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Viscoelastic liquid film flowing down a flexible tube

Published online by Cambridge University Press:  05 August 2016

Zhi-Qiang Zhou ,
Jie Peng ,
Yang-Jun Zhang  and
Wei-Lin Zhuge
Zhi-Qiang Zhou
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing, PR China
Jie Peng*
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing, PR China
Yang-Jun Zhang
Affiliation:
State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, PR China
Wei-Lin Zhuge
Affiliation:
State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, PR China
*
†Email address for correspondence: peng-jie@tsinghua.edu.cn
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Abstract

This investigation addresses the dynamics of annular viscoelastic films flowing down a flexible tube. The fluid viscoelasticity is assumed to be weak in order to obtain approximate explicit expressions for the stresses. Based on Shkadov’s integral boundary layer method (Fluid Dyn., vol. 2(1), 1967, pp. 29–34), a set of nonlinear evolution equations is derived that is valid for flows with moderate Reynolds numbers. The linear stability property of the system is examined by using normal-mode analysis, which is verified by comparing the results with those resulting from the linearization of the full Navier–Stokes equations. The results indicate that the fluid viscoelasticity plays an unstable role in the stability of the annular film flow. The tube flexibility, which includes wall damping and wall tension, plays a dual role. A bifurcation analysis is performed, and the families of steady travelling waves are catalogued. It is found that the stiffness of the tube tends to stimulate the interfacial capillary ripples. The fluid viscoelasticity acts to strengthen the dispersion of the interfacial waves but weakens the interfacial capillary ripples. The spatio-temporal evolutions of the system are also solved numerically. When the tube radius is small enough, tube closure can be observed due to the Plateau–Rayleigh instability. The fluid viscoelasticity acts to promote tube closure while the tube radius is relatively small. However, it plays a role in postponing the closure of the tube with a large radius.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Interfacial Flows (free surface): Thin films Non-Newtonian Flows: Viscoelasticity

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 802 , 10 September 2016 , pp. 583 - 610
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© 2016 Cambridge University Press 

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References

Andersson, H. I. & Dahl, E. N. 1999 Gravity-driven flow of a viscoelastic liquid film along a vertical wall. J. Phys. D: Appl. Phys. 32 (14), 1557–1562.Google Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45 (1), 150–155.CrossRefGoogle Scholar
Boulogne, F., Pauchard, L. & Giorgiutti-Dauphiné, F. 2012 Instability and morphology of polymer solutions coating a fibre. J. Fluid Mech. 704, 232–250.Google Scholar
Chang, H.-C. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26 (1), 103–136.Google Scholar
Chang, H.-C. & Demekhin, E. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier Science.Google Scholar
Craster, R. V. & Matar, O. K. 2006 On viscous beads flowing down a vertical fibre. J. Fluid Mech. 553, 85–106.Google Scholar
Dercole, F. & Rinaldi, S. 2011 Dynamical systems and their bifurcations. In Advanced Methods of Biomedical Signal Processing, pp. 291–325. IEEE-Wiley.Google Scholar
Dietze, G. F. & Ruyer-Quil, C. 2015 Films in narrow tubes. J. Fluid Mech. 762, 68–109.Google Scholar
Duprat, C., Ruyer-Quil, C. & Giorgiutti-Dauphiné, F. 2009 Spatial evolution of a film flowing down a fiber. Phys. Fluids 21 (4), 042109.Google Scholar
Gol’Denveizer, A. L. 2014 Theory of Elastic Thin Shells: Solid and Structural Mechanics. Elsevier.Google Scholar
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12 (30), 309–319.Google Scholar
Grotberg, J. B. 1994 Pulmonary flow and transport phenomena. Annu. Rev. Fluid Mech. 26 (1), 529–571.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Halpern, D., Fujioka, H. & Grotberg, J. B. 2010 The effect of viscoelasticity on the stability of a pulmonary airway liquid layer. Phys. Fluids 22, 011901.Google Scholar
Halpern, D. & Grotberg, J. B. 1992 Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244 (1), 615–632.Google Scholar
Halpern, D. & Grotberg, J. B. 1993 Surfactant effects on fluid-elastic instabilities of liquid-lined flexible tubes: a model of airway closure. Trans. ASME J. Biomech. Engng 115 (3), 271–277.Google Scholar
Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363–384.Google Scholar
Hu, F.-Q.1996 On perfectly matched layer as an absorbing boundary condition. AIAA Paper 96-1664.Google Scholar
Kalliadasis, S. & Chang, H.-C. 1994 Drop formation during coating of vertical fibres. J. Fluid Mech. 261, 135–168.Google Scholar
Kalliadasis, S., Ruyer-Quil, C. & Scheid, B. 2012 Falling Liquid Films. Springer.Google Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin viscous liquid films. Zh. Eksp. Teor. Fiz. 19, 105–120.Google Scholar
Kliakhandler, I. L., Davis, S. H. & Bankoff, S. G. 2001 Viscous beads on vertical fibre. J. Fluid Mech. 429, 381–390.Google Scholar
Matar, O. K., Craster, R. V. & Kumar, S. 2007 Falling films on flexible inclines. Phys. Rev. E 76 (5), 056301.Google Scholar
Padmanaban, A.2006 Film thickness measurements in falling annular films. PhD thesis, University of Saskatchewan, Saskatoon, SK.Google Scholar
Peng, J., Zhang, Y.-J. & Zhuge, W.-L. 2014 Falling film on flexible wall in the limit of weak viscoelasticity. J. Non-Newtonian Fluid Mech. 210, 85–95.Google Scholar
Ruyer-Quil, C. & Kalliadasis, S. 2012 Wavy regimes of film flow down a fiber. Phys. Rev. E 85 (4), 046302.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15 (2), 357–369.Google Scholar
Salamon, T. R., Armstrong, R. C. & Brown, R. A. 1994 Traveling waves on vertical films: numerical analysis using the finite element method. Phys. Fluids 6 (6), 2202–2220.Google Scholar
Saprykin, S., Koopmans, R. J. & Kalliadasis, S. 2007 Free-surface thin-film flows over topography: influence of inertia and viscoelasticity. J. Fluid Mech. 578, 271–293.Google Scholar
Shkadov, V. Y. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Fluid Dyn. 2 (1), 29–34.Google Scholar
Shkadov, V. Y. 1977 Solitary waves in a layer of viscous liquid. Fluid Dyn. 12 (1), 52–55.CrossRefGoogle Scholar
Shkadov, V. Y. & Sisoev, G. M. 2004 Waves induced by instability in falling films of finite thickness. Fluid Dyn. 35 (5), 357–389.Google Scholar
Sisoev, G. M., Craster, R. V., Matar, O. K. & Gerasimov, S. V. 2006 Film flow down a fibre at moderate flow rates. Chem. Engng Sci. 61 (22), 7279–7298.Google Scholar
Sisoev, G. M., Matar, O. K., Craster, R. V. & Kumar, S. 2010 Coherent wave structures on falling fluid films flowing down a flexible wall. Chem. Engng Sci. 65 (2), 950–961.Google Scholar
Sivashinsky, G. I. & Michelson, D. M. 1980 On irregular wavy flow of a liquid film down a vertical plane. Prog. Theor. Phys. 63, 2112–2114.Google Scholar
Trifonov, Y. Y. 1992 Steady-state traveling waves on the surface of a viscous liquid film falling down on vertical wires and tubes. AIChE J. 38 (6), 821–834.Google Scholar
Weibel, E. R. 1963 Geometry and Dimensions of Airways of Conductive and Transitory Zones. Springer.Google Scholar
Zhou, Z.-Q., Peng, J., Zhang, Y.-J. & Zhuge, W.-L. 2014 Instabilities of viscoelastic liquid film coating tube in the presence of surfactant. J. Non-Newtonian Fluid Mech. 204, 94–103.Google Scholar