Local flow around a tiny bubble under a pressure-oscillation field in a viscoelastic worm-like micellar solution
Introduction
One of the practical phenomena in many industrial processes is the rising of gas bubbles. There have been many fundamental studies related to the motion of gas bubbles; however, for the case of non-Newtonian fluids, interesting and unexpected observations remain unresolved [1]. For example, one well-known phenomenon in non-Newtonian fluids is the discontinuity in the bubble rising velocity. In a Newtonian fluid, the rising velocity of a gas bubble with a Reynolds number less than 1 (Stokes region [2]) is quadratically proportional to the bubble diameter, i.e., the gas bubble volume and rising velocity are related. However, for non-Newtonian liquids, many authors have reported a critical bubble volume, above which a rapid increase in the bubble rising velocity occurs. The rapid change of the bubble rising velocity is referred to as velocity jump.
Several groups have investigated the velocity jump in non-Newtonian fluids. For example, Astarita and Apuzzo [3] reported that the ratio of bubble rising velocity before and after the velocity jump is in the range of 2–6, depending on the polymer content in the solution. They argued that the velocity jump was the result of a transition from the Stokes region to the Hadamard region [4], i.e., a change from a rigid bubble interface to a free bubble interface. The velocity jump that results from such a change in boundary conditions could be up to 1.5 times. However, this transition cannot fully explain a six-fold increase in the bubble rising velocity.
Hassager [5] reported on the bubble rising velocity and the wake behind an air bubble in a non-Newtonian liquid, where laser Doppler anemometry was used to measure the liquid velocity. For small bubbles (with a volume smaller than the critical volume), the dominant flow in the wake behind the air bubble is towards the upper direction, i.e., the direction of rising. Such a wake is referred to as a positive wake and is similar to the flow of a bubble in Newtonian fluid. On the other hand, for large bubbles (with a volume larger than the critical volume), the flow is towards the lower direction, i.e., opposite to the bubble rising direction and thus termed a negative wake. In this case, the bubble shape has a cusped shape that seems to be sharpened at the tail of the bubble. To investigate the structure of a negative wake, Herrera-Velarde et al. [6] showed different types of wake behind a rising bubble before and after the critical volume using the particle image velocimetry (PIV) technique. Li and colleagues [7], [8], [9] used two practical methods, PIV and birefringence visualization. The latter method was performed by crossed Nicols measurements with two sheets of polarizer plate to visualize the stress field around the bubble. Moreover, a lattice Boltzmann (LB) simulation was used for the complex flow field, whereby they reported that the negative wake is related to the viscoelastic properties of the fluid. Pillapakkam et al. [10] indicated that a vortex ring is present in the negative wake of a larger bubble. They used direct numerical simulation (DNS) of the transient bubble for a viscoelastic fluid. The vortex ring located at the bottom of the bubble produces a negative wake, which was in agreement with the experimental result. Handzy and Belmonte [11] reported birefringent images of the negative wake behind a bubble rising in a wormlike micellar fluid, an aqueous solution of cetyl-pyridinium chloride and sodium salicylate (CPCl/NaSal). They showed that the residual stress in the tail of the cusped bubble by the crossed Nicols photographs. Belmonte [12] observed the motion of a rising cusped bubble in wormlike micellar aqueous solution of cetyl-trimethyl ammonium bromide and sodium salicylate (CTAB/NaSal) in the rotating toroidal cell, and found that the cusp-tip of the bubble is periodically elongated in the vertical direction for larger bubbles. The bubble shape appears to have a knife-edge shape [5], [13] in the shortest cusp-tip. The frequency of the oscillation varied slightly and increased with the bubble volume, which Belmonte referred to as bubble self-oscillation. Imaizumi et al. [14] used a new method of mesh deformation tracking and suggested that the cusped shape is related to the release of accumulated elastic energy from shear strain in the viscoelastic fluid.
Theoretical studies on nonlinear oscillations of bubbles in viscoelastic fluids subject to a pulsating pressure is one of the important themes related to ultrasonic cavitation. Fogler and Goddard [15] presented a numerical analysis with regard to collapse of a spherical cavity in a large body of an incompressible linear Maxwell model fluid. Shima et al. [16] theoretically investigated on nonlinear oscillations of gas bubbles in viscoelastic fluids of a three-constant Oldroyd model. Lind and Phillips [17] used boundary element method for a spherical bubble in several differential viscoelastic model fluids such as the Maxwell, the Jeffreys, the Rouse and the Doi-Edwards. The numerical scheme was extended to study the dynamics of non-spherical bubble near a rigid boundary [18] and a free surface [19]. Lind and Phillips [20] also used the spectral element method for the viscoelastic two-phase bubble near the rigid boundary, and showed good agreement with the dynamics observed in other numerical studies along with experimental observations. However, this method suffers from the high Weissenberg number problem. There is a limiting Weissenberg number for each of the fluid models beyond which the simulation fails to converge.
We measured motion of rising bubble under pressure-oscillating field, and reported a significant enhancement of the rising velocity up to approximately 400 times for a shear-thinning fluid with 1–5 mm3 bubbles in 0.7 wt% aqueous sodium polyacrylate (SPA) [21]. We identified two significant points in this case: (1) In the case of bigger amplitude and higher frequency of pressure-oscillation, the tiny bubble shape changed from spherical to cuspidal when the bubble size reached a minimum during the contraction phase, while the bubble shape became spherical again during the expansion phase. The tiny bubble shape without pressure-oscillation (i.e., natural rising bubble) is spherical due to a very low Reynolds number, such as the order of 10−3, which is much smaller than 1 (Stokes region). (2) A bubble with surrounding fluid undergoes alternating deformation due to bubble expansion and contraction under a pressure-oscillating field, which causes local shear and elongational flow around the bubble. The local shear rate on the surface of isotropically compressing/expanding bubble can be estimated according to the spherical model [21]. The shear rate on the bubble surface was reported to reach up to the order of several hundred, which indicates a sufficient decrease in shear viscosity and the generation of elastic stress. Interestingly, nonuniform radial velocities were observed around the bubble by measurement of motion of a sheet of fine pigments. The radial velocities were high in the lower region of the bubble than the upper region, because the fluid in upper region was surrounded by the rigid cell wall and the fluid was pressed from the downside through a sheet of rubber [22]. Such difference was boosted in the cuspidal bubble shape. The reason of cyclical change between spherical and cuspidal has not been identified in detail, however, should be related with the rheological property and local flow of the surrounding fluid. In this work, the local flow structure around a tiny bubble under a pressure-oscillation field is studied with both high time and space resolution using 2D polarization measurements by which the stress field can also be measured.
Section snippets
Test solution
A mixed solution of 0.03 M CTAB and 0.23 M NaSal was used. The solute was dissolved in distilled water using high-performance liquid chromatography. CTAB/NaSal aq. exhibits strong flow birefringence due to worm-like micelle orientation under stress conditions [23], [24]. This solution is known to obey the stress-optic rule. Fig. 1 shows the rheological properties of the CTAB/NaSal solution, where η is the shear viscosity, N1 is the normal stress, |η∗| is the complex viscosity, is the shear
Free rising velocity of an air bubble
Fig. 4 shows the free rising velocity of a tiny air bubble. In this study, the equivalent bubble diameter Deq, was calculated with consideration of the axial symmetry of the cusped bubble, given as Eq. (6), where DH and DV are the horizontal and vertical bubble diameters, respectively. The velocity jump is observed around a bubble diameter of Deq = 3 mm. For small bubbles (Deq < 3 mm), the flow around the bubble is similar to the creep flow. Moreover, before and after the velocity jump, the
Conclusions
The shape of a tiny bubble in CTAB/NaSal aq. was observed under a pressure-oscillating field. For free rising, the shape changes from spherical to a cusped shape as the bubble size increases. In this case, the critical volume is approximately V = 14 mm3 with Deq= 3 mm, the rising velocity of the bubbles can be accelerated by applying pressure-oscillation. A tiny bubble (V = 5.5 mm3, Deq= 2.2 mm) showed a cusped point at the tail and the rising velocity was accelerated by nearly seven-fold
Acknowledgements
The authors would like to thank Mr. Akinobu Nakayama and Mr. Hikaru Horiuchi for performing the birefringence high speed imaging experiments. The authors also would like to thank Ms. Akiko Onishi for technical assistance with the experiments. This work was supported by JSPS KAKENHI Grant Numbers (Nos. 23560195, 26420107, 17K06152) from the Japan Society for the Promotion of Science (JSPS).
References (28)
- et al.
Flow of non-Newtonian fluids around bubbles PIV measurements and birefringence visualization
Chem. Eng. Sci.
(2001) Bubbles in non-Newtonian fluids: Formation, interactions and coalescence
Chem. Eng. Sci.
(1999)- et al.
Origin of the negative wake behind a bubble rising in non-Newtonian fluids
Chem. Eng. Sci.
(2006) - et al.
Viscoelastic fluid behaviors around a rising bubble via a new method of mesh deformation tracking
Chem. Eng. Sci.
(2014) - et al.
Nonlinear oscillations of gas bubbles in viscoelastic fluids
Ultrasonics
(1986) - et al.
Spherical bubble collapse in viscoelastic fluids
J. Non-Newtonian Fluid Mech.
(2010) - et al.
Pressure-oscillation deforming for viscoelastic fluid
J. Non-Newtonian Fluid Mech.
(2008) - et al.
A development of two-dimensional birefringence distribution measurement system with a sampling rate of 1.3 MHz
Opt. Commun.
(2014) Bubbles, Drops, and Particles in Non-Newtonian Fluids
(2007)On the effect of the internal friction of fluids on the motion of pendulums
Trans. Cambridge Philos. Soc.
(1851)
Motion of gas bubble in non-Newtonian liquid
AIChE J
C.R. Hebd. Seances Acad. Sci.
Negative wake behind bubbles in non-newtonian liquids
Nature
The flow of non-Newtonian fluids around bubbles and its connection to the jump discontinuity
J. Non-Newtonian Fluid Mech.
Cited by (11)
Editorial for special issue on “Complex flows of viscoelastic wormlike micelle solutions”
2020, Journal of Non-Newtonian Fluid MechanicsThe Effect of Mechanical Shaking on the Rising Velocity of Bubbles in High-Viscosity Shear-Thinning Fluids
2023, Journal of Geophysical Research: Solid EarthFlow-To-Fracture Transition of Linear Maxwell-Type Versus Yield Strength Fluids by Air Injection—Implications for Magma Fracturing
2023, Geophysical Research LettersBubbles, Drops, and Particles in Non-Newtonian Fluids, Third Edition
2023, Bubbles, Drops, and Particles in Non-Newtonian Fluids, Third Edition