×

Inviscid instability of two-fluid free surface flow down an incline. (English) Zbl 1383.76138

Summary: The inviscid temporal stability analysis of two-fluid parallel shear flow with a free surface, down an incline, is studied. The velocity profiles are chosen as piecewise-linear with two limbs. The analysis reveals the existence of unstable inviscid modes, arising due to wave interaction between the free surface and the shear-jump interface. Surface tension decreases the maximum growth rate of the dominant disturbance. Interestingly, in some limits, surface tension destabilises extremely short waves in this flow. This can happen because of the interaction with the shear-jump interface. This flow may be compared with a corresponding viscous two-fluid flow. Though viscosity modifies the stability properties of the flow system both qualitatively and quantitatively, there is qualitative agreement between the viscous and inviscid stability analysis when the less viscous fluid is closer to the free surface.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Chandrasekhar, S.; Marshall, W. (ed.); Wilkinson, DH (ed.), Hydrodynamic and hydromagnetic stability (1961), Oxford · Zbl 0142.44103
[2] Drazin PG, Reid WH (1985) Hydrodynamic stability. Cambridge University Press, Cambridge · Zbl 0449.76027
[3] Friedlander S, Yudovich V (1999) Instabilities in fluid motion. Not Am Math Soc 46:1358-1367 · Zbl 0948.76003
[4] Lin CC (1944) On the stability of two-dimensional parallel flows. Proc Natl Acad Sci 30:316-324 · Zbl 0061.43502 · doi:10.1073/pnas.30.10.316
[5] Lin ZW (2005) Some recent results on instability of ideal plane flows. Contemp Math 371:217-229 · Zbl 1080.35086 · doi:10.1090/conm/371/06857
[6] Longuet-Higgins MS (1994) Shear instability in spilling breakers. Proc R Soc London Ser A 446:399 · Zbl 0832.76008 · doi:10.1098/rspa.1994.0111
[7] Duncan JH (2001) Spilling breakers. Annu Rev Fluid Mech 33:519 · Zbl 0988.76011 · doi:10.1146/annurev.fluid.33.1.519
[8] Triantafyllou GS, Dimas AA (1989) Interaction of two-dimensional separated flows with a free surface at low Froude numbers. Phys Fluids A 1:1813 · doi:10.1063/1.857507
[9] Olsson PJ, Henningson DS (1993) Direct optimal disturbance in watertable flow. Technical Report TRITA-MEK 1993:11, Royal Institute of Technology · Zbl 0821.76025
[10] Bakas NA, Ioannou PJ (2009) Modal and nonmodal growths of inviscid planar perturbations in shear flows with a free surface. Phys Fluids 21:024102 · Zbl 1183.76083 · doi:10.1063/1.3072617
[11] Renardy M (2009) Short wave stability for inviscid shear flow. SIAM J Appl Math 69:763-768 · Zbl 1165.76014 · doi:10.1137/080720905
[12] Kaffel A, Renardy M (2011) Surface modes in inviscid free surface shear flows. Z Angew Math Mech 91:649-652 · Zbl 1385.76003 · doi:10.1002/zamm.201000165
[13] Usha R, Tammisola O, Govindarajan R (2013) Linear stability of miscible two-fluid flow down an incline. Phys Fluids 25:104102 · Zbl 1320.76046 · doi:10.1063/1.4823855
[14] Kao TW (1968) Role of viscosity stratification in the instability of two-layer flow down an incline. J Fluid Mech 33:561-572 · Zbl 0159.58803 · doi:10.1017/S0022112068001515
[15] Sahu KC, Govindarajan R (2014) Instability of a free-shear layer in the vicinity of a viscosity-stratified layer. J Fluid Mech 752:626-648 · Zbl 1325.76082 · doi:10.1017/jfm.2014.361
[16] Yih C-S (1967) Instability due to viscosity stratification. J Fluid Mech 27:337-352 · Zbl 0144.47102 · doi:10.1017/S0022112067000357
[17] Ozgen S, Degrez G, Sarma GSR (1998) Two-fluid boundary layer instability. Phys Fluids 10:2746 · doi:10.1063/1.869798
[18] Renardy Y (1985) Instability at the interface between two shearing fluids in a channel. Phys Fluids 28:3441 · Zbl 0586.76097 · doi:10.1063/1.865346
[19] Hooper AP (1985) Long-wave instability at the interface between two viscous fluids: thin layer effects. Phys Fluids 28:1613 · Zbl 0586.76062 · doi:10.1063/1.864952
[20] Hooper AP, Boyd WGC (1983) Shear-flow instability at the interface between two viscous fluids. J Fluid Mech 128:507-528 · Zbl 0557.76044 · doi:10.1017/S0022112083000580
[21] Hooper AP, Boyd WGC (1987) Shear-flow instability due to a wall and a viscosity discontinuity at the interface. J Fluid Mech 179:201 · Zbl 0635.76033 · doi:10.1017/S0022112087001496
[22] Esch RE (1957) The instability of a shear layer between two parallel streams. J Fluid Mech 3:289-303 · Zbl 0080.39904 · doi:10.1017/S002211205700066X
[23] Miles JW (1959) On the generation of surface waves by shear flows. Part 3. J Fluid Mech 7:583-598 · Zbl 0090.42102 · doi:10.1017/S0022112059000842
[24] Lindsay KA (1984) The Kelvin-Helmholtz instability for a viscous interface. Acta Mech 52:51-61 · Zbl 0566.76041 · doi:10.1007/BF01175964
[25] Weinstein SJ, Ruschak KJ (2004) Coating flows. Annu Rev Fluid Mech 36:29 · Zbl 1081.76009 · doi:10.1146/annurev.fluid.36.050802.122049
[26] Kistler SF, Schweizer PM (1997) Liquid film coating. Chapman and Hall, London · doi:10.1007/978-94-011-5342-3
[27] Loewenherz DS, Lawrence CJ (1989) The effect of viscosity stratification on the instability of a free surface flow at low-Reynolds number. Phys Fluids A 1:1686 · Zbl 0683.76038 · doi:10.1063/1.857533
[28] Yih CS (1972) Surface waves in flowing water. J Fluid Mech 51:209-220 · Zbl 0243.76022 · doi:10.1017/S002211207200117X
[29] Hur VM, Lin Z (2008) Unstable surface waves in running water. Commun Math Phys 282:733-796 · Zbl 1170.35080 · doi:10.1007/s00220-008-0505-6
[30] Renardy M, Renardy Y (2012) On the stability of inviscid parallel shear flows with a free surface. J Math Fluid Mech 15:129-137 · Zbl 1408.76250
[31] Morland LC, Saffman PG, Yuen HC (1991) Waves generated by shear layer instabilities. Proc R Soc Lond A 433:441-450 · Zbl 0737.76024 · doi:10.1098/rspa.1991.0057
[32] Shirra VI (1993) Surface waves on shear currents: solution of the boundary-value problem. J Fluid Mech 252:565-584 · Zbl 0786.76012 · doi:10.1017/S002211209300388X
[33] Longuet-Higgins MS (1998) Instabilities of a horizontal shear flow with a free surface. J Fluid Mech 364:147-162 · Zbl 0929.76042 · doi:10.1017/S0022112098008957
[34] Voronovich AG, Lobanov ED, Rybak SA (1980) On the stability of gravitational-capillary waves in the presence of a vertically non-uniform current. Izv Atmos Ocean Phys 16:220-222
[35] Engevik L (2000) A note on the instability of a horizontal shear flow with free surface. J Fluid Mech 406:337-346 · Zbl 0987.76025 · doi:10.1017/S0022112099007612
[36] Bresch D, Renardy M (2013) Kelvin-Helmholtz instability with a free surface. Z Angew Math Phys 64:905-915 · Zbl 1271.76087 · doi:10.1007/s00033-012-0270-4
[37] Zalosh RG (1976) Discretized simulation of vortex sheet evolution with buoyancy and surface tension effects. AIAA J 14:1517-1523 · Zbl 0362.76053 · doi:10.2514/3.61493
[38] Rangel RH, Sirignano WA (1988) Nonlinear growth of Kelvin-Helmholtz instability: effect of surface tension and density ratio. Phys Fluids 31:1845-1855 · doi:10.1063/1.866682
[39] Drazin PG, Howard LN (1962) The instability to long waves of unbounded parallel inviscid flow. J Fluid Mech 14:257-283 · Zbl 0108.39502 · doi:10.1017/S0022112062001238
[40] Michalke A (1964) On the inviscid instability of the hyperbolic-tangent velocity profile. J Fluid Mech 19:543-556 · Zbl 0129.20302 · doi:10.1017/S0022112064000908
[41] Tatsumi T, Gotoh K, Ayukawa K (1964) The stability of a free boundary layer at large Reynolds numbers. J Phys Soc Jpn 19:1966-1980 · Zbl 0139.22001 · doi:10.1143/JPSJ.19.1966
[42] Pouliquen O, Chomaz JM, Huerre P (1994) Propagating Holmboe waves at the interface between two immiscible fluids. J Fluid Mech 266:277-302 · doi:10.1017/S002211209400100X
[43] Redekopp, LG; Grimshawet, R. (ed.); etal., Elements of instability theory for environmental flows (2002), Dordrecht
[44] Alabduljalil S, Rangel RH (2006) Inviscid instability of an unbounded shear layer: effect of surface tension, density and velocity profile. J Enge Math 54:99-118 · Zbl 1200.76068 · doi:10.1007/s10665-005-9017-y
[45] Yecko P, Zaleski S, Fullana J-M (2002) Viscous modes in two-phases mixing layers. Phys Fluids 14:4115 · Zbl 1185.76412 · doi:10.1063/1.1513987
[46] Boeck T, Zaleski S (2005) Viscous versus inviscid instability of two-phase mixing layers with continuous velocity profile. Phys Fluids 17:032106 · Zbl 1187.76057 · doi:10.1063/1.1862234
[47] Hinch EJ (1984) A note on the mechanism of the instability at the interface between two shearing fluids. J Fluid Mech 144:463 · doi:10.1017/S0022112084001695
[48] Otto T, Rossi M, Boeck T (2013) Viscous instability of a sheared liquid-gas interface: dependence on fluid properties and basic velocity profile. Phys Fluids 25:032103 · doi:10.1063/1.4792311
[49] Carpenter JR, Tedford EW, Heifetz E, Lawrence GA (2013) Instability in stratified shear flow: review of a physical interpretation based on interacting waves. Appl Mech Rev 64:060801 · doi:10.1115/1.4007909
[50] Guha A, Lawrence GA (2014) A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities. J Fluid Mech 755:336-364 · Zbl 1330.76022 · doi:10.1017/jfm.2014.374
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.