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

Abstract

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.

Motion and shape of bubbles rising through a yield-stress fluid

“…Our values of L/R max lie in the range from 4.2 to 10.3 for c = 1.8%, decreasing as V b increases. For a given bubble volume, L/R max is smaller (i.e., the bubble is closer to spherical) in the fluid with the lower yield stress, consistent with previous results in Carbopol dispersions with a much lower y [6] and also with recent numerical calculations [16].…”

“…They found for a Bingham fluid that motion occurred only for values of Y less than a yield criterion Y c . While recent experimental results [8] for spheres falling through simple (non-aging [9]) yieldstress fluids agree well with theoretical predictions [7,10,11], there are rather few previous papers concerned with the complementary problem of rising bubbles in yield-stress materials [4][5][6][12][13][14][15][16][17] or with the displacement of a yield-stress fluid by gas [18][19][20]. Theoretical treatment of this problem is complicated by the possibility of coupling among the fluid's rheological properties, the shape of the bubble, and its motion, as well as by the nonlinearity of model constitutive relations for viscoplastic fluids.…”

“…[16], although we note that other dimensionless quantities (the Bond number and Archimedes number; see Section 4) which are held constant in the numerical work in fact vary in our experiments. Note that our definition of Y is the same as that used by Beris et al [7], using R max in the numerator, but differs slightly from that used by Dubash and Frigaard [5,6] and Tsamopoulos et al [16]. This is discussed further in Section 4 below.…”

“…Estimates of the yield criterion Y c at which the buoyant force on the bubble is balanced by the yield stress have been derived by Dubash and Frigaard [5], and the same group has performed experiments on bubbles rising through a cylinder containing Carbopol, a yield-stress polymer gel [6]. Tsamopoulos et al have carried out a numerical study of the rise of bubbles through a viscoplastic fluid [16] using a regularized form of the Bingham model [21] and obtained detailed results for the bubble shape and rise velocity as functions of the dimensionless parameters which characterize the problem. They found that Y c depended strongly on the degree to which the bubble could be deformed: when the buoyant force was much larger than the force due to surface tension, the bubble took on a bullet-like shape which allowed it to more easily move through the material.…”

The dual effect of viscosity on bubble column hydrodynamics

“…This issue was discussed, at first, by Besagni and Inzoli and, later, by Besagni et al ) who analyzed bubble sizes and shapes in air-water bubble column and proposed correlations for the aspect ratio valid for dense bubbly flows. Despite the fact that some studies investigated the shapes of single rising bubbles in viscous media (Aoyama et al, 2016;Dimakopoulos et al, 2013;Fraggedakis et al, 2016;Gumulya et al, 2016;Sikorski et al, 2009;Tsamopoulos et al, 2008), the relations between the bubble size and shapes in dense bubbly flows with viscous media remains unanswered. In addition, the available databases of the bubble aspect ratios in viscous liquids are still insufficient, as observed by Aoyama et al (2016), and additional data are needed to better establish and extend the range of validity of the existing correlations to estimate the bubble shapes.…”