**3.4 Consideration of surface tension gradient**

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) differs from that of the conventional one. In this section, we discuss term (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 plane or spherical surface. Based on these concepts, we discuss term (d).

Fig. 7. Statistical local interface

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

$$f = f\_1 + f\_2 \tag{27}$$

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

Macroscopic Gas-Liquid Interfacial Equation

tensions of the *s*1 and *s*2 directions is imbalanced.

curvature. The concrete equation is as follows (Butt et al., 2003).

κ

flat and with a curvature, respectively. *H* [1/m3],

**4. Consideration of Kelvin equation** 

In this equation, \* *P* and *P*

flow equation.

same procedure.

smaller than that of <sup>2</sup>

the order of a macroscopic scale.

following equation is derived.

ρ ε

ερ

Based on Thermodynamic and Mathematical Approaches 73

In this equation, *t* is *t/t* . By replacing term (d) in Eq. (23) with Eq. (31), we establish that the thermodynamic jump condition agrees with the conventional condition except for terms (e)–(j) in Eq. (23). This result suggests that the conventional jump condition holds under the conditions of Eqs. (20) and (30). Therefore, the conventional jump condition is restricted to spherical bubbles or droplets and is inaccurate when the relationship between the surface

We consider the equilibrium state where the gas and liquid phases coexist and temperature is constant. The gas–liquid interface is a flat surface. In this situation, the amount of evaporated liquid from the gas phase to the liquid phase is determined by the relationship between the saturated vapor pressure and ambient pressure if the temperature is constant. The vapor in the gas phase condenses to the liquid phase when the ambient pressure is increased to greater than the saturated vapor pressure. On the other hand, the liquid evaporates to the gas phase when the ambient pressure is decreased to less than the saturated vapor pressure. This is formulated by considering the changes in chemical potential thermodynamically. The same discussion is applied to a bubble and droplet. However, a bubble or droplet has a curvature. The vapor pressure of a droplet takes a different value when the interface has a

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

<sup>=</sup> ⎜ ⎟ ⎝ ⎠

*T* [K] are the mean curvature, surface tension coefficient, specific volume, gas constant, and temperature, respectively. Equation (32) is mainly derived based on both the Gibbs–Duhem equation and Young-Laplace equation from the thermodynamic point of view. In this section, we show that the Kelvin equation can be derived from the multi-scale multiphase

The Chapman–Enskog expansion is applied to the conventional Navier–Stokes equation to derive the multi-scale multiphase flow equation from which the thermodynamic interfacial jump condition is finally derived. In this section, the Kelvin equation is derived using the

Figure 9 shows a schematic of a multi-scale concept around the interface. *O*(1) represents

DD D , DD D *tt t*

 ′ ′′ + + =− ⋅ − ⋅ − ⋅ + ′ ′′ *uu u* ∇

⎛ ⎞ κ

**4.1 Derivation of Kelvin equation from multi-scale multiphase flow equation** 

 and <sup>2</sup> ε

ε

each scale. Based on this assumption, we consider the Kelvin equation.

By considering the Chapman–Enskog expansion in Eq. (7) and

(0) (1) (2)

 ε ρ

R *P HVm P T*

σ

σ

represent the vapor pressures where the gas–liquid interface is

represent mesoscopic scales: the scale of

∇

ε

εε

∇

*T T Tg*

. However, we assume that the continuum approximation holds in

2 (0) (1) 2 (2)

εis

(33)

until the second order, the

 ρ

(32)

[N/m], *Vm* [m3/kg], R [J/kg K], and

$$\mathbf{f}\_1 = \mathbf{t}\_1 \frac{1}{2} \frac{\partial}{\partial \ s\_1} \frac{\sigma}{s\_1} \tag{28}$$

$$\mathbf{f}\_2 = \mathbf{t}\_2 \frac{\mathbf{l}}{2} \frac{\partial}{\partial s\_2} \mathbf{s}\_2 \tag{29}$$

Fig. 8. Vector diagram in the s1–s2 tangential plane: (a) Unit base vector *t*1, *t*2 (*t*=*t*1+*t*2), (b) Surface force *f*1, *f*2(*f*=*f*1+*f*2)

Therefore, an image of Eq. (27) is represented by a vector diagram, as shown in Fig. 8. The direction of *f* is arbitrary in the *s*1–*s*2 plane, as shown in Fig. 8b, because the magnitude of <sup>1</sup>*f* is not always equal to that of <sup>2</sup> *f* . However, if the magnitude of 1*f* is equal to that of <sup>2</sup> *f* , then

$$\frac{\stackrel{\partial}{\partial}\frac{\sigma}{\mathbf{s}\_{1}}=\frac{\stackrel{\partial}{\partial}\frac{\sigma}{\mathbf{s}\_{2}}}{\stackrel{\partial}{\partial}\mathbf{s}\_{2}}\tag{30}$$

This equation indicates that the direction of *f* is parallel to that of the *l* axis. Therefore, the resultant force *f* is revaluated by using vector *t* , which is a unit tangential vector of the *l* axis. Eventually, Equation (27) is transformed as follows:

$$\begin{split} \mathbf{f} &= \left( \mathbf{t}\_1 \frac{1}{2} \frac{\partial}{\partial} \frac{\sigma}{s\_1} + \mathbf{t}\_2 \frac{1}{2} \frac{\partial}{\partial} \frac{\sigma}{s\_2} \right) \\ &= \frac{1}{2} \frac{\mathbf{d}}{\mathbf{d}} \frac{\sigma}{l} \left( \mathbf{t}\_1 \frac{\partial}{\partial} \frac{l}{s\_1} + \mathbf{t}\_2 \frac{\partial}{\partial} \frac{l}{s\_2} \right) \\ &= \frac{\sqrt{2}}{2} \frac{\mathbf{d}}{\mathbf{d}} \frac{\sigma}{l} \left( \mathbf{t}\_1 + \mathbf{t}\_2 \right) \\ &= \frac{\mathbf{d}}{\mathbf{d}} \frac{\sigma}{l} \tilde{t} \end{split} \tag{31}$$

1 2 *s* ∂ σ

1 2 *s* ∂ σ

1

2

*s*2

 *f*<sup>2</sup>

<sup>=</sup> <sup>∂</sup> *f t* (28)

<sup>=</sup> <sup>∂</sup> *f t* (29)

*f*

<sup>=</sup> <sup>∂</sup> <sup>∂</sup> (30)

*f*<sup>1</sup> *s*<sup>1</sup>

(31)

*l*

1 1

2 2

Fig. 8. Vector diagram in the s1–s2 tangential plane: (a) Unit base vector *t*1, *t*2 (*t*=*t*1+*t*2),

∂ σ ∂ σ

Therefore, an image of Eq. (27) is represented by a vector diagram, as shown in Fig. 8. The direction of *f* is arbitrary in the *s*1–*s*2 plane, as shown in Fig. 8b, because the magnitude of <sup>1</sup>*f* is not always equal to that of <sup>2</sup> *f* . However, if the magnitude of 1*f* is equal to that

1 2 *s s*

This equation indicates that the direction of *f* is parallel to that of the *l* axis. Therefore, the resultant force *f* is revaluated by using vector *t* , which is a unit tangential vector of the *l*

1 2

σ

*l*

= +

*t*

σ

1 d 2 d 2 d 2 d d d

σ

*l*

=

σ

*ft t*

1 1 2 2

⎛ ⎞ ∂ ∂ = + ⎜ ⎟ ∂ ∂ ⎝ ⎠

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

( )

*t t*

1 2

1 2

*s s*

 σ

*l l*

1 2 1 2

*t t*

*ls s*

*s*1

*l*

*t*1

axis. Eventually, Equation (27) is transformed as follows:

(b) Surface force *f*1, *f*2(*f*=*f*1+*f*2)

of <sup>2</sup> *f* , then

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

*s*2

In this equation, *t* is *t/t* . By replacing term (d) in Eq. (23) with Eq. (31), we establish that the thermodynamic jump condition agrees with the conventional condition except for terms (e)–(j) in Eq. (23). This result suggests that the conventional jump condition holds under the conditions of Eqs. (20) and (30). Therefore, the conventional jump condition is restricted to spherical bubbles or droplets and is inaccurate when the relationship between the surface tensions of the *s*1 and *s*2 directions is imbalanced.
