, we solve two simple ODE problems and then treat a PDE problem - the inertial wave eigenvalue problem in a rotating cylinder, a sphere and spherical shell for small but non-zero Ekman number $E$. Where inviscid inertial waves exist (cylinder and sphere), the viscous flows in the Ekman boundary layer are $O(E^{1/2})$ weaker than for the corresponding stress-free layer and fully $O(E)$ weaker than in a non-slip layer. This reduction could allow precious numerical resources to focus on other areas of the flow and thereby make smaller, more realistic values of diffusion accessible to simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-05T10:08:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boundary condition",
      "diffusion-free",
      "inertial wave eigenvalue problem",
      "simple ode problems",
      "non-newtonian fluid mech"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}