×

Effect of wall deformability on the stability of shear-imposed film flow past an inclined plane. (English) Zbl 07205485

Summary: The linear stability of shear-imposed liquid film flow past an inclined plane is examined when a soft, deformable solid layer is attached to the inclined plane. A liquid film flowing past a rigid inclined plane exhibits a long-wave gas-liquid (GL) interfacial instability which is modified by the presence of an imposed shear stress at gas-liquid interface. This GL interfacial mode does not become unstable in creeping flow limit whether a GL interfacial shear is present or not. We demonstrate that the GL interface becomes unstable even at zero Reynolds number when the shear-imposed liquid film flows past an inclined plane which is coated with soft solid layer. This GL mode instability exists only when both shear and a deformable liquid-solid (LS) interface are simultaneously present. For non-zero Reynolds number, we show that there exists multiple unstable modes originating because of the presence of deformable LS interface. These unstable LS modes become important and practically realizable only for shear-imposed liquid film flow and become irrelevant for film flows in absence of imposed shear. We also show that this LS interfacial instability dominate the stability behavior of the composite fluid film-solid system in low Reynolds number regime. Our results suggest that the shear-imposed film flow can be made unstable by using a deformable solid coating in parameter regime where the film flow otherwise remains stable in rigid wall limit. Finally, we show that a deformable solid layer can be used to obtain a stable film flow configuration for the parameter regime where the GL interfacial mode is unstable for shear imposed film flow over a rigid incline. Thus, we demonstrate the capability of a soft solid layer in manipulation and control of instabilities for shear-imposed film flows.

MSC:

76-XX Fluid mechanics
80-XX Classical thermodynamics, heat transfer
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Achenbach, D. J., Wave propagation in elastic solids (1973), John Wiley: John Wiley North-Holland · Zbl 0268.73005
[2] Alekseenko, S.; Aktershev, S.; Cherdantsev, A.; Kharlamov, S.; Markovich, D., Primary instabilities of liquid film flow sheared by turbulent gas stream, Int. J. Multiph. Flow, 35, 7, 617-627 (2009)
[3] Baingne, M.; Sharma, G., Effect of wall deformability on the stability of surfactant-laden visco-elastic liquid film falling down an inclined plane, J. Non-Newtonian Fluid Mech., 269, 1-16 (2019)
[4] Beatty, M. F.; Zhou, Z., Universal motion for a class of viscoelastic materials of differential type, Continuum Mech. Thermodyn., 3, 169-191 (1991) · Zbl 0762.73028
[5] Benjamin, T. B., Wave formation in laminar flow down an inclined plane, J. Fluid Mech., 2, 554-574 (1957) · Zbl 0078.18003
[6] Bhat, F. A.; Samanta, A., Linear stability analysis of a surfactant-laden shear-imposed falling film, Phys. Fluids, 31, 5, 054103 (2019)
[7] Blyth, M. G.; Pozrikidis, C., Effect of surfactant on the stability of flow down an inclined plane, J. Fluid Mech., 521, 241-250 (2004) · Zbl 1065.76080
[8] Chang, H.-C.; Demekhin, E. A., Complex wave dynamics on thin films (2002), Elsevier: Elsevier Amsterdam
[9] Craster, R. V.; Matar, O. K., Dynamics and stability of thin liquid films, Rev. Mod. Phys., 81, 1131-1198 (2009)
[10] Demekhin, E. A.; Kalliadasis, S.; Velarde, M. G., Suppressing falling film instabilities by Marangoni forces, Phys. Fluids, 18, 042111, 1-16 (2006) · Zbl 1185.76604
[11] Destrade, M.; Saccocmandi, G., Finite-amplitude inhomogeneous waves in Mooney-Rivlin viscoelastic solids, Wave Motion, 40, 251-262 (2004) · Zbl 1163.74338
[12] Drazin, P., Introduction to hydrodynamic stability (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0997.76001
[13] Eggert, M. D.; Kumar, S., Observations of instability, hysterisis, and oscillation in low-Reynolds number flow past polymer gels, J. Colloid Interface Sci., 274, 234-242 (2004)
[14] Fosdick, R. L.; Yu, J. H., Thermodynamics, stability and non-linear oscillations of viscoelastic solids - I. Differential type solids of second grade, Int. J. Non-Linear Mech., 31, 495-516 (1996) · Zbl 0859.73008
[15] Gaurav; Shankar, V., Stability of gravity-driven free-surface flow past a deformable solid layer at zero and finite Reynolds number, Phys. Fluids, 19, 024105 (2007) · Zbl 1146.76387
[16] Gaurav; Shankar, V., Stability of fluid flow through deformable neo-Hookean tubes, J. Fluid Mech., 627, 291-322 (2009) · Zbl 1171.76363
[17] Gaurav; Shankar, V., Stability of pressure-driven flow in a deformable neo-Hookean channel, J. Fluid Mech., 659, 318-350 (2010) · Zbl 1205.76098
[18] Gkanis, V.; Kumar, S., Instability of creeping Couette flow past a neo-Hookean solid, Phys. Fluids, 15, 2864-2871 (2003) · Zbl 1186.76193
[19] Gkanis, V.; Kumar, S., Stabilty of pressure-driven creeping flows in channels lined with a nonlinear elastic solid, J. Fluid Mech., 524, 357-375 (2005) · Zbl 1065.76070
[20] Gkanis, V.; Kumar, S., Instability of gravity-driven free-surface flow past a deformable elastic solid, Phys. Fluids, 18, 044103 (2006) · Zbl 1185.76605
[21] Grotberg, J. B., Respiratory fluid mechanics, Phys. Fluids, 23, 021301, 1-15 (2011)
[22] Hayes, M. A.; Saccocmandi, G., Finite-amplitude waves superimposed on pseudoplanar motions for Mooney-Rivlin viscoelastic solids, Non Linear Mech., 37, 1139-1146 (2002) · Zbl 1346.74023
[23] Jain, A.; Shankar, V., Instability suppression in viscoelastic film flows down an inclined plane lined with a deformable solid layer, Phys. Rev. E, 76, 046314, 1-14 (2007)
[24] Jain, A.; Shankar, V., Elastohydrodynamic suppression of free-surface instabilities in annular liquid film flow outside wires and inside tubes,, Ind. Eng. Chem. Res., 47, 6473-6485 (2008)
[25] Jiang, W. Y.; Lin, S. P., Enhancement or suppression of instability in a two-layered liquid film flow, Phys. Fluids, 17, 054105 (2005) · Zbl 1187.76247
[26] Ju, P.; Liu, Y.; Brooks, C. S.; Ishii, M., Prediction of interfacial shear stress of vertical upward adiabatic annular flow in pipes, Int. J. Heat Mass Transf., 133, 500-509 (2019)
[27] Kapitza, P., Wave flow of thin layer of viscous fluid (in russian), Zh. Eksp. Teor. Fiz., 18, 3-28 (1948)
[28] Kumaran, V., Classification of instabilities in flow past flexible surfaces, Current Sci., 79, 766-773 (2000)
[29] Kumaran, V.; Fredrickson, G. H.; Pincus, P., Flow induced instability of the interface between a fluid and a gel at low Reynolds number, J. Phys. II Fr., 4, 893-904 (1994)
[30] Kumaran, V.; Muralikrishnan, R., Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface, Phys. Rev. Lett., 84, 3310-3313 (2000)
[31] Lavalle, G.; Li, Y.; Mergui, S.; Grenier, N.; Dietze, G. F., Suppression of the Kapitza instability in confined falling liquid films, J. Fluid Mech., 860, 608â639 (2019) · Zbl 1415.76036
[32] Lin, S. P.; Chen, J. N.; Woods, D. R., Suppression of instability in a liquid film flow, Phys. Fluids, 8, 12, 3247-3252 (1996) · Zbl 1027.76566
[33] Matar, O. K.; Kumar, S., Rupture of a surfactant-sovered thin liquid film on a flexible wall, SIAM J. Appl. Math., 64, 6, 2144-2166 (2004) · Zbl 1126.76007
[34] Neelamegam, R.; Giribabu, D.; Shankar, V., Instability of viscous flow over a deformable two-layered gel: Experiments and theory, Phys. Rev. E, 90, 043004, 1-13 (2014)
[35] Neelamegam, R.; Shankar, V., Experimental study of the instability of laminar flow in a tube with deformable walls, Phys. Fluids, 27, 024102, 1-18 (2015)
[36] Neelamegam, R.; Shankar, V.; Das, D., Suppression of purely elastic instabilities in the torsional flow of viscoelastic fluid past a soft solid, Phys. Fluids, 25, 12, 124102 (2013)
[37] Patne, R.; Giribabu, D.; Shankar, V., Consistent formulations for stability of fluid flow through deformable channels and tubes, J. Fluid Mech., 827, 31-66 (2017) · Zbl 1460.76957
[38] Peng, J.; Jiang, L. Y.; Zhuge, W. L.; Zhang, Y. J., Falling film on a flexible wall in the presence of insoluble surfactant, J. Eng. Math., 97, 33-48 (2016) · Zbl 1358.76011
[39] Sahu, S.; Shankar, V., Passive manipulation of free-surface instability by deformable solid bilayers, Phys. Rev. E, 94, 013111, 1-14 (2016)
[40] Samanta, A., Effect of surfactants on the instability of a two-layer film flow down an inclined plane, Phys. Fluids, 26, 9, 094105 (2014)
[41] Shankar, V.; Kumaran, V., Asymptotic analysis of wall modes in a flexible tube revisited, Euro. Phys. J. B., 19, 607-622 (2001)
[42] Shankar, V.; Sahu, A. K., Suppression of instability in liquid flow down an inclined plane by a deformable solid layer, Phys. Rev. E, 73, 016301, 1-12 (2006)
[43] Smith, M. K., The mechanism for the long-wave instability in thin liquid films, J. Fluid Mech., 217, 469â485 (1990) · Zbl 0706.76047
[44] Squires, T. M.; Quake, S. R., Microfluidics: Fluid physics at the nanoliter scale, Rev. Mod. Phys., 77, 977-1026 (2005)
[45] Tomar, D. S.; Baingne, M.; Sharma, G., Stability of gravity-driven free surface flow of surfactant-laden liquid film flowing down a flexible inclined plane, Chem. Eng. Sci., 165, 216-228 (2017)
[46] Tomar, D. S.; Sharma, G., Manipulation and control of instabilities for surfactant-laden liquid film flowing down an inclined plane using a deformable solid layer, Phys. Fluids, 30, 014104, 1-11 (2018)
[47] Verma, M. K.S.; Kumaran, V., A dynamical instability due to fluid-wall coupling lowers the transition Reynolds number in the flow through a flexible tube, J. Fluid Mech., 705, 322-347 (2012) · Zbl 1250.76200
[48] Verma, M. K.S.; Kumaran, V., A multifold reduction in the transition Reynolds number, and ultra-fast mixing, in a micro-channel due to a dynamical instability induced by a soft wall, J. Fluid Mech., 727, 407-455 (2013) · Zbl 1291.76161
[49] Verma, M. K.S.; Kumaran, V., Stability of the flow in a soft tube deformed due to an applied pressure gradient, Phys. Rev. E, 91, 043001 (2015)
[50] Wei, H. H., Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane, Phys. Fluids, 17, 012103 (2005) · Zbl 1187.76555
[51] Weideman, J. A.; Reddy, S. C., A Matlab differentiation matrix suite, ACM Trans. Math. Softw., 26, 3270-3282 (2000)
[52] Weinstein, S. J.; Ruschak, K. J., Coating flows, Annu. Rev. Fluid Mech., 36, 29-53 (2004) · Zbl 1081.76009
[53] Whitaker, S., Effect of surface active agents on the stability of falling liquid films, Ind. Eng. Chem. Fun., 3, 2, 132-142 (1964)
[54] Yih, C. S., Stability of liquid flow down an inclined plane, Phys. Fluids, 6, 321-334 (1963) · Zbl 0116.19102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.