A Review on Hydrodynamical Stability of Thin Film Flowing Along an Inclined Plane

Year 2019, , 133 - 142, 30.08.2019

Souradip Chattopadhyay , Anandamoy Mukhopadhyay Amlan Barua

https://doi.org/10.33187/jmsm.458359

Cited By: 8

Abstract

The dynamics and stability of thin liquid films have fascinated scientists over many decades. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics and biophysics. These include nanofluidics and microfluidics, lava flows, coating flows, tear-film rupture, dynamics of continental ice sheets and surfactant replacement therapy. Study of falling film instability has its wide applications in the practical field of industry and engineering. Practical applications in industrial processing motivate the recent research to investigate the factors which may affect the formation of waves on the surface of the coating layers and/or to determine the ways to overcome or to minimize the unwanted factors within the desired limit of tolerance. The dynamics of a liquid film flowing down a plane under the action of gravity is a problem which appears in many technological and natural systems, namely large scale geophysical environments such as lava flows or spillways, daily life scenarios such as water flowing down a window pane or a slippery road on a rainy day, chemical engineering processes such as evaporators, heat exchanges and falling film reactors or surface coating. The aim of this paper is to throw light on the studies conducted on hydrodynamical stability.

Keywords

Hydrodynamic stability, Thin film, Inclined plane

References

• [1] H. Schlichting, Boundary Layer Theory, McGraw-Hill Book Company Inc, New York., (1955), 308-337
• [2] C.C. Lin, Hydrodynamic Stability, Cambridge University Press, New York (1955)
• [3] D.Y. Hsieh, S.P. Ho, Wave and stability in fluids., World Scientific, Singapore (1994)
• [4] P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge(1981)
• [5] P.L. Kapitza, S.P. Kapitza, Wave flow of thin layers of a viscous fluid: III. Experimental study of undulatory flow conditions. Sov. Phys. J. Exp. Theor.Phys.,19 (1949), 105-120
• [6] T.B. Benjamin, Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2 (1957),554-574
• [7] C.S. Yih, Stability of liquid flow down an inclined plane. Phys. of Fluids 6(3) (1963), 321-334
• [8] D.J. Benny, Long waves on liquid films. J. Math. Phys. 45 (1966), 150-155
• [9] V.Ya. Shkadov, Wave models in the flow of a thin layer of a viscous liquid under the action of gravity. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 1 (1967), 43-50
• [10] V.Ya. Shkadov, Theory of wave flow of a thin layer of a viscous liquid. Izv. Akad. NaukSSSR, Mekh. Zhidk. Gaza 2 (1968), 20-25
• [11] G.I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flame, I. - Derivation of the basic equation, Acta Astronautica 4 (1977), 1177-1206
• [12] A. Pumir, P. Manneville, Y. Pomeau, On solitary waves running down an inclined plane. J. Fluid Mech. 135 (1983), 27-50
• [13] B. Gjevik, Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids. 13 (1970), 1918-1925
• [14] S.P. Lin, Finite-amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36 (1969), 113-126
• [15] C. Nakaya, Long waves on a thin fluid layer flowing down an inclined plane. Phys. Fluids. 18 (1975), 1407-1412
• [16] S.P. Lin, Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63 (1974), 417- 429
• [17] M.V.G. Krishna, S.P. Lin, Nonlinear stability of a viscous film with respect to three-dimensional side-band disturbances. Phys. Fluids 20 (1977), 1039-1044
• [18] J. Liu, J.D. Paul, J.P. Gollub, Measurements of the primary instabilities of film flows. J.Fluid Mech. 250 (1993), 69-101
• [19] J. Liu, J.P. Gollub, Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett. 70(15) (1993), 2289-2292
• [20] J. Liu, J.P. Gollub, Solitary wave dynamics of film flows. Phys. Fluids 6 (1994), 1702-1712
• [21] S.V. Alekseenko, V.E. Nakoryakov, B.G. Pokusaev, Wave formation on a vertical falling liquid film. AIChE J. 31 (1985), 1446-1460
• [22] H.C. Chang, Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26 (1994), 103-136
• [23] G.D. Fulford, The flow of liquid in thin films. Adv. Chem. Engg. 5 (1964), 151-236
• [24] S.P. Lin, C.Y. Wang, Modeling Wavy film flows. In: Encyclopedia of Fluid Mechanics (ed. Cheremisinoff, N. P.) (1985), 931-951
• [25] T.J. Hanratty, Interfacial instabilities caused by airflow over a thin liquid layer. In waves on Fluid Interfaces, Academic Press, London (1983), 221-259
• [26] T.J. Hanratty, Separated flow modeling and interfacial transport phenomena. Appl. Sci.Res. 48 (1991), 353-390
• [27] G.F. Hewitt, M.S. Hall-Taylor, Annular Two-Phase Flow. Pergamon Press, Oxford (1970)
• [28] V.E. Nakaroyakov, B.G. Pokusaev, I.R. Schreiber, Wave Propagation in Gas-Liquid Media.Begell, New York (1992)
• [29] J.R.A. Pearson, On convection cells induced by surface tension. J. Fluid Mech 4 (1958), 489- 500
• [30] L.E. Scriven, C.V. Sternling, On cellular convection driven by surface tension gradients: effects of mean surface-tension and surface viscosity. J. Fluid Mech. 19 (1964), 321-340
• [31] K.A. Smith, On convective instability induced by surface-tension gradients. J. Fluid Mech.14 (1966), 401-414
• [32] S.G. Bankoff, Significant questions in thin liquid-film heat-transfer. ASME J. Heat Transfer 116 (1994), 10-16
• [33] J.B. Burelbach, S.G. Bankoff, S.H. Davis, Nonlinear stability of evaporating/condensing liquid films. J. Fluid Mech. 195 (1988), 463-494
• [34] D.A. Goussis, R.E. Kelly, Surface waves and thermocapillary instabilities in a liquid film flow. J. Fluid Mech. 223 (1991), 24-45
• [35] S.W. Joo, S.H. Davis, S.G. Bankoff, Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers. J. Fluid Mech. 230 (1991), 117-146
• [36] P.G. Lopez, S.G. Bankoff, M.J. Miksis, Non-isothermal spreading of a thin liquid film on an inclined plane. J. Fluid Mech. 324 (1996), 261-286
• [37] O.A. Kabov, I.V. Marchuk, V. Chupin, Thermal imaging study of the liquid film flowing on a vertical surface with local heat source. Russ. J. Engng. Thermophys. 6 (1996), 105-138
• [38] O.A. Kabov, Formation of regular structures in a falling liquid film upon local heating. Thermophys. Aeromech. 5 (1998), 547-551
• [39] B. Scheid, O.A. Kabov, C. Minetti, P. Colinet, J.C. Legros, Measurement of free surface deformation by the reflectance-Schlieren method. In Proc. 3rd. European Thermal Sciences conference, (ed. E. W. P. Hahne et al.), Heidelberg (2000)
• [40] S. Kalliadasis, A. Kiyashko, E.A. Demekhin, Marangoni instability of a thin liquid film heated from below by a local heat source. J. Fluid Mech. 475 (2003), 377-408
• [41] S. Kalliadasis, E.A. Demekhin, C. Ruyer-Quil, M.G. Velarde, Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J. Fluid Mech. 492 (2003),303-338
• [42] E.A. Demekhin, V.Ya. Shkadov, Three-dimensional waves in a liquid flowing down a wall.Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 5 (1984), 21-27
• [43] C. Ruyer-Quil, B. Scheid, S. Kalliadasis, M.G. Velarde, R.Kh. Zeytounian, Thermocapillary long waves in a liquid film flow. Part 1 Low-dimensional formulation. J . FluidMech. 538 (2005), 199-222
• [44] P.M.J. Trevelyan, S. Kalliadasis, Wave dynamics on a thin-liquid film falling down a heated wall. J. Engineering Mathematics 50 (2004), 177-208
• [45] S. Miladinova, S. Slavtchev, G. Lebon, J.C. Legros, Long-wave instabilities of non- uniform heated falling films. J. Fluid Mech. 453 (2002), 153-175
• [46] S. Miladinova, D. Staykova, G. Lebon, B. Scheid, Effect of nonuniform wall heating on the three-dimensional secondary instability of falling films. Acta Mechanica 30 (2002), 1-13
• [47] A. Mukhopadhyay, A. Mukhopadhyay, Nonlinear stability of viscous film flowing down an inclined plane with linear temperature variation. Journal of Physics D Applied Physics. 40(18) (2007), 5683-5690
• [48] L.Y. Yeo, R.V. Craster, O.K. Matar, Marangoni instability of a thin liquid film resting on a locally heated horizontal wall. Physical Review E 67 (2003), 056315.1-056315.14
• [49] E.A. Demekhin, S. Kalliadasis, M.G. Velarde, Suppressing falling film instabilities by Marangoni forces. Phys. Fluids 18 (2006), 042111-1-16
• [50] B. Scheid, C. Ruyer-Quil, U. Thiele, O.A. Kabov, J.C. Legros, P. Colinet, Validity domain of the Benney equation including Marangoni effect for closed and open flows. J. Fluid Mech. 527 (2004), 303-335
• [51] B. Scheid, C. Ruyer-Quil, S. Kalliadasis, M.G. Velarde, R.Kh. Zeytounian, Thermocapillary long waves in a liquid film flow. Part 2. Linear stability and nonlinear waves. J.Fluid Mech. 538 (2005), 223-244
• [52] A. Oron, O. Gottlieb, Subcritical and supercritical bifurcations of the first and second- order Benney equations. J. Engineering Mathematics 50 (2004), 121-140
• [53] B.S. Dandapat, A. Samanta, Bifurcation analysis of first and second order Benney equations for viscoelastic fluid flowing down a vertical plane. J.Phys. D: Appl. Phys. 41 (2008), 095501
• [54] A. Oron, Nonlinear dynamics of thin evaporating liquid films subject to internal heat generation. In Fluid Dynamics at Interfaces, (ed. W. Shyy and R. Narayanan), Cambridge University Press (1999)
• [55] P. Colinet, J.C. Legros, M.G. Velarde, Nonlinear dynamics of surface-tension-driven instabilities, Wiley VCH (2001)
• [56] A.A. Nepomnyashchy, M.G. Velarde, P. Colinet, Interfacial phenomena and convection.Chapman and Hall (2002)
• [57] M.G. Velarde, R.Kh. Zeytounian, Interfacial Phenomena and the Marangoni effect. Springer (2002)
• [58] J.R. Melcher, W.J. Schwartz, Interfacial relaxation over stability in a tangential electric field. Phys. Fluids 11 (1968), 2604-2616
• [59] H. Kim, S.G. Bankoff, M.J. Misksis, The effect of an electrostatic field on film flowing down an inclined plane. Phys. Fluids 4 (1992), 2117-2130
• [60] A. Gonzalez, A. Castellanos, Nonlinear electrodynamic waves on film falling down an inclined plane. Phys. Rev. E 53 (1996), 3573-3578
• [61] A. Mukhopadhyay, B.S. Dandapat, Nonlinear stability of conducting viscous film flowing down an inclined plane at moderate Reynolds number in the presence of a uniform normal electric field. J. Phys. D: Appl. Phys. 38 (2005), 138-143
• [62] A. Mukhopadhyay, A. Mukhopadhyay, Stability of conducting viscous film flowing down an inclined plane with linear temperature variation in the presence of a uniform normal electric field. Int. J. Heat and Mass Transfer. 52 (2009), 709-715
• [63] D.Y. Hsieh, Stability of a conducting fluid flowing down an inclined plane in a magnetic field. Phys. Fluids 8 (1965), 1785-1791
• [64] Yu.P. Ladikov, Flow stability of a conducting liquid flowing down an inclined plane in the presence of a magnetic field. Fluid Dynamics 8 (1966), 1-4
• [65] P.C. Lu, G.S.R. Sarma, Magnetohydrodynamic gravity-capillary waves in a liquid film.Phys. Fluids 10 (1967), 2339-2344
• [66] A.S. Gupta, L. Rai, Hydrodynamic stability of a liquid film flowing down an inclined conducting plane. J. Phys. Soc. Japan 24 (1968), 626-632
• [67] Yu.N. Gordeev, V.V. Murzenko, Wave film flows of a conducting viscous fluid in the tangential magnetic field. Appl. Math. Theor.Phys. 3 (1990), 96-100
• [68] S. Korsunsky, Long waves on a thin layer of conducting fluid flowing down an inclined plane in an electromagnetic field. Eur. J. Mech. B/Fluids 18 (1999), 295-313
• [69] B.S. Dandapat, A. Mukhopadhyay, Finite amplitude long wave instability of a film of conducting fluid flowing down an inclined plane in presence of an electromagnetic field. Int. J.Appl. Mech. Engg. 8 (2003), 379-383
• [70] A. Mukhopadhyay, A. Mukhopadhyay, Stability of conducting liquid film flowing down an inclined plane at moderate Reynolds number in the presence of a constant electromagnetic field. Int. J. Non-Linear Mechanics. 43(7) (2008), 632-642
• [71] L.A. Davalos-Orozco, G. Ruiz-Chavarria, Hydrodynamic stability of a fluid layer flowing down a rotating inclined plane. Phys. Fluids A 4 (1992), 1651-1665
• [72] L.A. Davalos-Orozco, F.H. Busse, Instability of a thin film on a rotating horizontal or inclined plane. Physical Review E 65 (2002), 026312.1-026312.10
• [73] A. Mukhopadhyay, A. Mukhopadhyay, Stability of a thin viscous fluid film flowing down a rotating non-uniformly heated inclined plane. Acta Mechanica 216(1) (2011), 225-242
• [74] A. Mukhopadhyay, A. Mukhopadhyay, S. Mukhopadhyay, Instabilities of thin viscous film flowing down a uniformly heated inclined plane. J. of heat and mass transfer research 3 (2015), 1-22
• [75] A. Mukhopadhyay, S. Chattopadhyay, Long wave instability of thin film flowing down an inclined plane with the linear variation of thermophysical properties for very small Biot number. Int J. of Non-Linear Mechanics 100 (2018), 20-29
• [76] P.A. Sturrock, Non-linear effects in electron plasmas. A Royal Society journal 242 (1957), issue 1230
• [77] J.T. Stuart, On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behavior in plane Poiseuille flow. J. Fluid Mech. 9 (1960), 353-370
• [78] L. Debnath, Nonlinear partial differential equations for scientists and engineers. Springer (2005)
• [79] C.C. Mei, Nonlinear gravity waves in a thin sheet of viscous fluid. J. Math. Phys. 45 (1966), 266-288
• [80] P.L. Kapitza, Wave flow of thin layers of a viscous fluid. Sov. Phys. J. Exp. Theor. Phys. 18 (1948), 3-28