×

zbMATH — the first resource for mathematics

The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow. (English) Zbl 1381.76020
Summary: The viscoelastic analogue to the Newtonian Orr amplification mechanism is examined using linear theory. A weak, two-dimensional Gaussian vortex is superposed onto a uniform viscoelastic shear flow. Whilst in the Newtonian solution the spanwise vorticity perturbations are simply advected, the viscoelastic behaviour is markedly different. When the polymer relaxation rate is much slower than the rate of deformation by the shear, the vortex splits into a new pair of co-rotating but counter-propagating vortices. Furthermore, the disturbance exhibits a significant amplification in its spanwise vorticity as it is tilted forward by the shear. Asymptotic solutions for an Oldroyd-B fluid in the limits of high and low elasticity isolate and explain these two effects. The splitting of the vortex is a manifestation of vorticity wave propagation along the tensioned mean-flow streamlines, while the spanwise vorticity growth is driven by the amplification of a polymer torque perturbation. The analysis explicitly demonstrates that the polymer torque amplifies as the disturbance becomes aligned with the shear. This behaviour is opposite to the Orr mechanism for energy amplification in Newtonian flows, and is therefore labelled a ’ ‘reverse-Orr’ mechanism. Numerical evaluations of vortex evolutions using the more realistic FENE-P model, which takes into account the finite extensibility of the polymer chains, show the same qualitative behaviour. However, a new form of stress perturbation is established in regions where the polymer is significantly stretched, and results in an earlier onset of decay.

MSC:
76A10 Viscoelastic fluids
76A05 Non-Newtonian fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1016/j.jnnfm.2006.01.005 · Zbl 1195.76174
[2] Bender, Advanced Mathematical Methods for Scientists and Engineers (1978)
[3] DOI: 10.1063/1.4851295 · Zbl 06485889
[4] DOI: 10.1017/S0022112095001157 · Zbl 0842.76027
[5] DOI: 10.1017/S0022112003005305 · Zbl 1063.76580
[6] DOI: 10.1103/PhysRevLett.110.174502
[7] DOI: 10.1017/jfm.2013.686
[8] DOI: 10.1017/S0022112003004610 · Zbl 1054.76041
[9] DOI: 10.1017/S0022112088001302
[10] DOI: 10.1017/jfm.2013.114 · Zbl 1287.76101
[11] DOI: 10.1017/S0022112080000122 · Zbl 0428.76049
[12] DOI: 10.1016/0377-0257(93)85014-2 · Zbl 0812.76036
[13] DOI: 10.1016/j.jnnfm.2005.03.002 · Zbl 1187.76658
[14] DOI: 10.1016/S0377-0257(98)00095-0 · Zbl 0946.76020
[15] DOI: 10.1017/S0022112007006611 · Zbl 1175.76069
[16] DOI: 10.1016/j.jnnfm.2011.02.010 · Zbl 1282.76052
[17] Jovanović, Phys. Fluids 22 (2010) · Zbl 1183.76263
[18] DOI: 10.1088/1367-2630/9/10/360
[19] Jiménez, Phys. Fluids 25 (2013)
[20] DOI: 10.1017/S0022112009006223 · Zbl 1171.76364
[21] DOI: 10.1017/jfm.2013.572 · Zbl 1294.76119
[22] DOI: 10.1017/S0022112008000633 · Zbl 1151.76372
[23] DOI: 10.1017/jfm.2011.541 · Zbl 1250.76127
[24] DOI: 10.1063/1.861735
[25] DOI: 10.1017/S0022112010000066 · Zbl 1189.76326
[26] DOI: 10.1146/annurev.fluid.40.111406.102156 · Zbl 1229.76043
[27] DOI: 10.1063/1.1345882 · Zbl 1184.76137
[28] DOI: 10.1017/S0022112000008818 · Zbl 0948.76521
[29] DOI: 10.1098/rsta.1991.0064 · Zbl 0719.76518
[30] DOI: 10.1063/1.858386
[31] DOI: 10.1017/S0022112097007568 · Zbl 0905.76037
[32] DOI: 10.1002/aic.14328
[33] Bird, Dynamics of Polymeric Liquids 1 (1987)
[34] DOI: 10.1038/35011019
[35] DOI: 10.1017/S0022112090001045
[36] DOI: 10.1063/1.4895780 · Zbl 06500102
[37] Townsend, The Structure of Turbulent Shear Flow (1976) · Zbl 0325.76063
[38] DOI: 10.1103/PhysRevLett.57.2160
[39] DOI: 10.1175/1520-0469(1987)044<2191:DDIS>2.0.CO;2
[40] DOI: 10.1080/14685248.2014.952430
[41] DOI: 10.1017/S0022112094001254 · Zbl 0810.76023
[42] DOI: 10.1017/S0022112004000291 · Zbl 1067.76052
[43] DOI: 10.1063/1.869229
[44] DOI: 10.1017/S0022112006000607 · Zbl 1095.76021
[45] DOI: 10.1007/s10494-005-9002-6 · Zbl 1200.76106
[46] DOI: 10.1073/pnas.1219666110
[47] DOI: 10.1017/jfm.2014.586
[48] DOI: 10.1063/1.4820142 · Zbl 06480304
[49] Saffman, Vortex Dynamics (1992)
[50] Drazin, Hydrodynamic Stability (1995)
[51] DOI: 10.1146/annurev.fl.23.010191.003125
[52] DOI: 10.1063/1.4906441 · Zbl 06518760
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.