{
  "annotation_id": "f6c828c7-ea58-4034-b045-6ea94d22255c",
  "date_created": "2026-03-02T18:01:14.579000Z",
  "date_modified": "2026-03-02T18:01:14.579000Z",
  "file_hash": "05f1f751eb550ad0d9bd9df6124587a7f698f86fe8d74a42e1acaf887a68ef84",
  "private": false,
  "record": {
    "abstract": "Brenner has recently proposed modifications to the Navier-Stokes equations\nthat are based on theoretical arguments but supported only by experiments\nhaving a fairly limited range. These modifications relate to a diffusion of\nfluid volume that would be significant for flows with high density gradients.\nSo the viscous structure of shock waves in gases should provide an excellent\ntest case for this new model. In this paper we detail the shock structure\nproblem and propose exponents for the gas viscosity-temperature relation based\non empirical viscosity data that is independent of shock experiments. We then\nsimulate shocks in the range Mach 1.0-12.0 using the Navier-Stokes equations,\nboth with and without Brenner\u0027s modifications. Initial simulations showed\nBrenner\u0027s modifications display unphysical behaviour when the coefficient of\nvolume diffusion exceeds the kinematic viscosity. Our subsequent analyses\nattribute this behaviour to both an instability to temporal disturbances and a\nspurious phase velocity-frequency relationship. On equating the volume\ndiffusivity to the kinematic viscosity, however, we find the results with\nBrenner\u0027s modifications are significantly better than those of the standard\nNavier-Stokes equations, and broadly similar to those from the family of\nextended hydrodynamic models that includes the Burnett equations. Brenner\u0027s\nmodifications add only two terms to the Navier-Stokes equations, and the\nnumerical implementation is much simpler than conventional extended\nhydrodynamic models, particularly in respect of boundary conditions. We\nrecommend further investigation and testing on a number of different benchmark\nnon-equilibrium flow cases.",
    "arxiv_id": "physics/0611275",
    "authors": [
      "Christopher J. Greenshields",
      "Jason M. Reese"
    ],
    "categories": [
      "physics.flu-dyn"
    ],
    "doi": "10.1017/S0022112007005575",
    "title": "The structure of shock waves as a test of Brenner\u0027s modifications to the   Navier-Stokes equations",
    "url": "https://arxiv.org/abs/physics/0611275"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "a06f4c9c-e11c-49b4-abe3-e6f6c6625910",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}