Hydroelastic instability of viscoelastic fluids in developing flow through a compliant channel
Tóm tắt
Linear stability of a viscoelastic fluid obeying the Walters’ B model is analytically and numerically investigated in the entrance region of a plane channel formed between two parallel plates. The plates are compliant and obey the two degree-of-freedom von Karman solid model. Having obtained the base-flow velocity profiles using the boundary-layer theory, their vulnerability to infinitesimally small varicose disturbances is investigated using the temporal, normal-mode, linear stability analysis. The results obtained show that a fluid’s elasticity has a stabilizing effect on the developing velocity profiles. The distance at which the flow becomes unstable shifts further downstream (i.e., towards the fully-developed section) of the channel when the Deborah number is increased. An increase in the flexural rigidity of the plates is shown to have a stabilizing effect on the short waves (i.e., the flutter modes) whereas an increase in its mass can dramatically destabilize such modes. The flow becomes more stable when the stiffness of the soft matter restraining the vertical movement of the plates is increased with the effect being more significant on the long waves (i.e., flow-induced modes). Boosting the dissipating effect of this material is predicted to have a stabilizing effect on the short waves.
Tài liệu tham khảo
Ariel, P.D., 2008, Two dimensional stagnation point flow of an elastico-viscous fluid with partial slip, ZAMM-Z. Angew. Math. Mech.88, 320–324.
Bauchau, O.A. and J.I. Craig, Structural Analysis, 2009, Springer.
Beard, D.W. and K. Walters, 1964, Elastico-viscous boundary-layer flows I. Two-dimensional flow near a stagnation point, Math. Proc. Camb. Philos. Soc.60, 667–674.
Bewersdorff, H.W., 1991, Turbulence structure of dilute polymer and surfactant solutions in artificially roughened pipes, 6th European Drag Reduction Working Meeting, Eindhoven, Netherlands, 347-368.
Biau, D., 2008, Linear stability of channel entrance flow, Eur. J. Mech. B-Fluids27, 579–590.
Bird, R.B., R.C. Armstrong, and O. Hassager, 1987, Dynamics of Polymeric Liquids Vol. 1: Fluid Mechanics, 2nd ed., John Wiley and Sons Inc., New York.
Burrell, G.L., N.F. Dunlop, and F. Separovic, 2010, Non-Newtonian viscous shear thinning in ionic liquids, Soft Matter6, 2080–2086.
Carpenter, P.W. and A.D. Garrad, 1985, The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien-Schlichting instabilities, J. Fluid Mech.155, 465–510.
Carpenter, P.W. and A.D. Garrad, 1986, The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities, J. Fluid Mech.170, 199–232.
Carpenter, P.W. and J.S.B. Gajjar, 1990, A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls. Theor. Comput. Fluid Dyn.1, 349–378.
Carpenter, P.W., C. Davies, and A.D. Lucey, 2000, Hydrodynamics and compliant walls: Does the dolphin have a secret?, Curr. Sci.79, 758–765.
Chen, T.S. and E.M. Sparrow, 1967, Stability of the developing laminar flow in a parallel-plate channel, J. Fluid Mech.30, 209–224.
Chaudhary, I., P. Garg, V. Shankar, and G. Subramanian, 2019, Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow, J. Fluid Mech.881, 119–163.
Chun, D.H. and W.H. Schwarz, 1968, Stability of a plane Poiseuille flow of a second-order fluid, Phys. Fluids11, 5–9.
Davey, A., 1973, A simple numerical method for solving Orr-Sommerfeld problems, Q. J. Mech. Appl. Math.26, 401–411.
Davies, C. and P.W. Carpenter, 1997a, Instabilities in a plane channel flow between compliant walls, J. Fluid Mech.352, 205–243.
Davies, C. and P.W. Carpenter, 1997b, Numerical simulation of the evolution of Tollmien-Schlichting waves over finite compliant panels, J. Fluid Mech.335, 361–392.
Dervaux, J., P. Ciarletta, and M.B. Amar, 2008, Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Foppl-von Karman limit, J. Mech. Phys. Solids57, 458–471.
Draad, A.A., G.D.C. Kuiken, and F.T.M. Nieuwstadt, 1998, Laminar-turbulent transition in pipe flow for Newtonian and non-Newtonian fluids, J. Fluid Mech.377, 267–312.
Drazin, P.G., 2002, Introduction to Hydrodynamic Stability, Cambridge University Press, New York.
Duan, W.H. and C.M. Wang, 2009, Nonlinear bending and stretching of a circular graphene sheet under a central point load, Nanotechnology20, 075702.
Eggleton, C.D., J.H. Ferziger, and T.H. Pulliam, 1994, Moderate Reynolds number entry flow and boundary-layer approximations for a viscoelastic fluid, Phys. Fluids6, 700–709.
Garg, V.K. and S.C. Gupta, 1981, Nonparallel effects on the stability of developing flow in a channel, Phys. Fluids24, 1752–1754.
Gavriely, N., T.R. Shee, D.W. Cugell, and J.B. Grotberg, 1989, Flutter in flow-limited collapsible tubes: A mechanism for generation of wheezes, J. Appl. Physiol.66, 2251–2261.
Grotberg, J.B. and auS.H. Davis}, 1980, Fluid-dynamic flapping of a collapsible channel: Sound generation and flow limitation, J. Biomech.13, 219–230.
Hifdi, A., M.O. Touhami, and J.K. Naciri, 2004, Channel entrance flow and its linear stability, J. Stat. Mech.-Theory Exp.2004, P06003.
Hsiao, K.W., J. Dinic, Y. Ren, V. Sharma, and C.M. Schroeder, 2017, Passive non-linear microrheology for determining extensional viscosity, Phys. Fluids29, 121603.
Hwang, S.H., M. Litt, and W.C. Forsman, 1969, Rheological properties of mucus, Rheol. Acta8, 438–448.
Johnson, M.W. and T.J. Urbanik, 1984, A Nonlinear theory for elastic plates with application to characterizing paper properties, J. Appl. Mech.-Trans. ASME51, 146–152.
Labropulu, F., I. Husain, and M. Chinichian, 2004, Stagnation-point flow of the Walters’ B’ fluid with slip, Int. J. Math. Math. Sci.61, 3249–3258.
Larose, P. and J.B. Grotberg, 1997, Flutter and long-wave instabilities in compliant channels conveying developing flows, J. Fluid Mech.331, 37–58.
Lucey, A.D. and P.W. Carpenter, 1995, Boundary layer instability over compliant walls: Comparison between theory and experiment, Phys. Fluids7, 2355–2363.
Macosko, C.W., 1994, Rheology: Principles, Measurements, Applications, VCH publishers.
Madani-Tonekaboni, S.A., R. Abkar, and R. Khoeilar, 2012, On the study of viscoelastic Walters’ B fluid in boundary layer flows, Math. Probl. Eng.2012, 861508.
Metzner, A.B. and J.L. White, 1965, Flow behavior of viscoelastic fluids in the inlet region of a channel, AICHE J.11, 989–995.
Naebe, M. and K. Shirvanimoghaddam, 2016, Functionally graded materials: A review of fabrication and properties, Appl. Mater. Today5, 223–245.
Patne, R. and V. Shankar, 2019, Stability of flow through deformable channels and tubes: Implications of consistent formulation, J. Fluid Mech.860, 837–885.
Prescher, S., F. Polzer, Y. Yang, M. Siebenbürger, M. Ballauff, and J. Yuan, 2014, Polyelectrolyte as solvent and reaction medium, J. Am. Chem. Soc.136, 12–15.
Quijano, G., A. Couvert, and A. Amrane, 2010, Ionic liquids: Applications and future trends in bioreactor technology, Bioresour. Technol.101, 8923–8930.
Sadri, R.M. and J.M. Floryan, 2002, Entry flow in a channel, Comput. Fluids31, 133–157.
Sajid, M., T. Javed, Z. Abbas, and N. Ali, 2013, Stagnation-point flow of a viscoelastic fluid over a lubricated surface, Int. J. Nonlinear Sci. Numer. Simul.14, 285–290.
Sapir, M.H. and E.L. Reiss, 1979, Dynamic buckling of a nonlinear Timoshenko beam, SIAM J. Appl. Math.37, 290–301.
Shankar, V., and V. Kumaran, 2002, Stability of wall modes in fluid flow past a flexible surface, Phys. Fluids14, 2324–2338.
Smith, F.T. and R.J. Bodonyi, 1980, On the stability of the developing flow in a channel or circular pipe, Q. J. Mech. Appl. Math.33, 293–320.
Teipel, I., 1988, Stagnation point flow of a non-Newtonian second order fluid, Trans. Can. Soc. Mech. Eng.12, 57–61.
Tsigklifis, K. and A.D. Lucey, 2017, The interaction of Blasius boundary-layer flow with a compliant panel: Global, local and transient analyses, J. Fluid Mech.827, 155–193.
Weiss, E. and R. Abu-Reziq, 2017, Ionic liquid-based polymeric microreactors and their applicability, J. Mater. Sci.52, 10637–10647.
Yue, Y.M., C.Q. Ru, and K.Y. Xu, 2017, Modified von Kármán equations for elastic nanoplates with surface tension and surface elasticity, Int. J. Non-Linear Mech.88, 67–73.