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An instability criterion for viscoelastic flow past a confined cylinder  

Dou, Hua-Shu (Temasek Laboratories, The National University of Singapore)
Phan-Thien, Nhan (Department of Mechanical Engineering, The National University of Singapore)
Publication Information
Korea-Australia Rheology Journal / v.20, no.1, 2008 , pp. 15-26 More about this Journal
Abstract
It has been known that there is a viscoelastic instability in the channel flow past a cylinder at high Deborah (De) number. Some of our numerical simulations and a boundary layer analysis indicated that this instability is related to the shear flow in the gap between the cylinder and the channel walls in our previous work. The critical condition for instability initiation may be related to an inflection velocity profile generated by the normal stress near the cylinder surface. At high De, the elastic normal stress coupling with the streamline curvature is responsible for the shear instability, which has been recognized by the community. In this study, an instability criterion for the flow problem is proposed based on the analysis on the pressure gradient and some supporting numerical simulations. The critical De number for various model fluids is given. It increases with the geometrical aspect ratio h/R (half channel width/cylinder radius) and depends on a viscosity ratio ${\beta}$(polymer viscosity/total viscosity) of the model. A shear thinning first normal stress coefficient will delay the instability. An excellent agreement between the predicted critical Deborah number and reported experiments is obtained.
Keywords
instability; criterion; viscoelastic; normal stress; curvature; flow past a cylinder;
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  • Reference
1 Byars, J. A., 1996, Experimental characterization of viscoelastic flow instabilities, Ph.D thesis, MIT, Cambridge, MA 02139, USA
2 Coala, A. E., Y. L.Joo, R. C. Armstrong and R. A. Brown, 2001, Highly parallel time integration of viscoelastic flows, J. Non-Newt. Fluid Mech. 100, 191-216   DOI   ScienceOn
3 Dou, H.-S. and N. Phan-Thien, 2007, Viscoelastic flow past a confined cylinder: instability and velocity inflection, Chemical Engineering Science 62, 3909-3929   DOI   ScienceOn
4 Kumar, S. and G. M. Homsy, 1999, Direct numerical simulation of hydrodynamic instabilities in two-and three-dimensional viscoelastic free shear layers, J. Non-Newt. Fluid Mech. 83, 249-276   DOI   ScienceOn
5 Oliveira, P. J. and A. I. P. Miranda, 2005, A numerical study of steady and unsteady viscoelastic flow past bounded cylinders, J. Non-Newt. Fluid Mech. 127, 51-66   DOI   ScienceOn
6 Schlichting, H. and K. Gersten, 2000, Boundary layer theory, Springer, 8th Ed., Berlin, 415-494
7 Huilgol, R. R. and N. Phan-Thien, 1997, Fluid mechanics of viscoelasticity: general principles, constitutive modelling and numerical techniques, Rheology series vol. 6, Elsevier, Amsterdam
8 Smith, M. D., R. C. Armstrong, R. A. Brown and R. Sureshkumar, 2000, Finite element analysis of stability of two-dimensional viscoelastic flows to three-dimensional perturbations, J. Non-Newt. Fluid Mech. 93, 203-244   DOI   ScienceOn
9 Olagunju, D. O. and L. P. Cook, 1993, Linear-stability analysis of cone and plate flow of an Oldroyd-B fluid, J. Non-Newt. Fluid Mech. 47, 93-105   DOI   ScienceOn
10 Dou, H-S., 2006b, Physics of flow instability and turbulent transition in shear flows, Technical report, National university of singapore, 2006. http://www.arxiv.org/abs/physics/0607004. Also as part of the invited lecture: H.-S. Dou, Secret of Tornado, International workshop on geophysical fluid dynamics and scalar transport in the tropics, NUS, Singapore, 13 Nov. - 8 Dec., 2006
11 Wilson, H. J., 2006, Instabilities and constitutive modelling, Phil. Trans. R. Soc. A 364, 3267-3283
12 Dou, H.-S., B. C. Khoo and K. S.Yeo, 2007, Instability of Taylor- Couette flow between concentric rotating cylinders, Inter. J. Thermal Science, accepted and in press, http://arxiv.org/abs/physics/0502069
13 Avagliano, A. and N. Phan-Thien, 1998, Torsional flow: effect of second normal stress difference on elastic instability in a finite domain, J. Fluid Mech. 359, 217-237   DOI   ScienceOn
14 Larson, R. G., E. S. G. Shaqfeh and S. J. Muller, 1990, A purely elastic instability in Taylor-Couette flow, J. Fluid Mech. 218, 573-600   DOI
15 Pakdel, P. and G. H. McKinley, 1996, Elastic instability and curved streamlines, Phys. Rev. Lett. 77, 2459-2462   DOI   ScienceOn
16 Shaqfeh, E. S. G., 1996, Purely elastic instabilities in viscoelastic flows, Annu. Rev. Fluid Mech. 28, 129-186   DOI   ScienceOn
17 White, F. M., 1991, Viscous fluid flow, McGraw-Hill, New York, 2nd Ed., 335-393
18 Dou, H-S., 2006a, Mechanism of flow instability and transition to turbulence, Inter. J. of Non-Linear Mech. 41, 512-517. http://arxiv.org/abs/nlin.CD/0501049   DOI   ScienceOn
19 Groisman, A. and V. Steinberg, 2000, Elastic turbulence in a polymer solution flow, Nature 405, 53-55   DOI   ScienceOn
20 Larson, R. G., 1992. Instabilities in viscoelastic flows, Rheol. Acta 31, 213-263   DOI
21 Kim, J. M., C. Kim, K. H. Ahn and S. J. Lee, 2004, An efficient iterative solver and high-resolution computations of the Oldroyd- B fluid flow past a confined cylinder, J. Non-Newt. Fluid Mech. 123, 161-173   DOI   ScienceOn
22 Hulsen, M. A., R. Fattal and R. Kupferman, 2005, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non-Newt. Fluid Mech. 127, 27-39   DOI   ScienceOn
23 Avagliano, A. and N. Phan-Thien, 1999, Torsional flow stability of highly dilute polymer solutions, J. Non-Newt. Fluid Mech. 84, 19-44   DOI   ScienceOn
24 Bird, R.B., R.C. Armstrong and O. Hassager, 1987, Dynamics of polymeric liquids, vol.1: fluid mechanics, 2nd ed.,Wiley, New York
25 Dou, H.-S. and N. Phan-Thien, 1999, The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS-omega) formulation, J. Non-Newt. Fluid Mech. 87, 47-73   DOI   ScienceOn
26 Joo,Y. L. and E. S. G.Shaqfeh, 1992, A purely elastic instability in Dean and Taylor-Dean flow, Phys. Fluids A 4, 524-543
27 Sureshkumar, R., M. D. Smith, R. C. Armstrong and R. A. Brown, 1999, Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations, J. Non-Newt. Fluid Mech. 82, 57-104   DOI   ScienceOn
28 McKinley, G. H., R. C. Armstrong and R. A. Brown, 1993, The wake instability in viscoelastic flow past confined circular cylinders, Phil. Trans. R. Soc. Lond. A 344, 265-304
29 Dou, H-S., 2004, Viscous instability of inflectional velocity profile, Proc.of the foruth international conference on fluid mechanics, Ed. by F. Zhuang and J. Li, Tsinghua University Press & Springer-Verlag, Beijing, 76-79
30 Dou, H-S., 2007, Three important theorems for flow stability, Proc. of the fifth international conference on fluid mechanics, Ed. by F. Zhuang and J. Li, Tsinghua University press & Springer-Verlag, Beijing, 56-70. http://www.arxiv.org/abs/physics/0610082
31 Phan-Thien, N., 1985, Cone and plate flow of the Oldroyd-B fluid is unstable, J. Non-Newt. Fluid Mech. 17, 37-44   DOI   ScienceOn
32 McKinley, G. H., P. Pakdel and A. Oztekin, 1996, Rheological and geometric scaling of purely elastic flow instabilities, J. Non-Newt. Fluid Mech. 67, 19-47   DOI   ScienceOn
33 Groisman, A. and V. Steinberg, 1998, Mechanism of elastic instability in Couette flow of polymer solutions-experiment, Phys. Fluids 10, 2451-2463   DOI   ScienceOn
34 Byars, J. A., A. Oztekin, R. A. Brown and G. H. McKinley, 1994, Spiral instabilities in the flow of highly elastic fluids between rotating parallel disks, J. Fluid Mech. 271, 173-218   DOI   ScienceOn
35 Shiang, A. H., A.Oztekin and D. Rockwell, 2000, Hydroelastic instabilities in viscoelastic flow past a cylinder confined in a channel, Exp. Fluids 28, 128-142   DOI