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Kinetic Boundary Condition at the Interface

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Vapor-Liquid Interfaces, Bubbles and Droplets

Part of the book series: Heat and Mass Transfer ((HMT))

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Abstract

The vapor–liquid interface can exist only where the bulk vapor phase and the bulk liquid phase of the same molecules coexist side by side. Therefore, all the properties of the interface are inevitably affected by the bulk liquid and vapor phases, and vice versa. The relation among these three constituents still remains unresolved in general nonequilibrium states. However, at least in a weak nonequilibrium state, the relations can be simplified and reformulated into a form of Kinetic Boundary Condition (KBC) at the vapor–liquid interface. In this chapter, from the microscopic point of view, we explain how the two bulk phases of vapor and liquid are connected via the KBC at the interface. The main tools used here are the nonequilibrium molecular dynamics simulation of vapor–liquid twophase system and the Boltzmann equation for vapor. Our aims in this chapter are to establish the KBC at the interface by the molecular dynamics simulation and to reduce it into the boundary condition for the vapor flows in the fluid-dynamics region outside the Knudsen layer on the interface by the asymptotic analysis of the boundary-value problem of the Boltzmann equation for small Knudsen numbers.

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Notes

  1. 1.

    For simplicity, the fluid and the thermometer are assumed to be composed of monatomic molecules and at rest in the macroscopic sense. The formula \((m/2)\langle \xi_i^2 \rangle=kT/2\) \((i=x,y,z)\) is called the equipartition theorem or the law of equipartition of energy.

  2. 2.

    The sufficiently small volume in fluid dynamics is sufficiently large in molecular scales so that it may contain a number of molecules. Thus, the macroscopic variables defined by some kinds of averages can be regarded as continuous functions of the space coordinates and the time. If the fluid is an ideal gas in the standard state, the number of molecules in a cube with a side-length 1 ʼm is \(2.6867774\times10^7\). The number of molecules per unit volume is called the Loschmidt constant.

  3. 3.

    The actualization of local equilibrium requires a sufficient number of molecular interactions (intermolecular collisions).

  4. 4.

    Forces acting between atoms making up a molecule (covalent bonds, ionic bonds, and metallic bonds) are called the intramolecular forces.

  5. 5.

    The potential energy of the ith molecule is defined only formally. It is the total potential energy of all molecules that has the physical meaning.

  6. 6.

    In Chap. 5 and in Appendix B, the internal energy per unit mass of fluid is denoted by e.

  7. 7.

    The only requirement is that ρ and v j are continuously differentiable functions of x and t. This will be satisfied by choosing h in the function χ so that the cube with a side-length h centered at x contains a large number of molecules. If the fluid is a gas, this does not warrant the local equilibrium, because the mean free path of gas molecules can be very large compared with h. In the case that the fluid is a liquid, this may be a sufficient condition for the local equilibrium, because the mean free path of liquid molecules is usually comparable with or less than a typical diameter of a molecule.

  8. 8.

    The factor \(\frac{1}{3}\) in Eq. (2.34) is an ideal limit of \(N\to\infty\) in equilibrium states. The stress tensor in the original paper [15] is different from this and much more cumbersome.

  9. 9.

    The macroscopic properties in equilibrium states should not be different by the difference of simulation methods.

  10. 10.

    For example, it is natural that large-scale fluid flows are affected by the gravity.

  11. 11.

    The triple point temperature is 83.8 K for argon [28].

  12. 12.

    It is easy to confirm that the probability density of numerically obtained \(\hat{\mathcal{H}}\) approaches a Gaussian with the increase in N.

  13. 13.

    The relations \(p=\rho R T\) and \(e=(3R/2)T\) do not imply that the gas is in a (local) equilibrium state. They are formal extensions to nonequilibrium states, as well as the definition of temperature T.

  14. 14.

    The inverse of the mean collision frequency is called the mean free time.

  15. 15.

    Precisely, the mean free path corresponds to \(1/(n_0d_m^2)\), where n 0 is a characteristic number density of gas molecules, as shown by Eq. (2.72).

  16. 16.

    The Boltzmann H-theorem corresponds to the entropy inequality extended to nonequilibrium states [35]. The theorem states that the H function or the integral \(\hbox{}\overline{H}\) of H function over a domain D,

    $$H=\int f\ln (f/c)\,{\mathrm{d}}{\boldsymbol{\xi}} \quad\text{or}\quad \overline{H}=\int_D H\, {\mathrm{d}}\textbf{\textit{x}},$$

    never increases by an inequality

    $$\frac{{\mathrm{d}} \overline{H}}{{\mathrm{d}} t}-\int_{\partial D}(H_i-Hv_{wi})n_i \,{\mathrm{d}} S =\int_D\left(\int[1+\ln (f/c)]J(f)\,{\mathrm{d}}{\boldsymbol{\xi}}\right)\,{\mathrm{d}} V\leqq 0,$$

    if \((H_i-Hv_{wi})n_i=0\) on the boundary ∂D, where c is a constant to make f/c dimensionless and

    $$H_i=\int \xi_i f\ln(f/c)\,{\mathrm{d}}{\boldsymbol{\xi}}.$$
  17. 17.

    The center of the interface x 0 is almost constant on the coordinate fixed to the interface in spite of the backward movement of interface due to evaporation.

  18. 18.

    In Eq. (2.81), we used the symbol T w for the temperature at the surface of the liquid phase.

  19. 19.

    The internal motions of a polyatomic molecule are the rotational and vibrational motions. Although the rotational motions are usually active at room temperature, the vibrational modes are activated at higher temperature. However, since n is constant in Eq. (2.88), the gas flows associated with the activation and deactivation of the vibrational modes cannot be treated by Eq. (2.88).

    The distribution of the energy of internal motion is discussed in Sect. 3.1.2. 19

  20. 20.

    In Ref. [39], the ghost effect [35] induced by the noncondensable gas is found. The ghost effect has first been found in Ref. [38], and means a finite effect produced by an infinitesimal quantity. For example, in Ref. [38], Sone et al. discussed the temperature field in the limit \(\textrm{Kn}\to0\) affected by the thermal creep flow that has already vanished in the limit \(\textrm{Kn}\to0\).

  21. 21.

    Actually, the equations governing the macroscopic quantities outside the Knudsen layer are not always equal to the set of Navier–Stokes equations and the set of Euler equations. Depending upon the macroscopic situation of the flow considered, the derived equations can contain some terms that are not included in the set of Navier–Stokes equations, and hence these equations are sometimes called the fluid-dynamics-type equations [35].

  22. 22.

    In Ref. [2], the Gaussian–BGK models for both cases of monatomic and polyatomic molecules are discussed. The Boltzmann H-theorem is proved for both cases.

  23. 23.

    See Footnote 16.

  24. 24.

    Sects. 2.5.2 and 2.5.3 are concerned with the time-independent problem. However, even in a time-dependent problem, if the characteristic time scale t 0 of the problem is large so that \(\ell_0/t_0\sqrt{2RT_0}=O(k)\), then the time derivative term in the Boltzmann equation drops in the Knudsen layer in the leading order of approximation, and hence the vapor flow in the Knudsen layer can be treated as a time-independent flow in the leading order of approximation, where the time variable t is included as a parameter.

  25. 25.

    The fluid-dynamics parts are unchanged in the Knudsen layer and the Knudsen-layer corrections rapidly vanish \(y\to\infty\). Therefore, u iS1, τ S1, and P S1 in Eq. (2.125) and Eqs. (2.127), (2.128), (2.129), (2.130), and (2.131) are, respectively, equal to the velocity, temperature, and pressure of the vapor at the outer edge of the Knudsen layer in the approximation of O(k).

  26. 26.

    These values have recently been corrected by M. Inaba as follows (private communication): \(C_4^{*}=-2.0719, d_4^{*}=-0.1921, \Omega_4^{*}(0)=0.5006, \Theta_{4tr}^{*}(0)=-0.1090, \textrm{ and } \Theta_{4int}^{*}(0)=0.1190\).

  27. 27.

    See Footnote 24.

  28. 28.

    The dimensionless mass flux \(\rho(v_i-v_{wi})n_i/(\rho^\ast{\sqrt{2RT_w}})\) is approximately equal to \((v_i-v_{wi})n_i/{\sqrt{2RT_w}}\) in the accuracy of O(k) since \((v_i-v_{wi})n_i/{\sqrt{2RT_w}}=O(k)\) and \(\rho/\rho^\ast=1+O(k)\).

  29. 29.

    Apparently, the right-hand side of Eq. (2.142) is equal to the ratio of the mass flux given by Eq. (2.82) to that in the case of complete-condensation condition given by Eq. (2.80).

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Authors and Affiliations

  1. Dept. Mechanical & Space Engineering, Hokkaido University, Kita 13, Nishi 8, Sapporo, 060-8628, Japan

    Prof. Shigeo Fujikawa & Prof. Masao Watanabe

  2. Dept. Mechanical Engineering, Osaka University, Yamada-oka 2-1, Suita, 565-0871, Japan

    Prof. Takeru Yano

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  1. Prof. Shigeo Fujikawa
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Correspondence to Shigeo Fujikawa .

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Fujikawa, S., Yano, T., Watanabe, M. (2011). Kinetic Boundary Condition at the Interface. In: Vapor-Liquid Interfaces, Bubbles and Droplets. Heat and Mass Transfer. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18038-5_2

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