Three-Dimensional Long-Wave Instability of an Evaporation/Condensation Film
Abstract
:1. Introduction
2. Mathematical Formulation
2.1. Nondimensionalization
2.2. Long-Wave Model
3. Linear Stability Analysis
4. Nonlinear Evolution
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
Liquid (vapor) phase density | |
Liquid (vapor) phase kinematic viscosity | |
Liquid (vapor) phase thermal conductivity | |
Velocity in x, y, z component, respectively | |
() | Interface (vapor) velocity |
t, | Time |
p | Pressure |
Liquid dynamic viscosity | |
g | Acceleration of gravity |
Thermal diffusivity | |
∇ | Hamilton operator |
Surface gradient operator | |
Slope angle | |
h | Liquid film thickness |
Unit normal vector | |
Unit tangent vector | |
Aspect ratio | |
Complex growth rate | |
, | Streamwise and spanwise wavenumbers |
Surface tension | |
Interfacial free-energy | |
Entropy densities | |
Stress tensor | |
Identity tensor | |
Interface modulus | |
Dimensionless temperature | |
Thermodynamic equilibrium parameter | |
Effective pressure coefficient | |
C | Crispation number |
D | Density ratio |
E | Evaporation number |
G | Dimensionless gravity |
J | Mass flux conservation |
K | Mean curvature |
L | Latent heat |
Marangoni number | |
N | Energy flux coefficient |
T, , | Temperature, the wall temperature, the saturation temperature |
Prandtl number | |
S | Dimensionless surface tension |
Appendix A. The First-Order Growth Rate
References
- Grotberg, J.B. Pulmonary flow and transport phenomena. Annu. Rev. Fluid Mech. 1994, 26, 529–571. [Google Scholar] [CrossRef]
- Goldstein, R.J. Film cooling. In Advances in Heat Transfer; Elsevier: Amsterdam, The Netherlands, 1971; Volume 7, pp. 321–379. [Google Scholar]
- Mu, X.; Yang, Y.; Shen, S.; Liang, G.; Gong, L. Experimental study of heat transfer characteristics for horizontal-tube falling film evaporation. In Heat Transfer Summer Conference; American Society of Mechanical Engineers: New York, NY, USA, 2012; Volume 44786, pp. 865–871. [Google Scholar]
- Dai, Z.; Zhang, Y.; Wang, S.; Nawaz, K.; Jacobi, A. Falling-film heat exchangers used in desalination systems: A review. Int. J. Heat Mass Transf. 2022, 185, 122407. [Google Scholar] [CrossRef]
- Burelbach, J.P.; Bankoff, S.G.; Davis, S.H. Nonlinear stability of evaporating/condensing liquid films. J. Fluid Mech. 1988, 195, 463. [Google Scholar] [CrossRef]
- Davis, S.H. Thermocapillary instabilities. Annu. Rev. Fluid Mech. 1987, 19, 403–435. [Google Scholar] [CrossRef]
- Joo, S.W.; Davis, S.H.; Bankoff, S.G. Long-wave instabilities of heated falling films: Two-dimensional theory of uniform layers. J. Fluid Mech. 1991, 230, 117–146. [Google Scholar] [CrossRef]
- Reisfeld, B.; Bankoff, S.G. Nonlinear stability of a heated thin liquid film with variable viscosity. Phys. Fluids Fluid Dyn. 1990, 2, 2066–2067. [Google Scholar] [CrossRef]
- Oron, A.; Davis, S.H.; Bankoff, S.G. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 1997, 69, 931–980. [Google Scholar] [CrossRef]
- Craster, R.V.; Matar, O.K. Dynamics and stability of thin liquid films. Rev. Mod. Phys. 2009, 81, 1131. [Google Scholar] [CrossRef]
- Chattopadhyay, S.; Mukhopadhyay, A.; Barua, A. A review on hydrodynamical stability of thin film flowing along an inclined plane. J. Math. Sci. Model. 2019, 2, 133–142. [Google Scholar] [CrossRef]
- Sheludko, A. Thin liquid films. Adv. Colloid Interface Sci. 1967, 1, 391–464. [Google Scholar] [CrossRef]
- Ruckenstein, E.; Jain, R.K. Spontaneous rupture of thin liquid films. J. Chem. Soc. Faraday Trans. Mol. Chem. Phys. 1974, 7, 132–147. [Google Scholar] [CrossRef]
- Vrij, A. Possible mechanism for the spontaneous rupture of thin, free liquid films. Discuss. Faraday Soc. 1966, 42, 23–33. [Google Scholar] [CrossRef]
- Yih, C.-S. Stability of parallel laminar flow with a free surface. J. Appl.-Mech.-Trans. ASME 1954, 21, 281. [Google Scholar]
- Yih, C. Stability of liquid flow down an inclined plane. Phys. Fluids 1963, 6, 321–334. [Google Scholar] [CrossRef]
- Benjamin, T. Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 1957, 2, 554–573. [Google Scholar] [CrossRef]
- Benney, D.J. Long waves on liquid films. J. Math. Phys. 1966, 45, 150–155. [Google Scholar] [CrossRef]
- Williams, M.B.; Davis, S.H. Nonlinear theory of film rupture. J. Colloid Interface Sci. 1982, 90, 220–228. [Google Scholar] [CrossRef]
- Sharma, A.; Ruckenstein, E. An analytical nonlinear theory of thin film rupture and its application to wetting films. J. Colloid Interface Sci. 1986, 113, 456–479. [Google Scholar] [CrossRef]
- Mezger, M.; Reichert, H.; Schöer, S.; Okasinski, J.; Schrxoxer, H.; Dosch, H.; Palms, D.; Ralston, J.; Honkimäi, V. High-resolution in situ X-ray study of the hydrophobic gap at the Octadecyl-Trichlorosilane Interface. Proc. Natl. Acad. Sci. USA 2006, 103, 401–404. [Google Scholar] [CrossRef]
- Lin, F.-Y.; Steffen, W. Capillary wave dynamics of thin liquid polymer films. J. Chem. Phys. 2014, 141, 104903. [Google Scholar] [CrossRef]
- Bankoff, S.G. Stability of liquid flow down a heated inclined plane. Int. J. Heat Mass Transf. 1971, 14, 377–385. [Google Scholar] [CrossRef]
- Abderrahmane, H.A. Stability of an evaporating and condensing liquid film flowing down an inclined plane. Energy Procedia 2017, 142, 3944–3949. [Google Scholar] [CrossRef]
- Sreenivasan, S.; Lin, S. Surface tension driven instability of a liquid film flow down a heated incline. Int. J. Heat Mass Transf. 1978, 21, 1517–1526. [Google Scholar] [CrossRef]
- Kelly, R.E.; Davis, S.H.; Goussis, D.A. On the instability of heated film flow with variable surface tension. Int. Heat Transf. Conf. 1986, 8, 1937–1942. [Google Scholar]
- Kalliadasis, S.; Demekhin, E.; Ruyer-Quil, C.; Velarde, M. Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J. Fluid Mech. 2003, 492, 303–338. [Google Scholar] [CrossRef]
- Thiele, U.; Knobloch, E. Thin liquid films on a slightly inclined heated plate. Phys. Nonlinear Phenom. 2004, 190, 213–248. [Google Scholar] [CrossRef]
- Mukhopadhyay, A.; Mukhopadhyay, S.; Mukhopadhyay, A. Instabilities of thin viscous liquid film flowing down a uniformly heated inclined plane. J. Heat Mass Transf. Res. 2016, 3, 77–87. [Google Scholar]
- Scheid, B.; Oron, A.; Colinet, P.; Thiele, U.; Legros, J.C. Nonlinear evolution of nonuniformly heated falling liquid films. Phys. Fluids 2002, 14, 4130–4151. [Google Scholar] [CrossRef]
- Gambaryan-Roisman, T. Marangoni convection, evaporation and interface deformation in liquid films on heated substrates with non-uniform thermal conductivity. Int. J. Heat Mass Transf. 2010, 53, 390–402. [Google Scholar] [CrossRef]
- Ding, Z. Falling liquid films on a slippery substrate with marangoni effects. Int. J. Heat Mass Transf. 2015, 90, 689–701. [Google Scholar] [CrossRef]
- Chattopadhyay, S.; Mukhopadhyay, A.; Barua, A.K.; Gaonkar, A.K. Thermocapillary instability on a film falling down a non-uniformly heated slippery incline. Int. J.-Non-Linear Mech. 2021, 133, 103718. [Google Scholar] [CrossRef]
- Moussy, C.; Lebon, G.; Margerit, J. Influence of evaporation on bénard-marangoni instability in a liquid-gas bilayer with a deformable interface. Eur. Phys. J.-Condens. Matter Complex Syst. 2004, 40, 327–335. [Google Scholar] [CrossRef]
- Liu, R.; Liu, Q. The convective instabilities in a liquid–vapor system with a non-equilibrium evaporation interface. Microgravity Sci. Technol. 2009, 21, 233–240. [Google Scholar] [CrossRef]
- Liu, R.; Kabov, O. Instabilities in a horizontal liquid layer in cocurrent gas flow with an evaporating interface. Phys. Rev. 2012, 85, 066305. [Google Scholar] [CrossRef] [PubMed]
- Palmer, H.J. The hydrodynamic stability of rapidly evaporating liquids at reduced pressure. J. Fluid Mech. 1976, 75, 487. [Google Scholar] [CrossRef]
- Kanatani, K. Interfacial instability induced by lateral vapor pressure fluctuation in bounded thin liquid-vapor layers. Phys. Fluids 2010, 22, 012101. [Google Scholar] [CrossRef]
- Kanatani, K.; Oron, A. Nonlinear dynamics of confined thin liquid-vapor bilayer systems with phase change. Phys. Fluids 2011, 23, 032102. [Google Scholar] [CrossRef]
- Cammenga, H.K. Evaporation mechanisms of liquids. Curr. Top. Mater. Sci. 1980, 5, 335–446. [Google Scholar]
- Shklyaev, O.E.; Fried, E. Stability of an evaporating thin liquid film. J. Fluid Mech. 2007, 584, 157–183. [Google Scholar] [CrossRef]
- Anderson, D.M.; Cermelli, P.; Fried, E.; Gurtin, M.E.; Mcfadden, G.B. General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids. J. Fluid Mech. 2007, 581, 323–370. [Google Scholar] [CrossRef]
- Fried, E.; Shen, A.Q.; Gurtin, M.E. Theory for solvent, momentum, and energy transfer between a surfactant solution and a vapor atmosphere. Phys. Rev. E 2006, 73, 061601. [Google Scholar] [CrossRef] [PubMed]
- Wayner, P.C. Intermolecular forces in phase-change heat transfer: 1998 kern award review. AIChE J. 1999, 45, 2055–2068. [Google Scholar] [CrossRef]
- Wei, T.; Duan, F. Interfacial stability and self-similar rupture of evaporating liquid layers under vapor recoil. Phys. Fluids 2016, 28, 124106. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Jiang, W.; Huang, R.; Yang, Q.; Ding, Z. Three-Dimensional Long-Wave Instability of an Evaporation/Condensation Film. Fluids 2024, 9, 143. https://doi.org/10.3390/fluids9060143
Jiang W, Huang R, Yang Q, Ding Z. Three-Dimensional Long-Wave Instability of an Evaporation/Condensation Film. Fluids. 2024; 9(6):143. https://doi.org/10.3390/fluids9060143
Chicago/Turabian StyleJiang, Weiyang, Ruiqi Huang, Qiang Yang, and Zijing Ding. 2024. "Three-Dimensional Long-Wave Instability of an Evaporation/Condensation Film" Fluids 9, no. 6: 143. https://doi.org/10.3390/fluids9060143