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Multiphase Flows with Moving Interfaces and Contact Line—Balance Laws

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Abstract

A general continuum description for thermodynamic immiscible multiphase flows is presented with intersecting dividing surfaces, and three-phase common contact line, taking the contribution of the excess surface and line thermodynamic quantities into account. Starting with the standard postulates of continuum mechanics and the general global balance statement for an arbitrary physical quantity in a physical domain of three bulk phases including singular material or nonmaterial phase interfaces and a three-phase contact line, the local conservation equations on the phase interfaces and at the contact line are derived, in addition to the classical local balance equations for each bulk phase. Then, these general additional interface and line balance laws are specified for excess surface and line physical quantities, e.g., excess mass, momentum, angular momentum, energy, and entropy, respectively. Some simplified forms of these balance laws are also presented and discussed. In particular, for the massless phase interfaces and contact line, these balance laws reduce to the well-known jump conditions.

This chapter heavily draws from Wang and Oberlack (2011) [56].

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Notes

  1. 1.

    The term epitaxy refers to the growth of a crystalline layer above (epi) the surface of a crystalline substrate, whose crystallographic orientation imposes a crystalline order (taxis) onto the thin film.

  2. 2.

    It is tacitly assumed that the flux depends only on the outer normal vector at the surface points and not on differential geometric properties of the surface such as mean or Gauss ian curvature. This assumption has first been spelled out by Cauchy and is referred to as the Cauchy assumption.

  3. 3.

    The derivation of the result (27.11) requires a number of clarifying remarks to put it into the proper perspectives. The result was obtained by applying the balance law for a physical quantity to a material tetrahedronal volume with sharp edges and points for which unit normal vectors cannot uniquely be defined. Thus, to make the above argument mathematically “clean” the tetrahedron must be smoothed out such that the edges and vertices become “diffuse” with uniquely defined normal vectors. This smoothing can formally be done and the limit to a tetrahedron with sharp edges and vertices can be performed such that in the limit (27.11) is obtained. The assumption that is needed is that the flux \(\varvec{\phi }\) is nontrivially defined with respect to surface measure, but that there is no specific flux quantity defined along the edges or at the vertices.

    A historical account on Cauchy ’s Lemma is given by Truesdell and Toupin [52], Sect. 203, more general treatments, in which nontrivial edge fluxes and vertex fluxes are allowed for, are given by Noll [32], Noll and Virga [33] and Dell’Isola and Seppecher [9].

  4. 4.

    For a brief biography of Osborne Reynolds (1842–1912), see Vol. 2 of this treatise [25], Fig. 15.2 on p. 230.

  5. 5.

    For brief biographical sketches of Jean Frédéric Frenet (1816–1900) and Joseph Alfred Serret (1819–1885), see Fig.  27.4 .

  6. 6.

    For a brief biographical sketch of Carlo Giuseppe Matteo Marangoni (1840–1925), see Fig.  27.6 .

  7. 7.

    For a brief biographical sketch of Irving Langmuir (1881–1957), see Fig.  27.7 .

  8. 8.

    For a brief biographical sketch of Thomas Young (1773–1829), see Fig.  27.8 .

  9. 9.

    The derivation of these expressions is not difficult as it follows the procedure already used in simpler situations, but it is a bit long and tedious.

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Authors and Affiliations

  1. c/o Laboratory of Hydraulics, Hydrology, Glaciology, ETH Zürich, Zürich, Switzerland

    Kolumban Hutter

  2. Department of Mechanical Engineering, Darmstadt University of Technology, Darmstadt, Hessen, Germany

    Yongqi Wang

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  1. Kolumban Hutter
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Correspondence to Kolumban Hutter .

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Hutter, K., Wang, Y. (2018). Multiphase Flows with Moving Interfaces and Contact Line—Balance Laws. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77745-0_27

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