**1. Introduction**

58 Mass Transfer - Advanced Aspects

The procedure developed here for calculating the drift is especially important for applications associated with phenomena on the free surface of a liquid with the participation of viscous stresses. In particular, the approach proposed here makes it possible to calculate analytically the velocity of surface drift caused by wave perturbation for various surface substances (surface charge, surface-active substances, etc.) distributed over the free surface. If there is a total motion of the upper liquid along the interface between liquids (like wind along surface of ocean) then is interesting to study influence of the total upper liquid velocity on the arising of drift flow in lower liquid. Especially interesting if total upper liquid velocity is sufficient to excite an oscillatory instability which in the case of ideal liquids is known as Kelvin-Helmholtz instability. For all cases the qualitative preliminary analysis is helpful which is based on what the velocity of drift caused by the wave propagation is proportional to frequency of the wave motion and hence supporting information about behavior of the

Baily A.G. Electrostatic atomization of liquids (rev.)// Sci. Prog., Oxf. 1974. V.61. P. 555-581. Belonozhko D.F. & Grigor'ev A.I. (2003) Finite-amplitude waves on the surface of a viscous

Belonozhko D.F. & Grigor'ev A.I. (2004) Nonlinear periodic waves on the charged surface of a deep low-viscosity conducting liquid// *Tech. Phys.* Vol.49. N.3. pp.5-13. Belonozhko D.F. & Grigor'ev A.I. (2004) Nonlinear periodic waves on the charged surfactantcovered surface of a viscous fluid//*Tech. Phys.* Vol.49. N.11. pp.1422-1430*.*  Belonozhko D.F. & Grigor'ev A.I. (2008) Thickness of a boundary layer attributed to the

Belonozhko D.F., Shiryaeva S.O. & Grigor'ev A.I. (2005) Nonlinear periodic waves on the

Le Méhauté B. (1976) An Introduction to Hydrodynamics and Water Waves, Springer -

Longuet-Higgens M.S. Mass transport in water waves//*Phil. Trans. Roy. Soc. London. Ser. A.* 

Longuet-Higgens M.S. *(1986)* Eulerian and Lagragian aspects of surface waves //*J. Fluid* 

Melcher J.R. Field-coupled surface waves. A comparative study of surface coupled

Polyanin A.D. (2002), HandBook of Linear Differential Equations for Engineers and Scientist. Stokes G.G. (1847). On the theory of oscillatory waves *Transactions of the Cambridge* 

Stokes G.G. (1880). *Mathematical and Physical Papers*, Vol. I. Cambridge University Press.

Taylor G.I. & McEwan A.D. The stability of horizontal fluid interface in a vertical electric

electrohydrodynemics and magnetohydrodynemics systems. Cambridge. 1963. 190 p.

Faber T. E. Fluid Dynamics for Physicists, Cambridge University Press, 1997. 440 p. Le Blon P. H. & Mysak L. A. (1978) Waves in Ocean. *Amsterdam: Elsevier.* 602 p.

Lokenath D. (1994.) Nonlinear water waves. San Diego. Academiv Press. Inc. 544 p.

wave motion on the charged free surface of a viscous liquid // *Tech. Phys.* Vol.53.

charged surface of a viscous finite-conductivity fluid//*Tech. Phys.* Vol.50. N.2.

drift is contained in dispersion equation of the analyzed problem.

deep liquid// *Tech. Phys.* Vol.48. N.4. pp.404-414.

**4. References** 

N.3. pp. 306-313.

pp.177-184.

Verlag, 322 p.

pp. 197–229.

1953. V.245. N 903. pp.535-581*.*

*Philosophical Society* Vol.8. pp. 441–455.

field//*J. Fluid Mech*. 1965. V.22. N 1. pp. 1-15

*Mech.* Vol.173. pp.683-707.

At a gas–liquid interface, many complicated phenomena such as evaporation, condensation, electrokinesis, and heat and mass transfer occur. These phenomena are widely seen in various industrial and chemical systems. In chemical or biochemical reactive operations, bubble columns are used for increasing the mass transfer through the interface and for enhancing the separation of mixtures by rectification and water purification (Hong and Brauer 1989; Álvarez et al. 2000). However, the interfacial phenomena have various time and space scales (multi-scale) that are interrelated at the interface. Therefore, modeling gas– liquid interfaces over a wide range of scales spanning molecular motion to vortical fluid motion is very difficult, and this has remained one of the key unresolved issues in multiphase flow science and engineering since a long time. In particular, the mechanism for bubble coalescence/repulsion behaviour is unknown, although it is a superficially simple behaviour and fundamental phenomena in bubbly flows. In order to evaluate the interfacial interactions such as bubble coalescence and repulsion quantitatively, we need a new gas– liquid interfacial model based on the multi-scale concept which is expressed mathematically and that takes into account physical and chemical phenomena and heat and mass transfer at the interface.

In the theoretical point of view, the interfacial equation for a macroscopic-scale gas–liquid interface is mainly characterized by a jump condition. The macroscopic interface is discontinuous, and its physical properties such as density, viscosity, and temperature have discontinuous values. The jump condition has been discussed in terms of the mechanical energy balance (Scriven, 1960; Delhaye, 1974) using Stokes' theorem, the Gauss divergence theorem, differential geometry and so on. In these theorems, a test volume is considered at the interface between two continuous phases. In the derivation, the surface force acting on the discontinuous interface is modeled using the Young–Laplace equation. However, in such a mechanical approach, the definition of the curvature is unclear at the interface, and the surface tension coefficient is treated as a macroscopic experimental value. The interfacial model, which is based only on the mechanical energy balance, cannot take into account detailed physical and chemical phenomena occurring at the interface. In particular, the contamination at the interface, which is related to electric charges, is important for an

Macroscopic Gas-Liquid Interfacial Equation

Here,

NS equation is expressed as

Based on Thermodynamic and Mathematical Approaches 61

(mesoscopic interface). The concept of the gas–liquid interface is shown in Fig. 1. Experimental observations of bubble interactions (Craig, 2004; Henry & Craig, 2008) revealed that contamination at the interface may be an important factor. With this in mind, the contamination at the interface was associated with an electrostatic potential due to an electric double layer at the gas–liquid interface, and the free-energy equation, including the

( ) d <sup>2</sup>

 ψ

(1)

*cz eV* (2)

<sup>−</sup> <sup>=</sup> (3)

*<sup>l</sup>* <sup>=</sup> (4)

*<sup>l</sup>* <sup>=</sup> (5)

*<sup>l</sup>* <sup>=</sup> (6)

*g* (7)

ψ

[-] are

 ψ *<sup>V</sup>* ⎡ ⎤ = + ⎢ ⎥ ⎣ ⎦ <sup>∫</sup> ∇

2 4 <sup>0</sup> ( ) 2 4 *i e*

> 3 *U kT sum B* <sup>4</sup> *<sup>a</sup> <sup>l</sup>*

> > 3 16 3 *<sup>B</sup> k T <sup>b</sup>*

> > > 3 1 2 *c*

3 2 *Usum <sup>d</sup>*

The first and the second terms on the right-hand side of Eq. (1) denote the free energy in homogeneous and inhomogeneous systems, respectively. Coefficients *a* [J/m3], *b* [J/m3], *c*  [1/m3], and *d* [J/m] in Eqs. (1) and (2) have constant values. In Eqs. (3) and (6), *Usum* is related to an intermolecular potential. These values include microscopic physical

the charge number, elementary charge, electrostatic potential, and order parameter, respectively. The third term on the right-hand side of Eq. (2) is related to the contamination at the interface. In general, the surface tension is evaluated by the surface tension coefficient and the curvature characterizing the macroscopic shape of the interface. As mentioned earlier, in our model, the interface has a finite thickness similar to that of a fluid membrane (Fig. 1). For an interface with finite thickness, it is very difficult to determine the geometric shape of the interface. Therefore, the surface tension should be considered from the free energy at the interface rather than from the curvature (Fialkowski et at., 2000; Yonemoto et al., 2005). Moreover, there are many physical and chemical processes at the interface that are characterized by various time and space scales. Therefore, in order to consider the various scale interactions that may arise among interfacial phenomena, the Chapman–Enskog expansion (Chapman & Cowling, 1970) was applied to the NS equation. Here, the original

information (Yonemoto & Kunugi, 2010a). The symbols of *z*i [-], *e* [C], *V*e [V], and

D D *t* ρ

*u*

 =− ⋅ + ∇ Τ ρ

electrostatic potential, was then derived from a lattice-gas model (Safran, 1994):

0( ) <sup>2</sup> *<sup>V</sup> <sup>d</sup> F f* ψ

*a b <sup>f</sup>*

 =− + − ψψ

ψ

interfacial interaction. Craig (2004) and Henry and Craig (2008) have discussed the effects of specific ions on bubble interactions. They have reported that bubble coalescence is affected by the ions adsorbed at the interface, combination of these ions, and the electrolyte concentration. Others have similarly reported the importance of electrolytes in bubble coalescence (Marčelja, 2004; Ribeiro & Mewes, 2007; Tsang et al., 2004; Lessaard & Zieminski, 1971). In our experimental research on microbubble flow (Yonemoto et al., 2008), some patterns have been observed with respect to microbubble coalescence. Microbubble coalescence has been estimated analytically using the film thinning theory. Results have shown that microbubble coalescence cannot be explained based only on hydrodynamics. That is to say, our results indicate the importance of mass transfer, which is related to contamination at the interface, for interfacial interaction.

Recently, the phase field theory (Cahn & Hilliard, 1958) and van der Waals theory (Rowlinson & Widom, 1984), wherein the interface is assumed to be a diffuse interface with a finite thickness, have been applied to perform numerical research on multiphase flow. The surface force is evaluated by the free energy defined at the interface and depends on both concentration (or density) and its gradient. Anderson et al. (1998) have reviewed the diffuseinterface models of hydrodynamics and their application to various interfacial phenomena. In their study, the diffuse-interface model was associated with the sharp-interface model (Delhaye, 1974). But the multi-scale concept was not expressed in concrete terms.

In a previous study (Yonemoto & Kunugi, 2010a), a thermodynamic and mathematical interfacial model that takes into account the multi-scale concept has been developed on the basis of the phase field theory (Cahn & Hilliard, 1958). In this model, we assumed that the interface has a finite thickness and that free energy is defined at the interface. In particular, the free energy is derived on a microscopic scale (Hamiltonian); this includes the electrostatic potential due to contamination at the interface. The free energy is incorporated into the Navier–Stokes (NS) equation by using the Chapman–Enskog expansion (Chapman & Cowling, 1970), which mathematically discriminates the time and space scales of the interfacial phenomena. Finally, a new equation governing the fluid motion, called the multiscale multiphase flow equation, is derived. The multi-scale multiphase flow equation has been proven to have the potential to simulate interfacial interactions (Yonemoto & Kunugi, 2010b). In the simulation, microbubble interaction is simulated and a liquid film between them is observed when the bubbles interact with each other. In the present study, the multiscale multiphase flow equation, which is the mesoscopic interfacial equation, was further discussed, and a macroscopic interfacial equation was derived based on our interfacial model. In particular, an interfacial jump condition treated by thermodynamics was derived from the multi-scale multiphase flow equation, and the thermodynamic interfacial jump condition was then compared with the conventional jump condition. In addition, we derived the Kelvin equation based on both the multi-scale multiphase flow equation (Yonemoto & Kunugi, 2010a) and the thermodynamic jump condition. The present results indicate that our interfacial model can theoretically support various interfacial phenomena characterized by thermodynamics from a multi-scale viewpoint (micro to macro).

#### **2. Multi-scale multiphase flow equation**

In a previous study (Yonemoto & Kunugi, 2010a), the multi-scale multiphase flow equation was derived based on the phase field theory (Cahn & Hilliard, 1958). In the derivation, we assumed that the interface has a finite thickness similar to that of a fluid membrane (mesoscopic interface). The concept of the gas–liquid interface is shown in Fig. 1. Experimental observations of bubble interactions (Craig, 2004; Henry & Craig, 2008) revealed that contamination at the interface may be an important factor. With this in mind, the contamination at the interface was associated with an electrostatic potential due to an electric double layer at the gas–liquid interface, and the free-energy equation, including the electrostatic potential, was then derived from a lattice-gas model (Safran, 1994):

$$F = \int\_{V} \left[ f\_0(\boldsymbol{\nu}) + \frac{d}{2} (\boldsymbol{\nabla} \boldsymbol{\nu})^2 \right] \, \mathrm{d}V \tag{1}$$

Here,

60 Mass Transfer - Advanced Aspects

interfacial interaction. Craig (2004) and Henry and Craig (2008) have discussed the effects of specific ions on bubble interactions. They have reported that bubble coalescence is affected by the ions adsorbed at the interface, combination of these ions, and the electrolyte concentration. Others have similarly reported the importance of electrolytes in bubble coalescence (Marčelja, 2004; Ribeiro & Mewes, 2007; Tsang et al., 2004; Lessaard & Zieminski, 1971). In our experimental research on microbubble flow (Yonemoto et al., 2008), some patterns have been observed with respect to microbubble coalescence. Microbubble coalescence has been estimated analytically using the film thinning theory. Results have shown that microbubble coalescence cannot be explained based only on hydrodynamics. That is to say, our results indicate the importance of mass transfer, which is related to

Recently, the phase field theory (Cahn & Hilliard, 1958) and van der Waals theory (Rowlinson & Widom, 1984), wherein the interface is assumed to be a diffuse interface with a finite thickness, have been applied to perform numerical research on multiphase flow. The surface force is evaluated by the free energy defined at the interface and depends on both concentration (or density) and its gradient. Anderson et al. (1998) have reviewed the diffuseinterface models of hydrodynamics and their application to various interfacial phenomena. In their study, the diffuse-interface model was associated with the sharp-interface model

In a previous study (Yonemoto & Kunugi, 2010a), a thermodynamic and mathematical interfacial model that takes into account the multi-scale concept has been developed on the basis of the phase field theory (Cahn & Hilliard, 1958). In this model, we assumed that the interface has a finite thickness and that free energy is defined at the interface. In particular, the free energy is derived on a microscopic scale (Hamiltonian); this includes the electrostatic potential due to contamination at the interface. The free energy is incorporated into the Navier–Stokes (NS) equation by using the Chapman–Enskog expansion (Chapman & Cowling, 1970), which mathematically discriminates the time and space scales of the interfacial phenomena. Finally, a new equation governing the fluid motion, called the multiscale multiphase flow equation, is derived. The multi-scale multiphase flow equation has been proven to have the potential to simulate interfacial interactions (Yonemoto & Kunugi, 2010b). In the simulation, microbubble interaction is simulated and a liquid film between them is observed when the bubbles interact with each other. In the present study, the multiscale multiphase flow equation, which is the mesoscopic interfacial equation, was further discussed, and a macroscopic interfacial equation was derived based on our interfacial model. In particular, an interfacial jump condition treated by thermodynamics was derived from the multi-scale multiphase flow equation, and the thermodynamic interfacial jump condition was then compared with the conventional jump condition. In addition, we derived the Kelvin equation based on both the multi-scale multiphase flow equation (Yonemoto & Kunugi, 2010a) and the thermodynamic jump condition. The present results indicate that our interfacial model can theoretically support various interfacial phenomena

(Delhaye, 1974). But the multi-scale concept was not expressed in concrete terms.

characterized by thermodynamics from a multi-scale viewpoint (micro to macro).

In a previous study (Yonemoto & Kunugi, 2010a), the multi-scale multiphase flow equation was derived based on the phase field theory (Cahn & Hilliard, 1958). In the derivation, we assumed that the interface has a finite thickness similar to that of a fluid membrane

**2. Multi-scale multiphase flow equation** 

contamination at the interface, for interfacial interaction.

$$f\_0(\varphi) = -\frac{a}{2}\varphi^{\*^2} + \frac{b}{4}\varphi^{\*^4} - c\varepsilon\_i e V\_i \varphi \tag{2}$$

$$a = \frac{U\_{sau} - 4k\_B T}{l^3} \tag{3}$$

$$b = \frac{16k\_B T}{3l^3} \tag{4}$$

$$c = \frac{1}{2l^3} \tag{5}$$

$$d = \frac{U\_{sun}}{2l^3} \tag{6}$$

The first and the second terms on the right-hand side of Eq. (1) denote the free energy in homogeneous and inhomogeneous systems, respectively. Coefficients *a* [J/m3], *b* [J/m3], *c*  [1/m3], and *d* [J/m] in Eqs. (1) and (2) have constant values. In Eqs. (3) and (6), *Usum* is related to an intermolecular potential. These values include microscopic physical information (Yonemoto & Kunugi, 2010a). The symbols of *z*i [-], *e* [C], *V*e [V], and ψ [-] are the charge number, elementary charge, electrostatic potential, and order parameter, respectively. The third term on the right-hand side of Eq. (2) is related to the contamination at the interface. In general, the surface tension is evaluated by the surface tension coefficient and the curvature characterizing the macroscopic shape of the interface. As mentioned earlier, in our model, the interface has a finite thickness similar to that of a fluid membrane (Fig. 1). For an interface with finite thickness, it is very difficult to determine the geometric shape of the interface. Therefore, the surface tension should be considered from the free energy at the interface rather than from the curvature (Fialkowski et at., 2000; Yonemoto et al., 2005). Moreover, there are many physical and chemical processes at the interface that are characterized by various time and space scales. Therefore, in order to consider the various scale interactions that may arise among interfacial phenomena, the Chapman–Enskog expansion (Chapman & Cowling, 1970) was applied to the NS equation. Here, the original NS equation is expressed as

$$
\rho \frac{\text{D } \text{u}}{\text{D } t} = -\nabla \cdot \mathbf{T} + \rho \mathbf{g} \tag{7}
$$

Macroscopic Gas-Liquid Interfacial Equation

**3.1 Momentum jump condition** 

Fig. 2:

κ1 and κ

**3. Jump condition at gas–liquid interface** 

Based on Thermodynamic and Mathematical Approaches 63

An interfacial phenomenon is complex and interpreted as a discontinuous problem. The interface separates two continuous equilibrium phases. When the curvature radius is considerably larger than the thickness of the interface, the equilibrium force balance at the interface is given by the following equation based on the interfacial coordinates shown in

> <sup>d</sup> () () 2 0 <sup>d</sup> *P PH <sup>s</sup>*

> > *t*

τ σ

[N/m], *Pk* (k = G, L) [N/m2],

τ

, where

[N/m2] are

Μ

κ + κ

(11)

σ

*n*L

*n*G

σ

G L GG G G LL L L { } { } <sup>G</sup>

τ

*Liquid phase* 

*Gas phase* 

*Gas phase* 

Fig. 2. Interfacial coordinates at the gas–liquid interface

momentum jump condition at the interface. The symbols

**3.2 Derivation of thermodynamic jump condition** 

concept (Yonemoto & Kunugi, 2010a).

*Liquid phase* 

*Interface* 

*MM n n n n n t* + −− + ⋅ −− + ⋅ − − =

*s Interface* 

where the subscripts L and G represent liquid and gas phases, respectively. This is called the

and *s* [m] are the surface tension coefficient, pressure, and coordinate along the interface,

respectively. The mean curvature is denoted by the symbol *H*; here, 1 2 *H* = ( )/2

Therefore, we consider the jump condition to be a macroscopic interfacial equation.

<sup>2</sup> [1/m] are the principal curvatures. The bold symbols *n<sup>k</sup>* , *t* , and *<sup>k</sup>*

the unit normal, unit tangential vector, and shear stress, respectively. *Mk* [kg/ms] ( *<sup>k</sup>*

[N/m2] is the time derivative of *M<sup>k</sup>* ) denotes the term related to mass transfer through the interface. In this paper, we call this equation the conventional jump condition. The jump condition at the interface is characterized by the curvature related to the shape of the interface, which means that the interface is a mathematical interface with zero thickness.

Here, the interfacial jump condition treated by thermodynamics is derived using the multi-scale multiphase flow equation (Eq. (10)). We call this derived condition the "thermodynamic jump condition" because Equation (10) is based on the thermodynamic

where ρ [kg/m3], *u* [m/s], *T* [N/m2], *g* [m/s2], and *t* [s] represent the fluid density, velocity, stress tensor, acceleration due to gravity, and time, respectively.

Fig. 1. Concept of a gas–liquid interface: macroscopic and mesoscopic image of the interface

Stress tensor *T* [N/m3] is defined as *T I* = *P* −τ . The shear stress is τ [N/m2]. Pressure *P* [N/m2] is the mechanical pressure, hereafter represented by *P*mech . At this point, the operators D/D*t* and ∇ must include the various time and space scales (multi-scale concept). Therefore, in order to discriminate their scales, the Chapman–Enskog expansion was applied to D/D*t* and ∇ in the NS equation. The operators D/D*t* and ∇ were decomposed into the following expressions by using the small parameter ε:

$$\mathbf{V} = \mathbf{V}^{(0)} + \varepsilon \mathbf{V}^{(1)} + \varepsilon^2 \mathbf{V}^{(2)} + \cdots + \varepsilon^k \mathbf{V}^{(k)} + \cdots \tag{8}$$

$$\frac{\mathbf{D}}{\mathbf{D}} = \frac{\mathbf{D}}{\mathbf{D}} + \varepsilon \frac{\mathbf{D}}{\mathbf{D}} + \varepsilon^2 \frac{\mathbf{D}}{\mathbf{D}} + \dots + \varepsilon^k \frac{\mathbf{D}}{\mathbf{D}} + \dots + \varepsilon^k \frac{\mathbf{D}}{\mathbf{D}} + \dots \tag{9}$$

Here, superscript *k* (*k* = 0, 1, 2...) represents the scale of the phenomena, which becomes smaller as (*k*) increases. For example, the superscript (0) corresponds to the macroscopic scale. In Eqs. (8) and (9), the small parameter ε is defined as ε = δ / *L* . The symbols of δ [m] and *L* [m] represent the characteristic lengths of the interface and the vortical fluid flow, respectively. After substituting Eqs. (8) and (9) into Eq. (7) and performing a simple tensor analysis, we obtain a new governing equation:

$$
\varepsilon \rho \frac{\text{D } \mathfrak{u}}{\text{D } t^{(0)}} + \varepsilon \rho \frac{\text{D } \mathfrak{u}'}{\text{D } t^{(l)}} = -\nabla^{(0)} \cdot \mathbf{T} + \varepsilon \nabla^{(l)} \cdot \left( f\_0(\nu) I \right) - \varepsilon d \nabla^{(l)} \psi \left( \nabla^{(l)} \cdot \mathbf{V}^{(l)} \nu \right) + \rho \mathbf{g} \tag{10}
$$

In the derivation, the free energy (Eq. (1)) is associated with thermodynamic pressure using the Maxwell relation. This equation is the multi-scale multiphase flow equation (Yonemoto & Kunugi, 2010a).
