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Perturbative expansions of the conformation tensor in viscoelastic flows. (English) Zbl 1415.76015
Summary: We consider the problem of formulating perturbative expansions of the conformation tensor, which is a positive definite tensor representing polymer deformation in viscoelastic flows. The classical approach does not explicitly take into account that the perturbed tensor must remain positive definite – a fact that has important physical implications, e.g. extensions and compressions are represented similarly to within a negative sign, when physically the former are unbounded and the latter are bounded from below. Mathematically, the classical approach assumes that the underlying geometry is Euclidean, and this assumption is not satisfied by the manifold of positive definite tensors. We provide an alternative formulation that retains the conveniences of classical perturbation methods used for generating linear and weakly nonlinear expansions, but also provides a clear physical interpretation and a mathematical basis for analysis. The approach is based on treating a perturbation as a sequence of successively smaller deformations of the polymer. Each deformation is modelled explicitly using geodesics on the manifold of positive definite tensors. Using geodesics, and associated geodesic distances, is the natural way to model perturbations to positive definite tensors because it is consistent with the manifold geometry. Approximations of the geodesics can then be used to reduce the total deformation to a series expansion in the small perturbation limit. We illustrate our approach using direct numerical simulations of the nonlinear evolution of Tollmien-Schlichting waves.

##### MSC:
 76A10 Viscoelastic fluids 76A05 Non-Newtonian fluids
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##### References:
 [1] Agarwal, A.; Brandt, L.; Zaki, T. A., Linear and nonlinear evolution of a localized disturbance in polymeric channel flow, J. Fluid Mech., 760, 278-303, (2014) [2] Avgousti, M.; Beris, A. N., Viscoelastic Taylor-Couette flow: bifurcation analysis in the presence of symmetries, Proc. R. Soc. Lond. A, 443, 1917, 17-37, (1993) · Zbl 0791.76024 [3] Benney, D. J.; Lin, C. C., On the secondary motion induced by oscillations in a shear flow, Phys. Fluids, 3, 4, 656-657, (1960) [4] Cardano, G.; Witmer, T. R., Ars Magna, or, The Rules of Algebra, (1993), Dover [5] Cioranescu, D.; Girault, V.; Rajagopal, K., Mechanics and Mathematics of Fluids of the Differential Type, 35, (2016), Springer International Publishing · Zbl 1365.76001 [6] Doering, C. R.; Eckhardt, B.; Schumacher, J., Failure of energy stability in Oldroyd-B fluids at arbitrarily low Reynolds numbers, J. Non-Newtonian Fluid Mech., 135, 2-3, 92-96, (2006) · Zbl 1195.76174 [7] Graham, M. D., Polymer turbulence with Reynolds and Riemann, J. Fluid Mech., 848, 1-4, (2018) [8] Groisman, A.; Steinberg, V., Elastic turbulence in a polymer solution flow, Nature, 405, 6782, 53-55, (2000) [9] Groisman, A.; Steinberg, V., Elastic turbulence in curvilinear flows of polymer solutions, New J. Phys., 6, 1, 29, (2004) [10] Hameduddin, I.; Meneveau, C.; Zaki, T. A.; Gayme, D. F., Geometric decomposition of the conformation tensor in viscoelastic turbulence, J. Fluid Mech., 842, 395-427, (2018) [11] Haward, S. J.; Page, J.; Zaki, T. A.; Shen, A. Q., Inertioelastic Poiseuille flow over a wavy surface, Phys. Rev. Fluids, 3, 9, (2018) [12] Higham, N. J., Functions of Matrices, (2008), Society for Industrial and Applied Mathematics · Zbl 1167.15001 [13] Hoda, N.; Jovanović, M. R.; Kumar, S., Energy amplification in channel flows of viscoelastic fluids, J. Fluid Mech., 601, 407-424, (2008) · Zbl 1151.76372 [14] Hoda, N.; Jovanović, M. R.; Kumar, S., Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids, J. Fluid Mech., 625, 411-434, (2009) · Zbl 1171.76364 [15] Joo, Y. L.; Shaqfeh, E. S. G., A purely elastic instability in Dean and Taylor-Dean flow, Phys. Fluids A, 4, 3, 524-543, (1992) · Zbl 0748.76047 [16] Jovanović, M. R.; Kumar, S., Transient growth without inertia, Phys. Fluids, 22, 2, (2010) · Zbl 1183.76263 [17] Jovanović, M. R.; Kumar, S., Nonmodal amplification of stochastic disturbances in strongly elastic channel flows, J. Non-Newtonian Fluid Mech., 166, 14-15, 755-778, (2011) · Zbl 1282.76052 [18] Lang, S., Fundamentals of Differential Geometry, 191, (2001), Springer · Zbl 0995.53001 [19] Larson, R. G.; Shaqfeh, E. S. G.; Muller, S. J., A purely elastic instability in Taylor-Couette flow, J. Fluid Mech., 218, 573-600, (1990) · Zbl 0706.76011 [20] Lee, S. J.; Zaki, T. A., Simulations of natural transition in viscoelastic channel flow, J. Fluid Mech., 820, 232-262, (2017) · Zbl 1383.76288 [21] Mckinley, G. H.; Pakdel, P.; Öztekin, A., Rheological and geometric scaling of purely elastic flow instabilities, J. Non-Newtonian Fluid Mech., 67, 19-47, (1996) [22] Meulenbroek, B.; Storm, C.; Morozov, A. N.; Van Saarloos, W., Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow, J. Non-Newtonian Fluid Mech., 116, 2, 235-268, (2004) · Zbl 1106.76367 [23] Page, J.; Zaki, T. A., Streak evolution in viscoelastic Couette flow, J. Fluid Mech., 742, 520-551, (2014) [24] Page, J.; Zaki, T. A., The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow, J. Fluid Mech., 777, 327-363, (2015) · Zbl 1381.76020 [25] Page, J.; Zaki, T. A., Viscoelastic shear flow over a wavy surface, J. Fluid Mech., 801, 392-429, (2016) · Zbl 1445.76009 [26] Pan, L.; Morozov, A.; Wagner, C.; Arratia, P. E., Nonlinear elastic instability in channel flows at low Reynolds numbers, Phys. Rev. Lett., 110, 17, (2013) [27] Qin, B.; Arratia, P. E., Characterizing elastic turbulence in channel flows at low Reynolds number, Phys. Rev. Fluids, 2, 8, (2017) [28] Rajagopal, K.; Srinivasa, A., A thermodynamic frame work for rate type fluid models, J. Non-Newtonian Fluid Mech., 88, 3, 207-227, (2000) · Zbl 0960.76005 [29] Stuart, J. T., On the non-linear mechanics of hydrodynamic stability, J. Fluid Mech., 4, 1, 1-21, (1958) · Zbl 0081.41001 [30] Suzuki, M., Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems, Commun. Math. Phys., 51, 2, 183-190, (1976) · Zbl 0341.47028 [31] Trefethen, L. N.; Embree, M., Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, (2005), Princeton University Press · Zbl 1085.15009 [32] Vaithianathan, T.; Robert, A.; Brasseur, J. G.; Collins, L. R., An improved algorithm for simulating three-dimensional, viscoelastic turbulence, J. Non-Newtonian Fluid Mech., 140, 1, 3-22, (2006) · Zbl 1143.76349 [33] Zhang, M.; Lashgari, I.; Zaki, T. A.; Brandt, L., Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids, J. Fluid Mech., 737, 249-279, (2013) · Zbl 1294.76119
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