**3.1 Momentum jump condition**

62 Mass Transfer - Advanced Aspects

velocity, stress tensor, acceleration due to gravity, and time, respectively.

*Gas phase* 

Stress tensor *T* [N/m3] is defined as *T I* = *P* −

∇

scale. In Eqs. (8) and (9), the small parameter

analysis, we obtain a new governing equation:

*t t*

ερ

ρ

& Kunugi, 2010a).

(0) (1) 0 D D ( ) D D *f d*

∇

∇

∇ =∇

decomposed into the following expressions by using the small parameter

εε

εε

ε

∇

∇

operators D/D*t* and

was applied to D/D*t* and

*Liquid phase* 

*Interface* 

[kg/m3], *u* [m/s], *T* [N/m2], *g* [m/s2], and *t* [s] represent the fluid density,

*Gas phase* 

2 *k k*

 ε

τ

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

*P* [N/m2] is the mechanical pressure, hereafter represented by *P*mech . At this point, the

concept). Therefore, in order to discriminate their scales, the Chapman–Enskog expansion

(0) (1) (2) ( )

∇

2 (0) (1) (2) ( )

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

[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

ε

*<sup>k</sup> tt t t t* = + + +⋅⋅⋅+ +⋅⋅⋅

DD D D D DD D D D

+ + +⋅⋅⋅+ +⋅⋅⋅

*Liquid phase* 

*Fluid-membrane* 

. The shear stress is

must include the various time and space scales (multi-scale

in the NS equation. The operators D/D*t* and

∇

*k*

ε = δ

 ψ ∇

∇

 ε

is defined as

( ) ( ) (0) (1) (1) (1) (1)

*u u T I <sup>g</sup>* (10)

 ε

 ψ

′ + =− ⋅ + ⋅ − ⋅ +

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

τ

ε:

 ψρ

∇

(8)

(9)

/ *L* . The symbols of δ

[N/m2]. Pressure

∇

were

where

ρ

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 Fig. 2:

$$
\dot{M}\_{\rm G} + \dot{M}\_{\rm L} - \left\{ (-P\_{\rm G}) \mathbf{n}\_{\rm G} + \mathbf{r}\_{\rm G} \cdot \mathbf{n}\_{\rm G} \right\} - \left\{ (-P\_{\rm L}) \mathbf{n}\_{\rm L} + \mathbf{r}\_{\rm L} \cdot \mathbf{n}\_{\rm L} \right\} - 2H \sigma \mathbf{n}\_{\rm G} - \frac{\mathbf{d} \sigma}{\mathbf{ds}} \mathbf{t} = 0 \tag{11}
$$

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

where the subscripts L and G represent liquid and gas phases, respectively. This is called the momentum jump condition at the interface. The symbols σ [N/m], *Pk* (k = G, L) [N/m2], 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 κ + κ , where κ1 and κ <sup>2</sup> [1/m] are the principal curvatures. The bold symbols *n<sup>k</sup>* , *t* , and *<sup>k</sup>* τ [N/m2] are the unit normal, unit tangential vector, and shear stress, respectively. *Mk* [kg/ms] ( *<sup>k</sup>* Μ [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. Therefore, we consider the jump condition to be a macroscopic interfacial equation.

#### **3.2 Derivation of thermodynamic jump condition**

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 concept (Yonemoto & Kunugi, 2010a).

Macroscopic Gas-Liquid Interfacial Equation

vector along the interface is not considered.

coordinate system, which leads to

∇ ⋅∇

Because *u ss n <sup>i</sup>* = ( 1 2 , , ) and *xi* = ( *xyz* , , ) , the operator

becomes

∇ ∇

axis. Unit vectors <sup>1</sup>

Thus, the operator

rewrite Eq. (12) as

Here, we assume (0)

where κ1 and κ

Based on Thermodynamic and Mathematical Approaches 65

respectively. The unit vectors are orthogonal to each other. The direction of *n*L is upward, and *n*G is opposite to *n*<sup>L</sup> (i.e., *n n tt* L G 12 = − =× ). In the present study, we focused on the local interface; therefore, general covariance (Aris, 1962) such as the change in the tangential

1 2

∇

κ κ

mech mech 1 L (0) (0)

*s n*

+⋅ + ⋅ + +

1

τ

ψ

*t n*

/ 0 ∂ = *s* and (0)

mech L L

*n n*

∫ ∫

∂ ∂ = − + ⋅

11 1

*t t*

*<sup>d</sup> s ns*

L1 L

*n n*

∂∂∂ ⎛ ⎞ <sup>−</sup> <sup>⎜</sup> ∂ ∂∂ <sup>⎝</sup>

 κ

1

*t n*

 κ

− +

0 0 1 L

ψψ

∫ ∫

 ε

∂ ∂

∂ ∂

+∞ +∞ −∞ −∞ +∞ +∞ −∞ −∞

d d

*f f n n s n*

∂ ∂

+∞ +∞ −∞ −∞

*<sup>P</sup> n n n n*

0 dd

∫ ∫

1 1

*t t*

1 1

∫ ∫

ψ ψ

ψψ

∫ ∫

+∞ +∞ −∞ −∞

ψε

∫ ∫

− + ⎜ ⎟ ⎜ ⎟ <sup>∂</sup> ∂ ∂ ⎝ ⎠ ⎝ ⎠

+∞ +∞ −∞ −∞

− + ⎜ ⎟ ⎜ ⎟ ∂ ∂∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠

+∞ +∞ −∞ −∞

∂ ∂ ∂ ∂⎛ ⎞ − − ⎜ ⎟ ∂ ∂ ∂ ∂⎝ ⎠

ε

<sup>1</sup> ∂τ

Eq. (15) from −∞ to +∞ along the *n* axis gives

ε

ε

ε

L

*n*

ε

ε

<sup>0</sup> *P P*

∂ ∂ =− − ∂ ∂

*t n*

1 2 *s s n* ∂ ∂ ∂ =+ + ∂ ∂ ∂

1 2 22 2 1 2

∂ ∂∂∂

<sup>2</sup> are the principal curvatures at the interface. In what follows, for the sake

∂∂ ∂ ∂

( ) *ns s n*

⋅= + + + +

of simplicity, only the *s*1–*n* coordinate system is considered. Using Eqs. (13) and (14), we can

0 0 1L 1L (0) (0) (1) (1) 1 1

∂ ∂∂ ∂

∂ ∂∂ ∂

 ψ

τ

*tn tn*

*s ns n*

ε

*<sup>d</sup> snn s n*

 κ

1L 1 (1) (1) (1) <sup>2</sup> <sup>2</sup> (1) (1) <sup>1</sup> <sup>1</sup>

1 1 1

2

*<sup>d</sup> dn n n sn s n*

*<sup>d</sup> dn n s n s s*

 ε

d d

d d

∂ ∂⎛ ⎞ <sup>⎟</sup> <sup>+</sup> ⎜ ⎟ ∂ ∂ <sup>⎠</sup> ⎝ ⎠

d d

*<sup>d</sup> n n n s*

 ψ

1 1

2

⎛ ⎞ ∂ ∂ ∂⎛ ⎞

*<sup>d</sup> dn n <sup>n</sup> n n*

∂∂ ∂ ⎛ ⎞ ∂ ∂⎛ ⎞

2

L

*n*

2

ε

2 2

d d

 ε

⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂∂ ∂ − + ⎜ ⎟⎜ ⎟ + + ⎝ ⎠ ∂∂∂ ∂ ∂ ⎝ ⎠

 ψψ

τ

*t* , 2*t* , and *n<sup>k</sup>* (*k* = G or L) are defined on the *s*1, *s*2, and *n* axes,

∇

222

*f f*

 ε

*tt n* (13)

2 2

mech 1 ∂*P s* / 0 ∂ = at the interface. Then, the integration of

2

 ψ

 ψ

2

1

 ψ 2

 ψ

is transformed into the interfacial

(14)

(15)

(16)

Fig. 3. Fluid membrane comprising elemental interfaces: (a) Fluid membrane, (b) Elemental interfaces. Interpretation of the fluid membrane based on interfacial coordinates

Fig. 4. Elemental interface {*i*} with interfacial coordinates in a fluid membrane

The interfacial jump condition is mainly considered in the moving coordinate system. This means that this jump condition is discussed under the condition that the relative velocity between the convected interface and fluid motion that convects the interface is neglected. In particular, assuming that the interface is in the steady state, the left-hand side of Eq. (10) (substantial derivative term) is neglected. In this study, phase changes such as evaporation and condensation and the effect of gravity on the interface are not considered. Therefore, Equation (10) reduces to

$$0 = -\mathbf{V}^{(0)} \cdot \mathbf{T} + \varepsilon \mathbf{V}^{(l)} \cdot \left( f\_0(\boldsymbol{\psi})\boldsymbol{I} \right) - \varepsilon d \mathbf{V}^{(l)} \boldsymbol{\psi} \left( \mathbf{V}^{(l)} \cdot \mathbf{V}^{(l)} \boldsymbol{\psi} \right). \tag{12}$$

Because Equation (11) is based on the interfacial coordinate system, Equation (12) is also transformed into an equation based on the interfacial coordinate system.

Before the concrete derivation of the thermodynamic jump condition, a mathematical operation is prepared. First, a test interface is defined as shown in Fig. 3. The test interface is divided into many elemental interfaces along the normal coordinate perpendicular to each elemental interface. We assume that the curvature is constant along the normal coordinate with respect to all elemental interfaces. Figure 4 shows representative interface {*i*} for a number of elemental interfaces. Next, the interfacial coordinate system is considered at point P on its interface. The tangential axes are represented by *s*1 and *s*2, and *n* is the normal axis. Unit vectors <sup>1</sup> *t* , 2*t* , and *n<sup>k</sup>* (*k* = G or L) are defined on the *s*1, *s*2, and *n* axes, respectively. The unit vectors are orthogonal to each other. The direction of *n*L is upward, and *n*G is opposite to *n*<sup>L</sup> (i.e., *n n tt* L G 12 = − =× ). In the present study, we focused on the local interface; therefore, general covariance (Aris, 1962) such as the change in the tangential vector along the interface is not considered.

Because *u ss n <sup>i</sup>* = ( 1 2 , , ) and *xi* = ( *xyz* , , ) , the operator ∇ is transformed into the interfacial coordinate system, which leads to

$$\mathbf{V} = \mathbf{t}\_1 \frac{\partial}{\partial \ s\_1} + \mathbf{t}\_2 \frac{\partial}{\partial \ s\_2} + \mathbf{n} \frac{\partial}{\partial \ m} \tag{13}$$

Thus, the operator ∇ ⋅∇becomes

64 Mass Transfer - Advanced Aspects

Fig. 3. Fluid membrane comprising elemental interfaces: (a) Fluid membrane, (b) Elemental

*s*2

2 *i* α

interfaces. Interpretation of the fluid membrane based on interfacial coordinates

*n* 

*<sup>n</sup> <sup>t</sup>*<sup>2</sup>

*P* 

1 *i* α

Fig. 4. Elemental interface {*i*} with interfacial coordinates in a fluid membrane

<sup>0</sup> 0 ( =− ⋅ + ⋅ −

transformed into an equation based on the interfacial coordinate system.

 *T I* ε∇

∇

The interfacial jump condition is mainly considered in the moving coordinate system. This means that this jump condition is discussed under the condition that the relative velocity between the convected interface and fluid motion that convects the interface is neglected. In particular, assuming that the interface is in the steady state, the left-hand side of Eq. (10) (substantial derivative term) is neglected. In this study, phase changes such as evaporation and condensation and the effect of gravity on the interface are not considered. Therefore,

( ) ( ) (0) (1) (1) (1) (1)

 εψ

 ) ∇

∇ ⋅∇

 *f d* ψ

Because Equation (11) is based on the interfacial coordinate system, Equation (12) is also

Before the concrete derivation of the thermodynamic jump condition, a mathematical operation is prepared. First, a test interface is defined as shown in Fig. 3. The test interface is divided into many elemental interfaces along the normal coordinate perpendicular to each elemental interface. We assume that the curvature is constant along the normal coordinate with respect to all elemental interfaces. Figure 4 shows representative interface {*i*} for a number of elemental interfaces. Next, the interfacial coordinate system is considered at point P on its interface. The tangential axes are represented by *s*1 and *s*2, and *n* is the normal

*s*1

*t*1

*Gas phase*

{*i*}

 ψ

. (12)

*0* 

*i … …*

*m* 

*Liquid phase* 

*Liquid phase* 

)a( )b(

*Fluid-membrane* 

*Gas phase* 

Equation (10) reduces to

$$\mathbf{V} \cdot \mathbf{V} = (\kappa\_1 + \kappa\_2) \frac{\partial}{\partial \ n} + \frac{\hat{\mathcal{O}}^2}{\hat{\mathcal{O}}} + \frac{\hat{\mathcal{O}}^2}{\hat{\mathcal{O}}} + \frac{\hat{\mathcal{O}}^2}{\hat{\mathcal{O}}} + \frac{\hat{\mathcal{O}}^2}{\hat{\mathcal{O}}} \tag{14}$$

where κ1 and κ <sup>2</sup> are the principal curvatures at the interface. In what follows, for the sake of simplicity, only the *s*1–*n* coordinate system is considered. Using Eqs. (13) and (14), we can rewrite Eq. (12) as

$$\begin{aligned} 0 &= -\mathfrak{t}\_{\text{l}} \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{P\_{\text{mech}}}{\mathfrak{s}\_{\text{l}}^{(0)}} - \mathfrak{n}\_{\text{L}} \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{P\_{\text{mech}}}{\mathfrak{n}^{(0)}} \\ &+ \mathfrak{t}\_{\text{l}} \cdot \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{\mathfrak{r}}{\mathfrak{s}\_{\text{l}}^{(0)}} + \mathfrak{n}\_{\text{L}} \cdot \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{\mathfrak{r}}{\mathfrak{n}^{(0)}} + \varepsilon \mathfrak{t}\_{\text{l}} \frac{\widehat{\mathscr{O}}}{\mathfrak{d}} \frac{f\_{0}}{\mathfrak{n}^{(1)}} + \varepsilon \mathfrak{n}\_{\text{L}} \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{f\_{0}}{\mathfrak{n}^{(1)}} \\ &- \varepsilon \mathcal{E} \Big( \mathfrak{t}\_{\text{l}} \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{\mathfrak{\nu}}{\mathfrak{s}\_{\text{l}}^{(1)}} + \mathfrak{n}\_{\text{L}} \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{\mathfrak{\nu}}{\mathfrak{n}^{(1)}} \Big) \Big( \kappa\_{\text{l}} \frac{\widehat{\mathscr{O}}}{\widehat{\mathscr{O}}} \frac{\mathfrak{\nu}}{\mathfrak{n}^{(1)}} + \frac{\widehat{\mathscr{O}}^{2} \nu}{\widehat{\mathscr{O}}} + \frac{\widehat{\mathscr{O}}^{2} \nu}{\widehat{\mathscr{O}}} \Big) \end{aligned} \tag{15}$$

Here, we assume (0) <sup>1</sup> ∂τ / 0 ∂ = *s* and (0) mech 1 ∂*P s* / 0 ∂ = at the interface. Then, the integration of Eq. (15) from −∞ to +∞ along the *n* axis gives

mech L L 0 0 1 L 1 2 11 1 1 1 1 2 1 1 1 1 L 1 1 0 dd d d d d 2 d d 2 *<sup>P</sup> n n n n f f n n s n <sup>d</sup> dn n s n s s <sup>d</sup> dn n n sn s n <sup>d</sup> s ns* ε ε ψψ ε ψ ε κ ψψ ε ψ ε ψ ψ ε +∞ +∞ −∞ −∞ +∞ +∞ −∞ −∞ +∞ +∞ −∞ −∞ +∞ +∞ −∞ −∞ ∂ ∂ = − + ⋅ ∂ ∂ ∂ ∂ − + ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ − − ⎜ ⎟ ∂ ∂ ∂ ∂⎝ ⎠ ∂∂ ∂ ⎛ ⎞ ∂ ∂⎛ ⎞ − + ⎜ ⎟ ⎜ ⎟ ∂ ∂∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ∂∂∂ ⎛ ⎞ <sup>−</sup> <sup>⎜</sup> ∂ ∂∂ <sup>⎝</sup> ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ τ *n n t n t t t t n* 2 L 1 2 2 L1 L d d 2 d d 2 *<sup>d</sup> n n n s <sup>d</sup> dn n <sup>n</sup> n n* ε ψ ψε ψ ε κ +∞ +∞ −∞ −∞ +∞ +∞ −∞ −∞ ∂ ∂⎛ ⎞ <sup>⎟</sup> <sup>+</sup> ⎜ ⎟ ∂ ∂ <sup>⎠</sup> ⎝ ⎠ ⎛ ⎞ ∂ ∂ ∂⎛ ⎞ − + ⎜ ⎟ ⎜ ⎟ <sup>∂</sup> ∂ ∂ ⎝ ⎠ ⎝ ⎠ ∫ ∫ ∫ ∫ *n n n* (16)

Macroscopic Gas-Liquid Interfacial Equation

0

ε

ε

∫

εκ

ε

2 ε

ε

+∞ −∞

12 G

*tt n*

i i 1 2 1 2

12 1 2

+∞ −∞

⎛ ⎞ ∂ ∂ + ++ ⎜ ⎟ ∂ ∂ ⎝ ⎠

( )

( )

+∞ −∞

∫

+∞ −∞

∫

 κ *h*

∫

 

π

 

 σ

ψ

 πε

*<sup>d</sup> <sup>n</sup> s s*

( )

*j*

 

++ + ⎜ ⎟⎜ ⎟ ∂ ∂∂ ⎝ ⎠⎝ ⎠

ψψ

*<sup>d</sup> s ss s ss*

( )

*k*

 

⎡ ⎤ ∂∂ ∂∂ ⎛ ⎞ ⎛⎞

*<sup>d</sup> <sup>n</sup> ss s s t t*

12 21

ψ

*t t*

*d n s sn*

1 2

*t t*

1 2

the fluid motion (gas phase side) on the interface

the fluid motion (liquid phase side) on the interface

c. The conventional surface tension, known as the Young–Laplace formula

conventional term in Eq. (11), and we discuss this difference in Section 3.4

− + ⎢ ⎥ ⎜ ⎟ ⎜⎟ ∂∂ ∂ ∂ ⎝ ⎠ ⎝⎠ ⎣ ⎦

( )

*l*

 

*tt t*

 

*P P*

*n n*

2 2

 

2

σ

επ

Based on Thermodynamic and Mathematical Approaches 67

In Eq. (21), although the curvature depends not only on the *s*1, *s*2 directions but also on the *n* direction, as shown in Fig. 5, we do not consider the change in the curvature in the normal direction because the interface is very thin. In Eq. (22), the replacement of <sup>1</sup> ∂∂ = / 0 *s* by <sup>1</sup> d/d 0 *s* = means that the dependency of the surface tension coefficient on the *s*2 coordinate is not considered. After substituting Eqs. (20), (21), and (22) into Eq. (19) and adding the

terms with respect to the *s*2 coordinate in Eq. (19), we derive the following equation:

= − − − + ⋅ −⎡ − − + ⋅ ⎤− + ⎡ ⎤ ⎣ ⎦ ⎣ ⎦

( ) ( )

G 1 2

*n t t*

 ψ

2

+∞ −∞

∫

 ψψ

⎛ ∂ ∂∂ ⎞⎛ ⎞

1 2

⎡ ⎤ ∂∂∂ ∂∂∂ ⎛ ⎞⎛ ⎞ + + ⎢ ⎥ ⎜ ⎟⎜ ⎟ ∂ ∂∂ ∂ ∂∂ ⎣ ⎦ ⎝ ⎠⎝ ⎠

2 12 1 12

ψψ

2 2

d

*d e*

 ε +∞ −∞

+∞ +∞ −∞ −∞

∫ ∫

τ

( ) G GG GG L LL LL G 1 2 ( ) () ( ) ( ) ( )

⎛ ∂ ∂ ⎞ ⎛ ⎞⎛ ⎞ ⎡ ⎤ ∂∂∂ ∂∂∂ − +− ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥ + ∂ ∂ ∂ ∂∂ ∂ ∂∂ <sup>⎝</sup> ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦ <sup>∫</sup>

 

1 2 1 12 2

⎡ ⎤ ∂∂ ∂ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ∂ ∂ + ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ ⎟ + − + −+ ∂∂ ∂ ⎢ ⎥ ∂ ∂ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎣ ⎦

*<sup>g</sup> <sup>f</sup>*

1 1 <sup>d</sup>

ψψ

( ) 2 2

11 2 2

*t*

*<sup>n</sup> ss s s*

⎡ ⎤ ∂∂ ∂∂ ⎛⎞ ⎛ ⎞ ⎢ ⎥ ⎜⎟ ⎜ ⎟ <sup>+</sup> ∂∂ ∂∂ ⎝⎠ ⎝ ⎠ ⎣ ⎦

ψψ

 

2 2

 ψ

τ

επ

*a b c*

 

*nn n*

*d n s s s ns s ns*

1 2 1 2 ( ) ( )

d

d

 ψψ

This equation is the thermodynamic jump condition at the interface and is obtained by assuming that many elemental interfaces lie on the interface with a finite thickness and integrating it over the normal direction. The free energy is defined at the interface and is derived from the microscopic viewpoint (Hamiltonian) (Yonemoto & Kunugi, 2010a). Therefore, Equation (23) considers multiple scales from microscopic to macroscopic. In order of appearance, the physical meaning of each term in Eq. (23) is discussed as follows: a. The effects of the pressure, contamination at the interface, and shear stress caused by

b. The effects of the pressure, contamination at the interface, and shear stress caused by

d. The gradient of the surface tension in the tangential direction; this term differs from the

ε

*d na <sup>b</sup> <sup>n</sup> ns s s s*

<sup>1</sup> d d

1 2

ψ

( )

*i*

 

d

*n*

εκ κ σ

 

3

(23)

ψ ψ

d

 ψψ

The superscripts in terms *n*(0) and *n*(1) in Eq. (15) can be omitted in Eq. (16) because there is no difference between the normal directions *n*(0) and *n*(1) over the integration. The first term on the right-hand side of Eq. (16) is calculated as follows:

$$-\mathfrak{n}\_{\rm L} \int\_{P\_{\rm G}}^{P\_{\rm L}} \mathrm{d}P\_{\rm mech} = -\mathfrak{n}\_{\rm L}P\_{\rm L} - \mathfrak{n}\_{\rm G}P\_{\rm G},\tag{17}$$

and the second term is

$$
\mathfrak{n}\_{\rm L} \cdot \int\_{\mathfrak{r}\_{\rm G}}^{\mathfrak{r}\_{\rm L}} \mathrm{d}\mathfrak{r} = \mathfrak{n}\_{\rm L} \cdot \mathfrak{r}\_{\rm L} + \mathfrak{n}\_{\rm G} \cdot \mathfrak{r}\_{\rm G} \tag{18}
$$

When ∂ ∂= ψ/ 0 *n* at *n* = −∞ or +∞ , we obtain

( ) ( ) ( ) G GG GG L LL LL 2 2 L1 1 1 2 L L 1 1 1 <sup>3</sup> <sup>i</sup> 1 1 1 1 1 0 d d 2 d d 2 d d 2 *P P <sup>d</sup> dn n n s n <sup>d</sup> dn n s ns n s ab n n s s d* επ επ ψε ψ ε κ ψψ ε ψ ε ψ π ε ψψ ε ε +∞ +∞ −∞ −∞ +∞ +∞ −∞ −∞ +∞ +∞ −∞ −∞ = −⎡ − − + ⋅ ⎤−⎡ − − + ⋅ ⎤ ⎣ ⎦ ⎣ ⎦ ⎛ ⎞ ∂ ∂ ⎛ ⎞ ∂ + − ⎜ ⎟ ⎜ ⎟ ∂ ∂ <sup>∂</sup> ⎝ ⎠ ⎝ ⎠ ∂∂∂ ⎛ ⎞ ∂ ∂⎛ ⎞ + − ⎜ ⎟ ⎜ ⎟ ∂ ∂∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ∂ ∂ − −+ + ∂ ∂ + ∫ ∫ ∫ ∫ ∫ ∫ *n n* τ *n n* τ *n t n n t t t* 2 1 1 1 1 1 d d *nd n s s s n* ψ ψ ψ ε κ +∞ +∞ −∞ −∞ ∂ ∂⎛ ⎞ ∂ ∂ ⎜ ⎟ <sup>+</sup> ∂ ∂ ∂ ∂ ⎝ ⎠ ∫ ∫ *<sup>t</sup>* (19)

where *i ie* π = *cz eV*ψ [N/m2]. The function π *<sup>i</sup>* (*i* = L or G) represents contamination at the interface (liquid or gas phase). The surface tension coefficient under the flat surface is defined as follows (Rowlinson & Widom, 1984):

$$
\sigma = d \int\_{-\alpha}^{\alpha} \left( \frac{\partial}{\partial \ n} \frac{\varphi}{n} \right)^2 \,\mathrm{d}n \tag{20}
$$

Equation (20) is substituted into Eq. (19) and is transformed in order to consider the relation between Eqs. (11) and (19).

Using Eq. (20), we can rewrite the third and fourth terms on the right-hand side of Eq. (19) as

$$\mathfrak{n}\_{\rm L} \varepsilon d \int\_{-\infty}^{+\infty} \kappa\_{\rm l} \left( \frac{\partial}{\partial \cdot n} \frac{\psi}{n} \right)^{2} d\mathfrak{n} = \mathfrak{n}\_{\rm L} \varepsilon \sigma \kappa\_{\rm l} \tag{21}$$

$$\begin{aligned} \mathbf{t}\_{\mathrm{l}} \frac{\varepsilon \mathrm{d}}{2} \int\_{-\infty}^{+\infty} \frac{\partial}{\partial \ \mathrm{s}\_{\mathrm{l}}} \left( \frac{\partial}{\partial \ \mathrm{n}} \frac{\boldsymbol{\mu}}{\boldsymbol{n}} \right)^{2} \mathrm{d}\boldsymbol{n} &= \mathbf{t}\_{\mathrm{l}} \frac{\varepsilon}{2} \frac{\partial}{\partial \ \mathrm{s}\_{\mathrm{l}}} \left[ \mathrm{d} \int\_{-\infty}^{+\infty} \left( \frac{\partial}{\partial \ \boldsymbol{n}} \frac{\boldsymbol{\mu}}{\boldsymbol{n}} \right)^{2} \mathrm{d}\boldsymbol{n} \right] \\ &= \mathbf{t}\_{\mathrm{l}} \frac{\varepsilon}{2} \frac{\mathrm{d}}{\mathrm{d}} \frac{\sigma}{\mathrm{s}\_{\mathrm{l}}} \end{aligned} \tag{22}$$

The superscripts in terms *n*(0) and *n*(1) in Eq. (15) can be omitted in Eq. (16) because there is no difference between the normal directions *n*(0) and *n*(1) over the integration. The first term

L mech L L G G d , *<sup>P</sup>*

L L L G <sup>G</sup> <sup>⋅</sup> <sup>d</sup> =⋅+ ⋅ ∫

τ

G GG GG L LL LL 2 2

2

*<sup>d</sup> dn n n s n*

= −⎡ − − + ⋅ ⎤−⎡ − − + ⋅ ⎤ ⎣ ⎦ ⎣ ⎦ ⎛ ⎞ ∂ ∂ ⎛ ⎞ ∂

d d

 ε

2

d d

1 1

2

 ε

d d *nd n*

 κ

1 1 1

∂∂∂ ⎛ ⎞ ∂ ∂⎛ ⎞

*<sup>d</sup> dn n s ns n s*

> *ab n n s s*

 *n nn* τ

( ) ( )

τ

+∞ +∞ −∞ −∞

+ − ⎜ ⎟ ⎜ ⎟ ∂ ∂∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠

*P P*

*n n*

ψε

+ − ⎜ ⎟ ⎜ ⎟ ∂ ∂ <sup>∂</sup> ⎝ ⎠ ⎝ ⎠

+∞ +∞ −∞ −∞

∫ ∫

*<sup>P</sup>* − =− − *P PP* ∫ *n nn* (17)

(18)

(19)

(22)

2

*<sup>i</sup>* (*i* = L or G) represents contamination at the

⎛ ⎞ <sup>∂</sup> <sup>=</sup> ⎜ ⎟ <sup>∂</sup> ⎝ ⎠ <sup>∫</sup> (20)

τ

 ψ

τ

1

*n n*

d d

 π

> ψ

ψ

επ

 ψ

on the right-hand side of Eq. (16) is calculated as follows:

/ 0 *n* at *n* = −∞ or +∞ , we obtain

ε

ε

0

1

defined as follows (Rowlinson & Widom, 1984):

ε

*t*

+

ψ

between Eqs. (11) and (19).

2

*d*

ε

ε

and the second term is

When ∂ ∂= ψ

where *i ie* π= *cz eV*

as

L G

> L G

τ

τ

( )

ψ

σ <sup>∞</sup> −∞

ε

1 1

2 2

∫ ∫ *t t*

κ

1 1

ψε

+∞ +∞ −∞ −∞

−∞

∫ ∫

+∞ +∞ −∞ −∞

 ψψ

*t t*

− −+ +

<sup>3</sup> <sup>i</sup> 1 1

2

 +∞ +∞ −∞ −∞

∂ ∂

∂ ∂

ψψ

∫ ∫

1 1

1 1 1

interface (liquid or gas phase). The surface tension coefficient under the flat surface is

*d n*<sup>d</sup> *<sup>n</sup>* ψ

Equation (20) is substituted into Eq. (19) and is transformed in order to consider the relation

Using Eq. (20), we can rewrite the third and fourth terms on the right-hand side of Eq. (19)

⎛ ⎞ ∂

2 L 1 L 1 *d n*<sup>d</sup> *<sup>n</sup>* ψ

+∞

1

*t*

=

*<sup>d</sup> nd n sn s n*

 εσκ

2 2

d d

1

*s*

d 2 d

<sup>⎡</sup> <sup>⎤</sup> ∂ ∂⎛ ⎞ ∂ ∂⎛ ⎞ <sup>=</sup> <sup>⎢</sup> <sup>⎥</sup> ⎜ ⎟ ⎜ ⎟ ∂∂ ∂ ∂ ⎝ ⎠ <sup>⎢</sup> ⎝ ⎠ <sup>⎥</sup> <sup>⎣</sup> <sup>⎦</sup>

ε σ

⎜ ⎟ <sup>=</sup> <sup>∂</sup> ⎝ ⎠ ∫ *n n* (21)

 ψ

π

∂ ∂⎛ ⎞ ∂ ∂ ⎜ ⎟ <sup>+</sup> ∂ ∂ ∂ ∂ ⎝ ⎠ ∫ ∫ *<sup>t</sup>*

*s s s n*

ε

L L

*n n*

L1 1

*n t*

επ

 κ

ψ

[N/m2]. The function

In Eq. (21), although the curvature depends not only on the *s*1, *s*2 directions but also on the *n* direction, as shown in Fig. 5, we do not consider the change in the curvature in the normal direction because the interface is very thin. In Eq. (22), the replacement of <sup>1</sup> ∂∂ = / 0 *s* by <sup>1</sup> d/d 0 *s* = means that the dependency of the surface tension coefficient on the *s*2 coordinate is not considered. After substituting Eqs. (20), (21), and (22) into Eq. (19) and adding the terms with respect to the *s*2 coordinate in Eq. (19), we derive the following equation:

( ) G GG GG L LL LL G 1 2 ( ) () ( ) ( ) ( ) 12 G 1 2 1 12 2 ( ) ( ) 0 1 1 <sup>d</sup> 2 2 επ επ εκ κ σ σ σ ψψ ψψ ε ε +∞ −∞ = − − − + ⋅ −⎡ − − + ⋅ ⎤− + ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎛ ∂ ∂ ⎞ ⎛ ⎞⎛ ⎞ ⎡ ⎤ ∂∂∂ ∂∂∂ − +− ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥ + ∂ ∂ ∂ ∂∂ ∂ ∂∂ <sup>⎝</sup> ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦ <sup>∫</sup> *a b c d e P P d n s s s ns s ns n n* τ *nn n* τ *tt n* ( ) 2 2 3 G 1 2 1 2 1 2 ( ) ( ) i i 1 2 1 2 ( ) <sup>1</sup> d d 2 d 2 ψ ψ ψψ ε ε ψ ψ π πε ε +∞ +∞ −∞ −∞ +∞ −∞ ⎡ ⎤ ∂∂ ∂ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ∂ ∂ + ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎜ ⎟ + − + −+ ∂∂ ∂ ⎢ ⎥ ∂ ∂ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎛ ⎞ ∂ ∂ + ++ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ∫ ∫ ∫ *<sup>g</sup> <sup>f</sup> h d na <sup>b</sup> <sup>n</sup> ns s s s <sup>d</sup> <sup>n</sup> s s n t t tt t* ( ) 2 2 1 2 11 2 2 ( ) 12 1 2 1 2 ( ) 1 2 2 12 1 12 d d d ψ ψ ψ ψψ εκ κ ψψ ψψ ε +∞ −∞ +∞ −∞ ⎡ ⎤ ∂∂ ∂∂ ⎛⎞ ⎛ ⎞ ⎢ ⎥ ⎜⎟ ⎜ ⎟ <sup>+</sup> ∂∂ ∂∂ ⎝⎠ ⎝ ⎠ ⎣ ⎦ ⎛ ∂ ∂∂ ⎞⎛ ⎞ ++ + ⎜ ⎟⎜ ⎟ ∂ ∂∂ ⎝ ⎠⎝ ⎠ ⎡ ⎤ ∂∂∂ ∂∂∂ ⎛ ⎞⎛ ⎞ + + ⎢ ⎥ ⎜ ⎟⎜ ⎟ ∂ ∂∂ ∂ ∂∂ ⎣ ⎦ ⎝ ⎠⎝ ⎠ ∫ ∫ *i j <sup>n</sup> ss s s d n s sn <sup>d</sup> s ss s ss t t t t t* ( ) 2 2 1 2 12 21 ( ) d 2 ε ψψ +∞ −∞ +∞ −∞ ⎡ ⎤ ∂∂ ∂∂ ⎛ ⎞ ⎛⎞ − + ⎢ ⎥ ⎜ ⎟ ⎜⎟ ∂∂ ∂ ∂ ⎝ ⎠ ⎝⎠ ⎣ ⎦ ∫ ∫ *k l n <sup>d</sup> <sup>n</sup> ss s s t t* (23)

This equation is the thermodynamic jump condition at the interface and is obtained by assuming that many elemental interfaces lie on the interface with a finite thickness and integrating it over the normal direction. The free energy is defined at the interface and is derived from the microscopic viewpoint (Hamiltonian) (Yonemoto & Kunugi, 2010a). Therefore, Equation (23) considers multiple scales from microscopic to macroscopic. In order of appearance, the physical meaning of each term in Eq. (23) is discussed as follows:


Macroscopic Gas-Liquid Interfacial Equation

First, Equation (23) is divided by <sup>2</sup>

This normalization reduces Eq. (23) to

( )

κ κσ

 

+∞ −∞

∫

+∞ −∞

∫

ε

*n*

ε

+∞ −∞

\*

∫

+∞ −∞

∫

ε

ε κκ

ε

ε

*d*

*d*

+

*d*

ε ∫

*n*

In particular, we normalize *n*,

Based on Thermodynamic and Mathematical Approaches 69

*U*0 and then normalized by other characteristic values.

 π

 

∗ ∗

*n n*

λ1 and λ

<sup>2</sup> , respectively.

(24)

τ

and *s*1 and *s*2 by

 σ

ψ ψ

ρ

κ <sup>2</sup> by δ

\* \* \* \*\*

\* 1 2 \*

*d n s ns s ns*

⎡ ⎤ ∂∂∂ ∂∂∂ ⎛⎞ ⎛⎞

1 2 1 2

*n*

We

1 1 <sup>d</sup>

*s sn*

2

We

\* \* 2 \* \*

1 2 11 22

Kn Kn <sup>d</sup>

2 2

ψ

2 2

2 \* \* 2 2 2

> ψψ

2 \*

ψ

1 \*

*n*

*n*

ψ ψ

*s ss*

2 1 12

Kn <sup>d</sup>

*n*

Kn <sup>d</sup>

*s s*

 ψ

> ψ

*ab n*

G 12 1 \* \* 2

 

G \* \*\* \* \*\*

− + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂ ∂∂ ∂ ∂∂ ⎣ ⎦ ⎝⎠ ⎝⎠

\* \* 1 2

Kn Kn <sup>d</sup>

 π

 ψ

ψ

*d n ns s*

( )

*e*

 

′

⎡ ⎤ ∂∂ ∂ ⎛⎞ ⎛⎞

( ) 1 2 \* \*3 \*

1 Kn Kn <sup>d</sup>

− +− + ⎜ ⎟ ∂ ∂ ⎝ ⎠

1 11 11 We We 2 We 2

( ) ( )

*<sup>c</sup> <sup>d</sup>*

G \*\* \*

1 2 \* \* 1 2 ( ) \* \* 1i 2i \*

⎛ ⎞ ∂ ∂ <sup>−</sup> ⎜ ⎟ + −+ ∂ ∂ ⎝ ⎠

E E

ψ

*t t*

1 2 \* \*

⎛ ⎞ ∂ ∂ + + ⎜ ⎟ ∂ ∂ ⎝ ⎠

′

Kn Kn <sup>d</sup>

( )

*h*

1 \* \*

 

4N 4N

π

*t t*

1 1 \* \* 1 1 1

> +∞ −∞

∫

Kn

2 We

+∞ −∞

( )

<sup>2</sup> \* <sup>2</sup>

+∞ −∞

∫

+∞ −∞

∫

*t*

Kn We

In Eq. (24), the dimensionless numbers are

+ ⎜

*t*

1 2

*s s*

 ψ

( )

*i*

 

′

⎡ ⎤ ∂ ∂ ⎛ ⎞ ⎢ ⎥ ⎟ ⎜⎟ <sup>+</sup> ∂ ∂⎝ ⎠ ⎣ ⎦

*t*

( )

*j*

 

+ + <sup>⎜</sup> <sup>+</sup> ⎟⎜ ⎟ <sup>⎜</sup> ∂ ∂∂ <sup>⎝</sup> ⎠⎝ ⎠

*t t*

\*\* \* \* 12 1 \* \*\* 2

ψ

*d n*

We We

′

( ) 2 2 \* 2 <sup>2</sup>

*k*

 

⎡ ⎤ ∂ ∂ ⎛⎞ ⎛⎞ ∂ ∂

′

1 2 \* \* \* \* 1 2 1 2 2 1

( )'

*l*

 

1 2 1 2

⎡ ⎤ ⎛⎞ ⎛⎞ ∂∂∂ ⎢ ⎥ ⎜⎟ ⎜⎟ <sup>+</sup> ∂ ∂∂ ⎣ ⎦ ⎝⎠ ⎝⎠

*t*

⎛ ∂ ∂∂ ⎞⎛ ⎞

2 1 \*

ψψ

Kn Kn <sup>d</sup>

*ss ss*

*t t* .

∂ ∂ ⎛ ⎞

<sup>⎜</sup> ∂ ∂⎝ ⎠

1 2 12

∂ ∂∂

2 We We

∂∂∂

ψ ψ

*s ss*

− + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂∂ ∂∂ ⎝⎠ ⎝⎠ ⎣ ⎦

*s s*

+ + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂∂ ∂ ⎝⎠ ⎝⎠ ⎣ ⎦

2 We We

( )

*f*

′

 

′

 

*s s*

*g*

′ ′

We We

ψ ψ

 ε

*n tt*

 

\* \* \* \*

τ

σ

11 1 11 1 <sup>0</sup>

*P P*

*n n*

G G G GG L L L LL

( ) ( )

*a b*

⎛ ⎞ ∂ ∂

′ ′

⎡ ⎤ ⎛ ⎞ ⎛⎞ ⎡ ⎤ =− − − + ⋅ − − − + ⋅ ⎢ ⎥ ⎜ ⎟ ⎜⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎝⎠ ⎣ ⎦

E 4N Re E 4N Re

1 2 1 2

*s s*

κ<sup>1</sup> , and

π


Fig. 5. Curvature through the interface

#### **3.3 Order estimation of jump condition**

We now consider the order estimation of the thermodynamic jump condition. The characteristic velocity, length, density, viscosity, interfacial thickness, surface tension coefficient, and electrostatic potential at the interface are *U*<sup>0</sup> [m/s], *L*[m], ( )/2 ρ = + ρ ρ *l g* [kg/m3], μ<sup>0</sup> [kg/ms], δ [m], σ <sup>0</sup> [N/m], and *V*<sup>0</sup> [V], respectively. The characteristic pressure is <sup>2</sup> ρ*U*0 or *P*0 [N/m2], and the interfacial wavelength is λ1 (*s*1 axis) or λ<sup>2</sup> (*s*2 axis) [m].

g. The Marangoni effect term, which depends on the temperature difference because of

j. The synergy effect, which is the same as term (e); however, this term is slightly different

tangential direction; this term may simply be the tensile force, and the tangential

*n* 

1 *( ) nL κ*

> 1 *( ) nG κ*

<sup>0</sup> [N/m], and *V*<sup>0</sup> [V], respectively. The characteristic pressure

1 (*s*1 axis) or

λ

λ

k. The synergy effect of the change in the tangential direction with respect to the

*0* 

We now consider the order estimation of the thermodynamic jump condition. The characteristic velocity, length, density, viscosity, interfacial thickness, surface tension coefficient, and electrostatic potential at the interface are *U*<sup>0</sup> [m/s], *L*[m], ( )/2

*m* 

*i …*

*…*

in the normal and tangential directions at the

gradient in the tangential

gradient in the tangential

ψ

*m*

*i* 

*Gas* 

ρ = + ρ ρ*l g*

<sup>2</sup> (*s*2 axis) [m].

*0* 

*Liquid* 

1 *( ) ni κ*

ψ

gradient in the

ψ

ψ

gradient changes in the tangential direction at the

ψ

ψ

e. The synergy effect of the change in

direction; this term arises if the

interface of the gas or liquid phase

f. The change in the normal direction with respect to the

the temperature-dependent coefficients *a* and *b* h. The effect of contamination in the tangential direction i. The change in the tangential direction with respect to the

gradient in the *s*1 and *s*2 tangential directions

directions are different from term (i).

Fig. 5. Curvature through the interface

[kg/m3],

is <sup>2</sup> ρ

μ

<sup>0</sup> [kg/ms],

**3.3 Order estimation of jump condition** 

δ [m], σ

*U*0 or *P*0 [N/m2], and the interfacial wavelength is

direction; this term may simply be the tensile force

from term (e) because this term includes the curvature

l. The effect of the change in the tangential direction with respect to the

interface

First, Equation (23) is divided by <sup>2</sup> ρ*U*0 and then normalized by other characteristic values. In particular, we normalize *n*, κ<sup>1</sup> , and κ <sup>2</sup> by δ and *s*1 and *s*2 by λ1 and λ<sup>2</sup> , respectively. This normalization reduces Eq. (23) to

( ) \* \* \* \* G G G GG L L L LL ( ) ( ) \* \* \* \*\* G 12 1 \* \* 2 1 2 1 2 ( ) ( ) 11 1 11 1 <sup>0</sup> E 4N Re E 4N Re 1 11 11 We We 2 We 2 π π σ σ κ κσ ε ∗ ∗ ′ ′ ′ ′ ⎡ ⎤ ⎛ ⎞ ⎛⎞ ⎡ ⎤ =− − − + ⋅ − − − + ⋅ ⎢ ⎥ ⎜ ⎟ ⎜⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎝⎠ ⎣ ⎦ ⎛ ⎞ ∂ ∂ − +− + ⎜ ⎟ ∂ ∂ ⎝ ⎠ *a b <sup>c</sup> <sup>d</sup> P P s s n n* τ *n n* τ *n tt* \* 1 2 \* G \* \*\* \* \*\* 1 2 11 22 ( ) 2 2 \* \* 1 2 G \*\* \* 1 2 1 2 ( ) Kn Kn <sup>d</sup> We We 1 Kn Kn <sup>d</sup> 2 We We ψ ψ ψ ψ ε ψ ψ ε +∞ −∞ ′ +∞ −∞ ′ ⎡ ⎤ ∂∂∂ ∂∂∂ ⎛⎞ ⎛⎞ − + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂ ∂∂ ∂ ∂∂ ⎣ ⎦ ⎝⎠ ⎝⎠ ⎡ ⎤ ∂∂ ∂ ⎛⎞ ⎛⎞ + + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂∂ ∂ ⎝⎠ ⎝⎠ ⎣ ⎦ ∫ ∫ *e f d n s ns s ns d n ns s n n* ( ) 1 2 \* \*3 \* 1 2 \* \* 1 2 ( ) \* \* 1i 2i \* 1 2 \* \* 1 2 ( ) \* 1 1 \* \* 1 1 1 Kn Kn <sup>d</sup> E E Kn Kn <sup>d</sup> 4N 4N Kn 2 We ψ ψ ε ψ ψ π π ε ψ +∞ −∞ ′ +∞ −∞ ′ ⎛ ⎞ ∂ ∂ <sup>−</sup> ⎜ ⎟ + −+ ∂ ∂ ⎝ ⎠ ⎛ ⎞ ∂ ∂ + + ⎜ ⎟ ∂ ∂ ⎝ ⎠ ∂ ∂ ⎛ ⎞ + ⎜ <sup>⎜</sup> ∂ ∂⎝ ⎠ ∫ ∫ *g h ab n s s n s s d s s t t t t t* ( ) 2 2 2 \* 2 \* \* 2 2 2 ( ) \*\* \* \* 12 1 \* \*\* 2 1 2 1 2 ( ) <sup>2</sup> \* <sup>2</sup> 1 \* \* 1 2 12 Kn <sup>d</sup> We 1 1 <sup>d</sup> We We Kn We ψ ψ ψψ ε κκ ψ ψ ε +∞ −∞ ′ +∞ −∞ ′ ⎡ ⎤ ∂ ∂ ⎛ ⎞ ⎢ ⎥ ⎟ ⎜⎟ <sup>+</sup> ∂ ∂⎝ ⎠ ⎣ ⎦ ⎛ ∂ ∂∂ ⎞⎛ ⎞ + + <sup>⎜</sup> <sup>+</sup> ⎟⎜ ⎟ <sup>⎜</sup> ∂ ∂∂ <sup>⎝</sup> ⎠⎝ ⎠ ∂∂∂ + ∂ ∂∂ ∫ ∫ *i j n s s d n s sn d s ss t t t t* 2 1 \* \* \* 2 \* \* 2 1 12 ( ) 2 2 \* 2 <sup>2</sup> 2 1 \* 1 2 \* \* \* \* 1 2 1 2 2 1 ( )' Kn <sup>d</sup> We Kn Kn <sup>d</sup> 2 We We ψ ψ ε ψψ +∞ −∞ ′ +∞ −∞ ⎡ ⎤ ⎛⎞ ⎛⎞ ∂∂∂ ⎢ ⎥ ⎜⎟ ⎜⎟ <sup>+</sup> ∂ ∂∂ ⎣ ⎦ ⎝⎠ ⎝⎠ ⎡ ⎤ ∂ ∂ ⎛⎞ ⎛⎞ ∂ ∂ − + ⎢ ⎥ ⎜⎟ ⎜⎟ ∂∂ ∂∂ ⎝⎠ ⎝⎠ ⎣ ⎦ ∫ ∫ *k l n s ss d n ss ss t t t* . (24)

In Eq. (24), the dimensionless numbers are

Macroscopic Gas-Liquid Interfacial Equation

**3.4 Consideration of surface tension gradient** 

Based on Thermodynamic and Mathematical Approaches 71

characteristic velocity is the bubble velocity). If the bubble velocity approaches zero and the bubble diameter decreases, the inverse of the dimensionless number N increases. This indicates that the electrostatic force due to contamination at the interface is significant when compared to the hydrodynamic force, although it depends on the magnitude of the electrostatic potential at the interface. In other words, it is difficult to deform the bubble; it maintains a spherical shape except for the effect of the electrostatic force or mass transfer at the interface. In this situation, the dimensionless number N roughly explains that the

breakup of a small bubble has difficult occurring if the bubble diameter is very small.

differs from that of the conventional one. In this section, we discuss term (d).

plane or spherical surface. Based on these concepts, we discuss term (d).

*n* 

*n*

*P* 

*s*1

Let *f* denote term (d) in Eq. (23). This term can be expressed as

on the *s*1 axis, and force <sup>2</sup> *f* is defined on the *s*2 axis as follows:

Fig. 7. Statistical local interface

*t1*

α1

The thermodynamic jump condition seems to be the same as the conventional jump condition when we focus on terms (a)–(d) in Eq. (23). However, the detailed formula of (d)

The interfacial coordinate system shown in Fig. 7 is reconsidered using the same notations as in Section 3.2: *n*, *s*1, and *s*2. Vector *t* is defined on the *l* axis and consists of 1*t* and 2*t* . Unit vectors 1*t* and 2*t* are the tangential vectors of the *s*1 and *s*2 axes, respectively. Here, = +1 2 *tt t* . However, the coordinate system in Fig. 7 has different implications from the implication in Fig. 4: the interface shown in Fig. 7 is a statistical interface obtained after integrating the elemental interfaces of Fig. 4. This means that the integrated interface is the macroscopic interface in which the interface can be recognized geometrically, such as a

*s*2

α*2*

*fff* = 1 2 + (27)

*t*2

*l*

*β*

Equation (27) is a resultant force with respect to the gradient of surface tension in the tangential direction. In this equation, force 1*f* is the gradient of the surface tension defined

$$\mathbf{E} = \frac{\overline{\rho}U\_0^2}{P\_0} \quad \mathbf{N} = \frac{L\delta^2 \overline{\rho}U\_0^2}{z\_i e V\_0} \quad \text{Re} = \frac{\overline{\rho}U\_0L}{\mu\_0} \quad \text{We} = \frac{\overline{\rho}U\_0^2 \delta}{\sigma\_0} \tag{25.a, b, c, d)}$$

$$\text{We}\_1 = \frac{\overline{\rho} U\_0^2 \lambda\_1}{\sigma\_0} \quad \text{We}\_2 = \frac{\overline{\rho} U\_0^2 \lambda\_2}{\sigma\_0} \quad \text{Kn}\_1 = \frac{\delta}{\lambda\_1} \quad \text{Kn}\_2 = \frac{\delta}{\lambda\_2} \tag{26.a, b, c, d)}$$

Of these equations, we especially focus on Eq. (25b).

Focusing on the electrical repulsive and hydrodynamic forces, we discuss a situation in which two bubbles either coalesce or bounce off each other. In terms (*a*)', (*b*)', and (*h*)' of Eq. (24), the dimensionless number N represents the relationship between the electrostatic force due to contamination at the interface and the hydrodynamic force. With respect to the dimensionless number N, we assume a situation in which bubbles A and B interact and there is a difference in the velocities of the bubbles, as shown in Fig. 6. Here, we simply take the interfacial electrostatic potential of an experimental value as the measured ζ potential of gas bubbles (Graciaa et al., 1995). The measured ζ potential is on the order of tens of millivolts. The density is 103 [kg/m3], and δ is 10−9 [m]. The elementary charge is 10−19 [C]. If the characteristic length *L* corresponding to the bubble diameter is 1 [mm], and the characteristic velocity corresponding to the relative bubble velocity between bubbles A and B is 1 [m/s], then the order of the inverse of the dimensionless number N in terms (*a*)', (*b*)', and (*h*)' becomes 10−2. This means that the electrostatic force due to contamination can be ignored when compared with the hydrodynamic force. However, if the characteristic velocity is 0.01 [m/s], the order of the inverse of the dimensionless number N becomes 102, and the electrostatic force due to contamination is significant when compared to the hydrodynamic force. This order estimation indicates that the dimensionless number N may be important for evaluating bubble coalescence and repulsion when focusing on both the electrostatic potential at the interface and the bubble velocity. In contrast, assume a situation where the relative bubble velocity between bubbles A and B is 0, as shown in Fig. 6 (i.e., the

*Vf*: Fluid velocity *u*1,2: Relative velocity *u*1,2: Relative velocity

Fig. 6. Schematic of two microbubbles

Re ρ*U L* μ<sup>=</sup>

2 0 2 <sup>2</sup> 0

σ<sup>=</sup> <sup>1</sup>

Focusing on the electrical repulsive and hydrodynamic forces, we discuss a situation in which two bubbles either coalesce or bounce off each other. In terms (*a*)', (*b*)', and (*h*)' of Eq. (24), the dimensionless number N represents the relationship between the electrostatic force due to contamination at the interface and the hydrodynamic force. With respect to the dimensionless number N, we assume a situation in which bubbles A and B interact and there is a difference in the velocities of the bubbles, as shown in Fig. 6. Here, we simply take

ρ*U* λ 0

Kn

ζ

We

Kn

1

λ<sup>=</sup> <sup>2</sup>

δ

2 0 0

2

λ

δ

<sup>=</sup> (25 a, b, c, d)

<sup>=</sup> (26 a, b, c, d)

ζ

potential is on the order of tens of

*u*1

A

*u*2 , *u*1=0

*u*2

B

is 10−9 [m]. The elementary charge is 10−19 [C].

potential

ρ*U* δ

σ

2 2 0 i 0 <sup>N</sup> *<sup>L</sup> <sup>U</sup> z eV* δ ρ= <sup>0</sup>

We

the interfacial electrostatic potential of an experimental value as the measured

δ

If the characteristic length *L* corresponding to the bubble diameter is 1 [mm], and the characteristic velocity corresponding to the relative bubble velocity between bubbles A and B is 1 [m/s], then the order of the inverse of the dimensionless number N in terms (*a*)', (*b*)', and (*h*)' becomes 10−2. This means that the electrostatic force due to contamination can be ignored when compared with the hydrodynamic force. However, if the characteristic velocity is 0.01 [m/s], the order of the inverse of the dimensionless number N becomes 102, and the electrostatic force due to contamination is significant when compared to the hydrodynamic force. This order estimation indicates that the dimensionless number N may be important for evaluating bubble coalescence and repulsion when focusing on both the electrostatic potential at the interface and the bubble velocity. In contrast, assume a situation where the relative bubble velocity between bubbles A and B is 0, as shown in Fig. 6 (i.e., the

*u*1

A

*u*2 > *u*<sup>1</sup>

*Vf*: Fluid velocity *u*1,2: Relative velocity *u*1,2: Relative velocity

B

*u*2

2 0 0 <sup>E</sup> *<sup>U</sup> P* ρ=

> ρ*U* λ

We

Of these equations, we especially focus on Eq. (25b).

of gas bubbles (Graciaa et al., 1995). The measured

millivolts. The density is 103 [kg/m3], and

*Vf*

Fig. 6. Schematic of two microbubbles

2 0 1 <sup>1</sup> 0

σ<sup>=</sup> characteristic velocity is the bubble velocity). If the bubble velocity approaches zero and the bubble diameter decreases, the inverse of the dimensionless number N increases. This indicates that the electrostatic force due to contamination at the interface is significant when compared to the hydrodynamic force, although it depends on the magnitude of the electrostatic potential at the interface. In other words, it is difficult to deform the bubble; it maintains a spherical shape except for the effect of the electrostatic force or mass transfer at the interface. In this situation, the dimensionless number N roughly explains that the breakup of a small bubble has difficult occurring if the bubble diameter is very small.
