Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-26T02:18:33.226Z Has data issue: false hasContentIssue false
  • English
  • Français

On the velocity discontinuity at a critical volume of a bubble rising in a viscoelastic fluid

Published online by Cambridge University Press:  19 January 2016

D. Fraggedakis ,
M. Pavlidis ,
Y. Dimakopoulos  and
J. Tsamopoulos
D. Fraggedakis
Affiliation:
Fluid Mechanics and Rheology Laboratory, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
M. Pavlidis
Affiliation:
Fluid Mechanics and Rheology Laboratory, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
Y. Dimakopoulos
Affiliation:
Fluid Mechanics and Rheology Laboratory, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
J. Tsamopoulos*
Affiliation:
Fluid Mechanics and Rheology Laboratory, Department of Chemical Engineering, University of Patras, Patras 26500, Greece
*
†Email address for correspondence: tsamo@chemeng.upatras.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the abrupt increase in the rise velocity of an isolated bubble in a viscoelastic fluid occurring at a critical value of its volume, under creeping flow conditions. This ‘velocity discontinuity’, in most experiments involving shear-thinning fluids, has been somehow associated with the change of the shape of the bubble to an inverted teardrop with a tip at its pole and/or the formation of the ‘negative wake’ structure behind it. The interconnection of these phenomena is not fully understood yet, making the mechanism of the ‘velocity jump’ unclear. By means of steady-state analysis, we study the impact of the increase of bubble volume on its steady rise velocity and, with the aid of pseudo arclength continuation, we are able to predict the stationary solutions, even lying in the discontinuous area in the diagrams of velocity versus bubble volume. The critical area of missing experimental results is attributed to a hysteresis loop. The use of a boundary-fitted finite element mesh and the open-boundary condition are essential for, respectively, the correct prediction of the sharply deformed bubble shapes caused by the large extensional stresses at the rear pole of the bubble and the accurate application of boundary conditions far from the bubble. The change of shape of the rear pole into a tip favours the formation of an intense shear layer, which facilitates the bubble translation. At a critical volume, the shear strain developed at the front region of the bubble sharply decreases the shear viscosity. This change results in a decrease of the resistance to fluid displacement, allowing the developed shear stresses to act more effectively on bubble motion. These coupled effects are the reason for the abrupt increase of the rise velocity. The flow field for stationary solutions after the velocity jump changes drastically and intense recirculation downstream of the bubble is developed. Our predictions are in quantitative agreement with published experimental results by Pilz & Brenn (J. Non-Newtonian Fluid Mech., vol. 145, 2007, pp. 124–138) on the velocity jump in fluids with well-characterized rheology. Additionally, we predict shapes of larger bubbles when both inertia and elasticity are present and obtain qualitative agreement with experiments by Astarita & Apuzzo (AIChE J., vol. 11, 1965, pp. 815–820).

JFM classification

Drops and Bubbles: Drops and Bubbles Interfacial Flows (free surface): Interfacial Flows (free surface) Non-Newtonian Flows: Non-Newtonian Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 789 , 25 February 2016 , pp. 310 - 346
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acharya, A., Mashelkar, R. A. & Ulbrecht, J. 1976 Mechanics of bubble motion and deformation in non-Newtonian media. Chem. Engng Sci. 32, 863–872.CrossRefGoogle Scholar
Arigo, M., Rajagopalan, D., Shapley, N. & McKinley, G. 1995 The sedimentation of a sphere through an elastic fluid. Part 1. Steady motion. J. Non-Newtonian Fluid Mech. 60, 225–258.Google Scholar
Arigo, M. T. & McKinley, G. H. 1998 An experimental investigation of negative wakes behind spheres settling in a shear-thinning viscoelastic fluid. Rheol. Acta 37, 307–327.Google Scholar
Astarita, G. & Apuzzo, G. 1965 Motion of gas bubbles in non-Newtonian liquids. AIChE J. 11, 815–820.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd edn. vol. 1. Wiley.Google Scholar
Bisgaard, C. 1983 Velocity fields around spheres and bubbles investigated by laser-Doppler anemometry. J. Non-Newtonian Fluid Mech. 12, 283–302.CrossRefGoogle Scholar
Bisgaard, C. & Hassager, O. 1982 An experimental investigation of velocity fields around spheres and bubbles moving in non-Newtonian liquids. Rheol. Acta 21, 537–539.Google Scholar
Brooks, A. N. & Hughes, T. J. 1982 Streamline upwind/Petrov–Galerkin formulations. Comput. Meth. Appl. Mech. Engng 32, 199–259.Google Scholar
Brown, R. B., Szady, M. J., Northey, P. J. & Armstrong, R. C. 1993 On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations. Theor. Comput. Fluid Dyn. 5, 77–106.Google Scholar
Bush, M. B. 1994 On the stagnation flow behind a sphere in a shear-thinning viscoelastic liquid. J. Non-Newtonian Fluid Mech. 55, 229–247.Google Scholar
Chatzidai, N., Giannousakis, A., Dimakopoulos, Y. & Tsamopoulos, J. 2009 On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations. J. Comput. Phys. 228, 1980–2011.CrossRefGoogle Scholar
Chilcott, M. D. & Rallison, J. M. 1988 Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newtonian Fluid Mech. 29, 381–432.Google Scholar
Dimakopoulos, Y. & Tsamopoulos, J. 2003 A quasi-elliptic transformation for moving boundary problems with large anusotropic deformations. J. Comput. Phys. 192, 494–522.Google Scholar
Dimakopoulos, Y. & Tsamopoulos, J. 2009 On the transient coating of a straight tube with a viscoelastic material. J. Non-Newtonian Fluid Mech. 159, 95–114.Google Scholar
Fraggedakis, D., Kouris, Ch., Dimakopoulos, Y. & Tsamopoulos, J. 2015 Flow of two immiscible fluids in a periodically constricted tube: transitions to stratified, segmented, churn, spray, or segregated flow. Phys. Fluids 27, 082102.CrossRefGoogle Scholar
Frank, X., Charpentier, J. C., Ma, Y., Midoux, N. & Li, H. Z. 2012 A multiscale approach for modeling bubbles rising in non-Newtonian fluids. Ind. Engng Chem. Res. 51, 2084–2093.Google Scholar
Harlen, O. G. 2002 The negative wake behind a sphere sedimenting through a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 108, 411–430.Google Scholar
Harlen, O. G., Rallison, J. M. & Chilcott, M. D. 1990 High-Deborah-number flows of dilute polymers. J. Non-Newtonian Fluid Mech. 34, 319–349.CrossRefGoogle Scholar
Hassager, O. 1979 Negative wake behind bubbles in non-Newtonian liquids. Nature 279, 402–403.Google Scholar
Herrera-Vélarde, J. R., Zenit, R., Chehata, D. & Mena, B. 2003 The flow of non-Newtonian fluids around bubbles and its connection to the jump discontinuity. J. Non-Newtonian Fluid Mech. 111, 199–209.Google Scholar
Jin, H., Phan-Thien, N. & Tanner, R. I. 1991 A finite element analysis of the flow past a sphere in a cylindrical tube. Comput. Mech. 9, 409–422.Google Scholar
Karapetsas, G. & Tsamopoulos, J. 2013 On the stick-slip flow from slit and cylindrical dies of a Phan-Tien and Tanner fluid model: II. Linear stability analysis to two- and three-dimensional disturbances. Phys. Fluids 25, 093105.CrossRefGoogle Scholar
Kopf-Sill, A. R. & Homsy, G. M. 1987 Bubble motion in a Hele-Shaw cell. Phys. Fluids 31, 18–26.Google Scholar
Leal, L. G., Skoog, J. & Acrivos, A. 1971 On the motion of gas bubbles in a viscoelastic liquid. Can. J. Chem. Engng 49, 569–575.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.Google Scholar
Lunsmann, W. J., Genieser, L., Armstrong, R. C. & Brown, R. A. 1993 Finite element analysis of steady viscoelastic flow around a sphere in a tube: calculations with constant viscosity models. J. Non-Newtonian Fluid Mech. 48, 63–99.CrossRefGoogle Scholar
Malaga, C. & Rallison, J. M. 2007 A rising bubble in a polymer solution. J. Non-Newtonian Fluid Mech. 141, 59–78.Google Scholar
Noh, D. S., Kang, I. S. & Gary Leal, L. G. 1993 Numerical solutions for the deformation of a bubble rising in dilute polymeric fluids. Phys. Fluids 5, 1315–1332.Google Scholar
Papaioannou, J., Giannousakis, A., Dimakopoulos, Y. & Tsamopoulos, J. 2014 Bubble deformation and growth inside viscoelastic filaments undergoing very large extensions. Ind. Engng Chem. Res. 53, 7548–7569.Google Scholar
Papanastasiou, T. C., Malamataris, N. & Ellwood, K. 1992 A new outflow boundary condition. Intl J. Numer. Meth. Fluids 14, 587–608.Google Scholar
Pavlidis, M., Dimakopoulos, Y. & Tsamopoulos, J. 2009 Analytical and numerical solutions for the flow of an underformed viscoelastic film down a vertical cylinder. Rheol. Acta 48, 1031–1048.Google Scholar
Phan-Thien, N. 1978 A nonlinear network viscoelastic model. J. Rheol. 22, 259–283.Google Scholar
Phan-Thien, N. & Tanner, R. I. 1977 A new constitutive equation derived from network theory. J. Non-Newtonian Fluid Mech. 2, 353–365.CrossRefGoogle Scholar
Pillapakkam, S. B., Singh, P., Blackmore, D. & Aubry, N. 2007 Transient and steady state of a rising bubble in a viscoelastic fluid. J. Fluid Mech. 589, 215–252.Google Scholar
Pilz, C. & Brenn, G. 2007 On the critical bubble volume at the rise velocity jump discontinuity in viscoelastic liquids. J. Non-Newtonian Fluid Mech. 145, 124–138.Google Scholar
Ramaswamy, S. & Leal, L. G. 1999 The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid. J. Non-Newtonian Fluid Mech. 85, 127–163.Google Scholar
Renardy, M. 1997 Imposing ‘No’ boundary condition at outflow: Why does it Work? J. Numer. Meth. Fluids 85, 413–417.Google Scholar
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique. J. Fluid Mech. 148, 1–17.CrossRefGoogle Scholar
Seydel, R. 1994 Practical Bifurcation and Stability Analysis, 2nd edn. Interdisciplinary Applies Mathematics, 5. Springer.Google Scholar
Sigli, D. & Coutanceau, M. 1977 Effect of finite boundaries on the slow laminar isothermal flow of a viscoelastic fluid around a spherical obstacle. J. Non-Newtonian Fluid Mech. 2, 1–21.Google Scholar
Soto, E., Goujon, C., Zenit, R. & Manero, O. 2006 A study of velocity discontinuity for single air bubbles rising in an associative polymer. Phys. Fluids 18, 121510.Google Scholar
Tiefenbruck, G. F. & Leal, L. G. 1980 A note on the slow motion of a bubble in a viscoelastic liquid. J. Non-Newtonian Fluid Mech. 7, 257.CrossRefGoogle Scholar
Tsamopoulos, J., Dimakopoulos, Y., Chatzidai, N., Karapetsas, G. & Pavlidis, M. 2007 Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment. J. Fluid Mech. 601, 123–164.Google Scholar
Tsouka, S., Dimakopoulos, Y., Mavrantzas, V. & Tsamopoulos, J. 2014 Stress-gradient induced migration of polymers in corrugated channels. J. Rheol. 58, 911–947.Google Scholar
Vélez-Cordero, J. R., Sámano, D. & Zenit, R. 2012 Study of the properties of bubbly flows in Boger-type fluids. J. Non-Newtonian Fluid Mech. 175–176, 1–9.Google Scholar
Wagner, A. J., Giraud, L. & Scott, C. E. 2000 Simulations of a cusped bubble rising in a viscoelastic fluid with a new numerical method. Comput. Phys. Commun. 129, 227–231.Google Scholar
You, R., Borhan, A. & Haj-Hariri, H. 2009 Stability analysis of cusped bubbles in viscoelastic flows. J. Fluid Mech. 621, 131–154.Google Scholar
Zana, E. & Leal, L. G. 1977 The dynamics and dissolution of gas bubbles in a viscoelastic fluid. Intl J. Multiphase Flow 4, 237–262.CrossRefGoogle Scholar