**3. Conclusion**

56 Mass Transfer - Advanced Aspects

<sup>4</sup> <sup>4</sup> \* \* \* \* <sup>3</sup> <sup>1</sup> r = = ; k = ; = . *g g*

 αγ

According to work [Belonozhko & Grigor'ev, 2008] the position of the reference points WA

k

; ( ) B B

*g*α ν

The fig. 5 show that for every value of wave number k there is a critical value of the surface charge density corresponding to condition W = WB(k). If the surface charge density larger than the critical value (W > WB), complex frequency is pure real and cyclic motion of liquid particle is absent (see curve 5: if W > WB then r′ > 0 and ω′ = 0). The liquid motion under these conditions can not be a wave. The real part of the complex frequency describes increment of well-known instability of the charged liquid surface relative to excess of a surface charge [Taylor, 1965; Baily, 1974]. Aperiodic growth over time is only an initial stage of the instability appearing owing to what electric forces on the free surface dominate over capillary and gravity forces. The experiments have shown that the instability leads to what a strongly nonlinear conical projections (Taylor's cones) are formed on the charged liquid surface [Taylor & McEwan, 1965]. From the peak of the cones the emission of small strongly charged droplet is occurring. Analyses of expression for WB(k) shows that if condition W > 2 is valid, there is a range of values k for which W > WB(k) and appropriated small wave perturbations (for example thermal fluctuating) are involved in formation of the Taylor's cones. This phenomenon is well-known and underlies the work of different devices for

Thereby if W > 2, we can not speak about any drift flow caused by the propagation of the

If WA < W < WB, real and imaginary parts of roots of the dispersion equation described by curves 3,4. It can be see that under these conditions there are two modes of liquid motion. Both modes are aperiodic damping (r′ < 0) and differ only in the damping rate. The cyclic motion of the liquid particle is absent (ω′ < 0). For both modes initially deformation of the liquid surface monotonically diminishes over time till complete disappearing. Liquid

If W < WB(k), there are two modes of wave liquid motion with same damping rate and ω′ ≠ 0 (see curves 1,2 at fig.5). One mode ω′1 > 0 (curve 1) correspond to wave that is propagating in direction Ox and another mode ω′2 = −ω′1 < 0 (curve 2) describes the wave that is traveling in the opposite direction. One can see that for the capillary-gravitational wave with wave number k the dimensionless circular frequency ω′ (in absolute value) decreasing with increasing value of W and vanishes if W ≥ WB(k). As was notice above the drift velocity

Summarizing the above-said we can conclude that horizontal drift caused by propagation of the periodic capillary-gravitational wave with wave number k is possible only under conditions W < 2 and W < WB(k) where W and WB(k) is defined by (43) and (44). The first condition ensures that liquid surface is not subjected to instability with respect to an excess of surface electric charge. The second condition provides circumstances under which frequency of cyclical motion of the liquid is not zero. The velocity of the drift flow decreasing with increasing of surface charge density and vanishes if values of surface charge

particles participate only in vertical motion and there is not any horizontal drift.

is proportional to the frequency and consequently behaves as it.

density is reached quantity so that W = WB(k).

2

ρ

( ) ( ) ( )

2

γ

ω

and WB depicted at the fig. 5 are defined by formulas:

AA B B

 ≡ −= ν

W W k = W 'k' W -

electrodispersion of various liquid [Baily, 1974].

capillary gravitational waves.

3 3

 ρ

 ν

<sup>=</sup>

*g*

1 1 W W k = + k' <sup>α</sup><sup>k</sup>

k' αk ≡ =+ . (44)

γ

 ρ

> Viscous forces play an important role in formation and evolution of the mean horizontal drift induced by periodic capillary-gravitational waves propagating over liquid surface. There are two components of the mean drift flow caused by the propagation of waves: main part that is called the Modified Stokes drift and supplementary part that is named the Additional flow or Additional drift.

> At low but nonzero viscosity the Modified Stokes drift behaves almost like classical Stokes drift (model of drift phenomenon without a viscosity) at all depths there is only exponential decrease in the rate of flow over time due to viscous dissipation. The physical mechanism responsible for an appearance of the Modified Stokes drift is same as that of the classical Stokes drift. For period of the wave motion a liquid particle makes approximately circular motion but returns not to initial position but is shifted a little in the direction of wave propagation. The shift is occurred due to that the lower part of the trajectory is shorter than upper since the motion decays with depth.

> A considerable contribution to the total drift flow comes from Additional drift into which the liquid is entrained by horizontal viscous stresses acting along the direction of propagation of the Modified Stokes drift. The horizontal viscous shear stresses appear between adjacent horizontal layers since velocity of Modified Stokes drift decreases with the depth. The phenomenon of the Additional drift appears exclusively in the model of a viscous liquid and is ruled out by the laws of an inviscid liquid flow.

> Velocities of the Modified Stokes drift and the Additional drift are values of the second order of smallness in wave amplitude. In present work we have offered an analytical procedure of calculation of both drift components.

> Expression for velocity of the Modified Stokes drift consists of two terms. The first term is calculated in the same way as in the case of the classical Stokes drift and it is a result of special manipulation with products of values of first order in wave amplitude. Thereby, first term of the Modified Stokes drift is expressed only via quantities that are found as result of calculating in the first order in wave amplitude velocity field caused by propagation of a capillary-gravitational wave over liquid surface.

> The second term of the Modified Stokes drift is a special particular solution of the problem of calculating of second order in wave amplitude corrections for the velocity field caused by propagation of a capillary-gravitational wave over liquid surface. This term is essential only in vicinity of liquid surface in the narrow field of viscous boundary layer and negligible in deeper layers. In the limit of almost vanishing viscosity the thickness of the surface viscous boundary layer becomes nearly zero. At the upper bounder of this layer the first term tends to zero and the second term takes care of the correct description of the drift. The best agreement of the properties classical and Modified drifts is archived only when one takes into account both terms.

> Expression for the Additional drift is derived as a special part of solution of the problem of calculating of second order in wave amplitude corrections for the velocity field caused by propagation of a capillary-gravitational wave over liquid surface.

**4** 

**Macroscopic Gas-Liquid Interfacial** 

**Mathematical Approaches** 

*1Japan Atomic Energy Agency* 

*2Kyoto University* 

*Japan* 

Yukihiro Yonemoto1 and Tomoaki Kunugi2

**Equation Based on Thermodynamic and** 

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

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

**1. Introduction** 

the interface.

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 drift is contained in dispersion equation of the analyzed problem.
