**5. Conclusion**

In a previous study, a new interfacial model of the gas–liquid interface was developed based on thermodynamics, assuming that the interface has a finite thickness, similar to a thin fluid membrane. In particular, the free energy was derived based on a lattice-gas model that includes the electrostatic potential due to contamination. The free energy was incorporated into the NS equation by using the Chapman–Enskog expansion. Finally, a multi-scale multiphase flow equation was derived that characterizes the mesoscopic scale. The interfacial equation for a macroscopic-scale gas–liquid interface is characterized by a jump condition. In the present study, the jump condition at the gas–liquid interface treated by thermodynamics was derived by using the multi-scale multiphase flow equation and compared with the conventional jump condition. Finally, we developed the multi-scale gas– liquid interfacial model; this model supports the interfacial phenomena from the microscopic to macroscopic scale theoretically to give the following results:

1. The thermodynamic interfacial jump condition was derived by using the multi-scale multiphase flow equation. The present study indicated the relationship between the

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In our recent study (Yonemoto & Kunugi, 2010c), the momentum jump condition was applied to a discussion on an equilibrium condition at a three-phase contact line of a sessile droplet on a smooth solid surface. Furthermore, the equilibrium condition of the sessile droplet was also considered by the thermodynamic approach (Yonemoto & Kunugi, 2009). Therefore, our new interfacial model may allow the development of a general multi-scale interfacial model that can treat two-phase and three-phase interfaces theoretically in the future.

### **6. References**


2. The Marangoni-effect terms, which are related to temperature differences and contamination at the interface, were included in some additional terms derived under

3. From the normalized thermodynamic jump condition, we obtained a new dimensionless number N that represents the relationship between the electrostatic force due to contamination at the interface and the hydrodynamic force. The order estimation of N suggests that we may be able to specifically classify bubble coalescence or breakup. 4. Considering term (d) of Eq. (23), we concluded that the conventional jump condition holds true for Eqs. (20) and (30). Therefore, the conventional jump condition is restricted to the case of spherical bubbles or droplets and is inaccurate when the relationship between the surface tensions in the *s*1 and *s*2 tangential directions is

5. On the basis of the multi-scale multiphase flow equation, we derived the Kelvin equation. This result indicates that equation (37) contains the physics for the evaporation/condensation of a curved surface and will support other interfacial phenomena characterized by thermodynamics. However, more detailed discussion of Eq. (51) is needed because other terms in Eq. (41) are omitted in the derivation of Eq.

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imbalanced.

(51).

future.

**6. References** 

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the thermodynamic jump condition.

mechanical and thermodynamic approaches with respect to the model of the gas–liquid


**0**

**5**

*South Africa*

**Heat and Mass Transfer from an Inverted Cone in a Porous Medium with Cross-Diffusion Effects**

The study of double-diffusive convection has received considerable attention during the last several decades since this occurs in a wide range of natural settings. The origins of these studies can be traced to oceanography when hot salty water lies over cold fresh water of a higher density resulting in double-diffusive instabilities known as "salt-fingers," Stern (35; 36). Typical technological motivations for the study of double-diffusive convection range from such diverse fields as the migration of moisture through air contained in fibrous insulations, grain storage systems, the dispersion of contaminants through water-saturated soil, crystal growth and the underground disposal of nuclear wastes. Double-diffusive convection has also been cited as being of particular relevance in the modeling of solar ponds (Akbarzadeh

Double-diffusive convection problems have been investigated by, among others, Nield (28) Baines and Gill (3), Guo et al. (14), Khanafer and Vafai (17), Sunil et al. (37) and Gaikwad et al. (13). Studies have been carried out on horizontal, inclined and vertical surfaces in a porous medium by, among others, Cheng (9; 10), Nield and Bejan (29) and Ingham and Pop (32). Na and Chiou (24) presented the problem of laminar natural convection in Newtonian fluids over the frustum of a cone while Lai (18) investigated the heat and mass transfer by natural convection from a horizontal line source in saturated porous medium. Natural convection over a vertical wavy cone has been investigated by Pop and Na (33). Nakyam and Hussain (25) studied the combined heat and mass transfer by natural convection in a porous medium

Chamkha and Khaled (4) studied the hydromagnetic heat and mass transfer by mixed convection from a vertical plate embedded in a uniform porous medium. Chamkha (5) investigated the coupled heat and mass transfer by natural convection of Newtonian fluids about a truncated cone in the presence of magnetic field and radiation effects and Yih (38) examined the effect of radiation in convective flow over a cone. Cheng (6) used an integral approach to study the heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and concentration. Khanafer and Vafai (17) studied the double-diffusive convection in a lid-driven enclosure filled with a fluid-saturated porous medium. Mortimer and Eyring (22) used an elementary transition state approach to obtain a simple model for Soret and Dufour effects in thermodynamically ideal mixtures of substances with molecules of nearly equal size. In their model the flow of heat in the Dufour effect was identified as the transport of the enthalpy change of activation as molecules diffuse.

and Manins (1)) and magma chambers (Fernando and Brandt (12)).

**1. Introduction**

by integral methods.

Faiz GA Awad, Precious Sibanda and Mahesha Narayana *School of Mathematical Sciences, University of KwaZulu-Natal*

