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Stability analysis of cusped bubbles in viscoelastic flows

Published online by Cambridge University Press:  12 February 2009

R. YOU ,
A. BORHAN  and
H. HAJ-HARIRI
R. YOU
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA
A. BORHAN
Affiliation:
Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, USA
H. HAJ-HARIRI*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA
*
†Email address for correspondence: [email protected]
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Abstract

Experiments have established that an axisymmetric sharp trailing edge of a gas bubble in a viscoelastic fluid can develop into a three-dimensional ‘knifelike’ shape under certain conditions (high capillary number, large bubble size). A numerical study is conducted to discover the physics of this phenomenon. The axisymmetric deformation of a bubble rising buoyantly in a viscoelastic fluid is simulated by solving the axisymmetric flow equations coupled with the constitutive equations of the finitely extensible nonlinear elastic Chilcott–Rallison (FENE-CR) model. The three-dimensional temporal linear stability analysis of this axisymmetric base state is carried out. The dominant eigenvalue which is indicative of the growth rate of the perturbations is computed. The only unstable eigenmode has azimuthal wavenumber m equal to 2. The corresponding eigenfunction shows that indeed a sharp axisymmetric tail develops a knife-edge form. A further investigation of the energy budget of the disturbances for m = 2 is performed to determine the production and dissipation terms affecting the growth of this instability. It is shown that the normal gradient of the base-state pressure along the free surface plays an important role in the evolution of the instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 621 , 25 February 2009 , pp. 131 - 154
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.CrossRefGoogle Scholar
Belmonte, A. 2000 Self-oscillations of a cusped bubble rising through a micellar solution. Rheol. Acta 39, 554–559.CrossRefGoogle Scholar
Chilcott, M. D. & Rallison, J. M. 1988 Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newton. Fluid Mech. 29, 381–432.Google Scholar
Chorin, A. 1967 A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2 (1), 12–26.Google Scholar
Hassager, O. 1979 Negative wake behind bubbles in non-Newtonian liquids. Nature 279, 402–403.CrossRefGoogle ScholarPubMed
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81, 206–229.CrossRefGoogle Scholar
Kim, I. & Pearlstein, A. J. 1990 Stability of the flow past a sphere. J. Fluid Mech. 211, 73–93.CrossRefGoogle Scholar
Kuhlmann, H. C. & Rath, H. J. 1993 Hydrodynamic instabilities in cylindrical thermocapillary liquid bridges. J. Fluid Mech. 247, 247–274.CrossRefGoogle Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users' Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM series in software, environments and tools. SIAM.Google Scholar
Levenstam, M., Amberg, G. & Winkler, C. 2001 Instabilities of thermocapillary convection in a half-zone at intermediate Prandtl numbers. Phys. Fluids 13 (4), 807–816.CrossRefGoogle Scholar
Liu, Y. J., Liao, T. Y. & Joseph, D. D. 1995 A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid. J. Fluid Mech. 304, 321–342.CrossRefGoogle Scholar
Ramanan, N. & Homsy, G. M. 1994 Linear stability of lid-driven cavity flow. Phys. Fluids 6, 2690–2701.CrossRefGoogle Scholar
Roe, P. L. 1981 Approximate riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372.CrossRefGoogle Scholar
Sorensen, D. C. 1992 Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. App. 13 (1), 357–385.CrossRefGoogle Scholar
You, R., Borhan, A. & Haj-Hariri, H. 2008 A finite volume formulation for simulating drop motion in a viscoelastic two-phase system. J. Non-Newton. Fluid Mech. 153, 109–129.Google Scholar
Zebib, A. 1987 Removal of spurious modes encountered in solving stability problems by spectral methods. J. Comput. Phys. 70, 521–525.CrossRefGoogle Scholar