Dynamics of Thin Film Under a Volatile Solvent Source Driven by a Constant Pressure Gradient Flow
Abstract
:1. Introduction
2. Methods
2.1. Mathematical Formulations
2.1.1. Modeling the Inhomogeneous Vapor Phase Region
2.1.2. On the Surface Tension and Its Derivative
2.1.3. Constitutive Equations
2.1.4. Initial and Boundary Conditions
2.2. Scaling Analysis
2.2.1. Asymptotic Approximations
2.3. Leading Order Model
2.3.1. Alternative Form for the Pressure Field in the Vapor Phase
2.3.2. Method of Solution
2.3.3. Evolution Equation for the Interface
2.4. Numerical Method
2.4.1. Preliminaries
2.4.2. Computing with the COMSOL Multiphysics Software
3. Results and Discussion
3.1. Preliminaries
3.2. Case 1. The h-Evolution Equation versus the Term
3.2.1. On Spatial Inhomogeneity
3.2.2. Of the Strength of the Source
3.2.3. A Liquid Film of Infinitesimal Thickness
3.2.4. On Capillary and Marangoni Effects
3.3. Case 2. The h-Evolution Equation Versus the Term
3.3.1. Effect of an Air Blow with
3.3.2. Effect of an Air Blow with
4. Conclusions
- The discrepancy between the several results is owing to the effect of the hitherto neglected term on the dyamics of the liquid film.
- The inclusion of a constant pressure-gradient-driven flow in the vapor phase domain might be of some advantage in supporting the thinning process.
Author Contributions
Funding
Conflicts of Interest
Appendix A. Modeling Methodology with COMSOL
References
- Kim, S.; Kim, J.; Kim, H.-H. Dewetting of Liquid Film via Vapour-Mediated Marangoni Effect. J. Fluid Mech. 2019, 872, 100–114. [Google Scholar] [CrossRef]
- Li, C.; Zhao, D.; Wen, J.; Cheng, J.; Lu, X. Evolution of Entrained Water Film Thickness and Dynamics of Marangoni Flow in Marangoni Drying. RSC Adv. 2018, 8, 4995–5004. [Google Scholar] [CrossRef]
- Hérnandez-Sánchez, J.H.; Eddi, A.; Snoeijer, J.H. Marangoni Spreading due to a Localized Alcohol Supply on a Thin Water Film. Phys. Fluids 2015, 27, 032003. [Google Scholar] [CrossRef]
- Marangoni, C. On the Expansion of Liquid Floating on the Surface of Another Liquid; Tipographia dei Fratelli Fusi: Pavia, Italy, 1865. [Google Scholar]
- Thomson, J. On Certain Curious Motions Observable at the Surfaces of Wine and Other Alcoholic Liquors. Philos. Mag. 1855, 10, 330–333. [Google Scholar] [CrossRef]
- Leenaars, A.F.M.; Huethorst, J.A.M.; Van Oekel, J.J. Marangoni Drying: A New Extremely Clean Drying Process. Am. Chem. Soc. 1990, 6, 1701–1703. [Google Scholar] [CrossRef]
- Matar, O.K.; Craster, R.V. Models for Marangoni Drying. Phys. Fluids 2001, 13, 1869–1883. [Google Scholar] [CrossRef]
- O’Brien, S.B.G.M. On Marangoni Drying: Nonlinear Kinematic Waves in a Thin Film. J. Fluid Mech. 1993, 254, 649–670. [Google Scholar] [CrossRef]
- O’Brien, S.B.G.M.; Schwartz, L.M. Theory and Modeling of Thin Film Flows. Encycl. Surf. Colloid Sci. 2002, 63, 52–83. [Google Scholar]
- Carles, P.; Cazabat, A.M. Spreading of Oil Drops under a Solvent Vapor: Influence of Marangoni Effect. Progr. Colloid Polym. Sci. 1990, 82, 76–81. [Google Scholar]
- Carles, P.; Cazabat, A.M. Spreading Involving the Marangoni Effect: Some Preliminary Results. Colloids Surf. 1989, 41, 97–105. [Google Scholar] [CrossRef]
- Oron, A.; Davis, S.H.; Bankoff, S.G. Long-Scale Evolution of Thin 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, 1131–1198. [Google Scholar] [CrossRef]
- Sultan, E.; Boudaoud, A.; Amar, M.B. Evaporation of a Thin Film: Diffusion of the Vapour and Marangoni Instability. J. Fluid Mech. 2005, 543, 183–202. [Google Scholar] [CrossRef]
- Burelbach, J.P.; Bankoff, S.G.; Davis, S.H. Nonlinear Stability of Evaporating-Condensing Liquid Film. J. Fluid Mech. 1988, 195, 463–494. [Google Scholar] [CrossRef]
- Bertozzi, A.; Shearer, M.; Buckingham, R. Thin Film Traveling Waves and the Navier Slip Condition. SIAM J. Appl. Math. 2003, 63, 722–744. [Google Scholar]
© 2019 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 (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Khodabocus, M.I.; Sellier, M.; Nock, V. Dynamics of Thin Film Under a Volatile Solvent Source Driven by a Constant Pressure Gradient Flow. Fluids 2019, 4, 198. https://doi.org/10.3390/fluids4040198
Khodabocus MI, Sellier M, Nock V. Dynamics of Thin Film Under a Volatile Solvent Source Driven by a Constant Pressure Gradient Flow. Fluids. 2019; 4(4):198. https://doi.org/10.3390/fluids4040198
Chicago/Turabian StyleKhodabocus, Mohammad Irshad, Mathieu Sellier, and Volker Nock. 2019. "Dynamics of Thin Film Under a Volatile Solvent Source Driven by a Constant Pressure Gradient Flow" Fluids 4, no. 4: 198. https://doi.org/10.3390/fluids4040198