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Multiphase Flows with Moving Interfaces and Contact Line—Constitutive Modeling

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Abstract

A thermodynamic analysis, based on the müller–Liu thermodynamic approach of the second law of thermodynamics, is performed to derive the expressions of the constitutive variables in thermodynamic equilibrium. Non-equilibrium responses are proposed by use of a quasi-linear theory. A set of constitutive equations for the surface and line constitutive quantities is postulated. Some restrictions for the emerging material parameters are derived by means of the minimum conditions of the surface and line entropy productions in thermodynamic equilibrium. Hence, a complete continuum mechanical model to describe excess surface and line physical quantities is formulated. Technically, in the exploitation of the entropy inequality, all field equations are incorporated with Lagrange parameters into the entropy inequality. In the process of its exploitation, the Lagrange parameter of the energy balance is identified with the inverse of the absolute temperature in the bulk, the phase interface, and in the three-phase contact line. Interesting results, among many others, are the Gibbs relations, which are formally the same in the bulk, on the interface and along the contact line, with the pressure in the compressible bulk replaced by the surface tension on the interface and by the line tension along the contact line, see (28.45), (28.87).

This chapter heavily draws from Wang, Oberlack and Zieleniewicz (2013) [40].

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Notes

  1. 1.

    Additional symbols are the same as in Chap. 27.

  2. 2.

    This argument must be applied with caution: If \({\varvec{\alpha }}^{({\mathfrak {s}}_{i})}\) has components which are symmetric, \(\alpha _{k \ell }^{({\mathfrak {s}}_{i})} = \alpha _{\ell k}^{({\mathfrak {s}}_{i})}\), then the corresponding component of \({\varvec{a}}^{({\mathfrak {s}}_{i})}\) must be skew-symmetric, \(a_{k \ell }^{({\mathfrak {s}}_{i})} = -a_{\ell k}^{({\mathfrak {s}}_{i})}\).

  3. 3.

    The surfacial Gibbs equation, (28.45), can be easily derived. By means of (28.21)\(_2\) and (28.44)\(_1\), one obtains

    $$\begin{aligned}\eta ^{({\mathfrak {s}}_i)}=\hat{\eta }^{({\mathfrak {s}}_i)}\left( \rho ^{({\mathfrak {s}}_i)},\varTheta ^{({\mathfrak {s}}_i)}\right) = \frac{1}{\varTheta ^{({\mathfrak {s}}_i)}}\left( u^{({\mathfrak {s}}_i)}-\psi ^{({\mathfrak {s}}_i)}\right) . \end{aligned}$$

    Taking the total differential of \(\eta ^{({\mathfrak {s}}_i)}\) yields

    $$\begin{aligned}\mathrm {d}\eta ^{({\mathfrak {s}}_i)}= & {} -\frac{1}{(\varTheta ^{({\mathfrak {s}}_i)})^2}\left( u^{({\mathfrak {s}}_i)}-\psi ^{({\mathfrak {s}}_i)}\right) \mathrm {d}\varTheta ^{({\mathfrak {s}}_i)} + \frac{1}{\varTheta ^{({\mathfrak {s}}_i)}}\left( \mathrm {d}u^{({\mathfrak {s}}_i)}-\mathrm {d}\psi ^{({\mathfrak {s}}_i)}\right) \\= & {} -\frac{1}{(\varTheta ^{({\mathfrak {s}}_i)})^2}\left( u^{({\mathfrak {s}}_i)}-\psi ^{({\mathfrak {s}}_i)}\right) \mathrm {d}\varTheta ^{({\mathfrak {s}}_i)} \\&+ \frac{1}{\varTheta ^{({\mathfrak {s}}_i)}}\left( \mathrm {d}u^{({\mathfrak {s}}_i)}-\frac{\partial \psi ^{({\mathfrak {s}}_i)}}{\partial \rho ^{({\mathfrak {s}}_i)}}\mathrm {d}\rho ^{({\mathfrak {s}}_i)}-\frac{\partial \psi ^{({\mathfrak {s}}_i)}}{\partial \varTheta ^{({\mathfrak {s}}_i)}}\mathrm {d}\varTheta ^{({\mathfrak {s}}_i)}\right) \\= & {} -\frac{1}{(\varTheta ^{({\mathfrak {s}}_i)})^2}\left( u^{({\mathfrak {s}}_i)}-\psi ^{({\mathfrak {s}}_i)}+\varTheta ^{({\mathfrak {s}}_i)} \frac{\partial \psi ^{({\mathfrak {s}}_i)}}{\partial \varTheta ^{({\mathfrak {s}}_i)}} \right) \mathrm {d}\varTheta ^{({\mathfrak {s}}_i)} \\&+ \frac{1}{\varTheta ^{({\mathfrak {s}}_i)}} \mathrm {d}u^{({\mathfrak {s}}_i)}+ \frac{1}{\varTheta ^{({\mathfrak {s}}_i)}}\left( (\rho ^{({\mathfrak {s}}_i)})^2\frac{\partial \psi ^{({\mathfrak {s}}_i)}}{\partial \rho ^{({\mathfrak {s}}_i)}}\right) \mathrm {d}\left( \frac{1}{\rho ^{({\mathfrak {s}}_i)}}\right) \\= & {} \frac{1}{\varTheta ^{({\mathfrak {s}}_i)}}\left( \mathrm {d} u^{({\mathfrak {s}}_i)}-\sigma ^{({\mathfrak {s}}_i)}\mathrm {d}\left( \frac{1}{\rho ^{({\mathfrak {s}}_i)}}\right) \right) . \end{aligned}$$

    In the last step, relations (28.21)\(_2\), (28.35), and (28.43) have been used.

  4. 4.

    The derivations of the first conditions, (28.68) and (28.69)\(_1\), are straight. However, the derivations of (28.69)\(_2\) and (28.70) are somewhat tedious and complicated by choosing various special functions for \(\varvec{f}\). Here, we refrain from providing more details.

  5. 5.

    The Gibbs equation for the contact line, (28.87), can also be proved as demonstrated for the surfacial Gibbs equation, (28.45) in footnote 3 on page 429.

  6. 6.

    Here, the material historic dependence on the material time derivative \(\dot{\varvec{D}}^{({\mathfrak s}_i)}\) does not satisfy the requirement of material objectivity. A more reasonable variable describing the dependence of the materially deformational history may be the material objective time derivatives of \({\varvec{D}} ^{({\mathfrak s}_i)}\), e.g., the upper-convected time derivative (or Oldroyd derivative) \({\mathop {\mathbf {D}}\limits ^{\triangledown }}\,\!\! ^{({\mathfrak s}_i)}\) , the lower-convected time derivative \({\mathop {\mathbf {D}}\limits ^{\vartriangle }}\,\!\! ^{({\mathfrak s}_i)}\) or Jaumann time derivatives \({\mathop {\mathbf {D}}\limits ^{\circ }}\,\!\! ^{({\mathfrak s}_i)}\), respectively, defined by

    $$\begin{aligned} \text {Upper-convected time derivative}&{\mathop {\mathbf {D}}\limits ^{\triangledown }}\,\!\! ^{({\mathfrak s}_i)}=\dot{\mathbf {D}}^{({\mathfrak s}_i)}- \mathbf {L}^{({\mathfrak s}_i)}\mathbf {D}^{({\mathfrak s}_i)} - \mathbf {D}^{({\mathfrak s}_i)}(\mathbf {L}^{({\mathfrak s}_i)})^T,\\ \text {Lower-convected time derivative}&{\mathop {\mathbf {D}}\limits ^{\vartriangle }}\,\!\! ^{({\mathfrak s}_i)}=\dot{\mathbf {D}}^{({\mathfrak s}_i)}+ (\mathbf {L}^{({\mathfrak s}_i)})^T\mathbf {D}^{({\mathfrak s}_i)} + \mathbf {D}^{({\mathfrak s}_i)}\mathbf {L}^{({\mathfrak s}_i)},\\ {\textsc {Jaumann}}\;\text {time derivative}&{\mathop {\mathbf {D}}\limits ^{\circ }}\,\!\! ^{({\mathfrak s}_i)}=\dot{\mathbf {D}}^{({\mathfrak s}_i)}- \mathbf {W}^{({\mathfrak s}_i)}\mathbf {D}^{({\mathfrak s}_i)} + \mathbf {D}^{({\mathfrak s}_i)}\mathbf {W}^{({\mathfrak s}_i)}, \end{aligned}$$

    where \(\mathbf {L}^{({\mathfrak s}_i)}\) is the velocity gradient, \(\mathbf {D}^{({\mathfrak s}_i)}\) the rate of deformation tensor and \(\mathbf {W}^{({\mathfrak s}_i)}\) is the spin tensor. If we would employ these materially objective time derivatives, the following evaluation of the entropy principle would become much more complicated, if they would be still achievable by hand. Here, we refrain doing this.

  7. 7.

    These derivations are fairly tedious and complicated. Here, we refrain from providing more details.

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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—Constitutive Modeling. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77745-0_28

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