Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment

Journal of Fluid Mechanics - Tập 601 - Trang 123-164 - 2008
John Tsamopoulos1, Yannis Dimakopoulos1, N. Chatzidai1, George Karapetsas1, M. Pavlidis1
1Laboratory of Computational Fluid Dynamics, Department of Chemical Engineering, University of Patras, Patras 26500, Greece

Tóm tắt

We examine the buoyancy-driven rise of a bubble in a Newtonian or a viscoplastic fluid assuming axial symmetry and steady flow. Bubble pressure and rise velocity are determined, respectively, by requiring that its volume remains constant and its centre of mass remains fixed at the centre of the coordinate system. The continuous constitutive model suggested by Papanastasiou is used to describe the viscoplastic behaviour of the material. The flow equations are solved numerically using the mixed finite-element/Galerkin method. The nodal points of the computational mesh are determined by solving a set of elliptic differential equations to follow the often large deformations of the bubble surface. The accuracy of solutions is ascertained by mesh refinement and predictions are in very good agreement with previous experimental and theoretical results for Newtonian fluids. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham Bn=$\tau_y^{\ast}/\rho^{\ast}g^{\ast} R_b^{\ast}$ Bond Bo =$\rho^{\ast}g^{\ast} R_b^{\ast 2}/\gamma^{\ast}$ and Archimedes Ar=$\rho^{\ast2}g^{\ast} R_b^{\ast3}/\mu_o^{\ast2}$ numbers, where ρ* is the density, μ*o the viscosity, γ* the surface tension and τ*y the yield stress of the material, g* the gravitational acceleration and R*b the radius of a spherical bubble of the same volume. If the fluid is viscoplastic, the material will not be deforming outside a finite region around the bubble and, under certain conditions, it will not be deforming either behind it or around its equatorial plane in contact with the bubble. As Bn increases, the yield surfaces at the bubble equatorial plane and away from the bubble merge and the bubble becomes entrapped. When Bo is small and the bubble cannot deform from the spherical shape the critical Bn is 0.143, i.e. it is a factor of 3/2 higher than the critical Bn for the entrapment of a solid sphere in a Bingham fluid, in direct correspondence with the 3/2 higher terminal velocity of a bubble over that of a sphere under the same buoyancy force in Stokes flow. As Bo increases allowing the bubble to squeeze through the material more easily, the critical Bingham number increases as well, but eventually it reaches an asymptotic value. Ar affects the critical Bn value much less.

Từ khóa


Tài liệu tham khảo

10.1002/aic.690110514

Rybczyński W. 1911 On the translatory motion of fluid sphere in a viscous medium. Bull. Intl l' Acad. Polonaise Scie. Lett. Classe des Sciences Mathématiques et Naturelles A, 40–46.

10.1063/1.2358090

10.1016/j.jnnfm.2006.06.006

Schenk, 2006, On fast factorization pivoting methods for symmetric indefinite systems, Elec. Trans. Numer. Anal., 23, 158

10.1017/S0022112085002622

10.1016/S0009-2509(01)00002-1

10.1017/S0022112095001546

10.1063/1.868515

Levich, 1949, The motion of bubbles at high Reynolds numbers, Zh. Eksper. Teor. Fiz., 19, 18

10.1063/1.861445

10.1063/1.1803391

10.1017/S0022112082002936

10.1063/1.2196451

10.1016/S0377-0257(03)00060-0

10.1016/j.jcp.2006.08.008

10.1016/j.jnnfm.2004.01.003

Hadamard, 1911, Mouvement permanent lent d'une sphère liquide et visqueuse dans un liquide visqueux, C. R. l'Acad. Sci. Paris, 152, 1735

10.1017/S0022112064000349

10.1016/S0377-0257(01)00177-X

10.1016/0021-9991(92)90307-K

10.1016/S0065-2156(08)70133-9

10.1017/S0022112065001660

10.1016/j.ijmultiphaseflow.2006.07.003

10.1016/j.jnnfm.2007.05.015

10.1063/1.2194931

10.1016/j.jnnfm.2006.08.002

10.1016/S0377-0257(96)01536-4

10.1016/j.jnnfm.2006.07.012

10.1063/1.1578634

10.1016/j.jnnfm.2005.10.010

10.1017/S0022112085002683

10.1122/1.550992

10.1016/j.jcp.2003.07.027

10.1016/0021-9991(80)90163-1

Glowinski, 1981, Numerical Analysis of Variational Inequalities

10.1017/S0022112096007781

10.1146/annurev.fluid.32.1.659

10.1063/1.857359

10.1002/1521-4001(200011)80:11/12<827::AID-ZAMM827>3.0.CO;2-5

10.1016/j.future.2003.07.011

10.1122/1.549926

10.1017/S0022112063000665

Haberman W. L. & Morton R. K. 1953 An experimental investigation of the drag and shape of air bubbles rising in various liquids. Report 802, Navy Dept., David W. Taylor Model Basin. 68 Washington, DC.

10.1017/S0022112084002226

10.1016/S0377-0257(01)00141-0

10.1017/S002211208100311X

10.2118/20431-PA

10.1016/j.jnnfm.2005.01.003

Clift, 1978, Bubbles, Drops and Particles.

Bingham, 1922, Fluidity and Plasticity.

Thông tin
Thông tin xuất bản

Journal of Fluid Mechanics

Tập 601

123-164

Thông tin tác giả