Bubbles, Drops, and Particles in Non-Newtonian Fluids, Second Edition (Chemical Industries)

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All experiments were carried out at room temperature and atmospheric pressure. Nitrogen was used as the gas phase and its density and viscosity are 1. Mass concentrations of the aqueous CMC solutions were 0. Liquid physical properties and bubble parameters are listed in Table 1.

In order to obtain the viscosity distribution around a rising bubble, the recorded instantaneous flow field around the bubble is divided into a series of grids to execute the interpolation, resulting in a grid size of 1. The interpolation is a method of constructing new data points within the range of a discrete set of known data points. Here, in executing the interpolation process, the Kriging algorithm was performed for the instantaneous flow field.

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The shear rate of each grid point is calculated based on a finite-difference approximation for the interpolated velocity field as follows:. The viscosity of CMC solutions could be calculated by a suitable constitutive equation according to the shear rate. As is well known, it is impossible to establish a general constitutive equation for all kinds of non-Newtonian fluids.

Therefore, selecting a proper constitutive equation for a specific non-Newtonian fluid is necessary. Among the existing constitutive equations, the simplest and most widely used one is the power-law model Schowalter, :. Another extensively used constitutive equation is the Carreau model Carreau, :. The Carreau model is a three parameters model and can be used at low shear rate. The change of apparent viscosity along with shear rate for CMC solutions is shown in Figure 3. It can be clearly seen from Figure 2 that the results predicted by the Carreau model exhibit good agreement with the experimental data.

Accordingly, the Carreau model was adopted to characterize the shear thinning behavior of CMC solutions in this work.

Additional information

The non-uniformity of the liquid flow field near a rising bubble is responsible for the viscosity unevenness at each local point around the bubble in a non-Newtonian fluid due to the shear-thinning effect. Information on the flow field is accordingly an important basis for a the better understanding of the local viscosity distribution around a rising bubble.

Figure 4 illustrates the measured flow field distributions around rising bubbles at different concentrations of CMC, where white represents the shape of the bubble.

Figure 4 shows that the flow field displays a typical structure: the rising bubble pushes the liquid ahead of it in up-flow and the liquid on both sides of the bubble flows back into the bubble wake, forming a pseudo-plug flow, as reported by Funfschilling et al. The bubble follows a spiral trajectory at low concentration of CMC solution, as shown in Figure 4 a.

The reason for the spiral trajectory of the bubbles is the unstable wake of the bubble, as reported by Zenit et al. The unstable wake induces a horizontal force, causing the sideways motion, which leads to the path instability of the bubble.


Chabbra & Richardson Non-Newtonian Flow

It also can be seen from Figure 4 a that a spiral vortex rope appears around the wake of the bubble. When the concentration of CMC solution increases, the velocity of the bubble decreases and the rising trajectory of the bubble varies from spiral to rectilinear. Consequently, the spiral vortex rope disappears and a vortex ring appears around the wake of the bubble. The vortex ring follows the bubble due to the upward pressure induced by the velocity gradient around it. At the same time, the bubble shape evolves from oblate to spherical, as shown in Figure 4 b and 4 c.

The velocity components in the x-direction and y-direction decrease gradually with the increase of the concentration of the CMC solution, but the velocity gradient in the y-direction increases gradually with the increase of the CMC solution concentration because high concentrations of CMC result in high viscosity, which leads to a large energy dissipation rate and prevents the momentum from transfering to far away regions. When a single bubble rises in a shear-thinning fluid, liquid motion results in the change of viscosity of the liquid around the bubble due to different shear stress effect.

Otherwise, the variation of the viscosity would lead to a change in the velocity and shape of the bubbles. Figure 6 illustrates the experimental results for the viscosity field around a single bubble rising in solutions with different concentrations of CMC. Figure 6 b and Figure 6 c indicate that the viscosity distributions of the liquid near the bubble are similar: a hollow cylindrical low viscosity region around the wake of the bubble and a cylindrical high viscosity region in the central wake.

This result also shows good conformity to the simulated results Zhang et al. Moreover, because the viscosity distribution around a bubble is closely related to the flow field, as shown in Figure 4 , the liquid behind the bubble has a high flow velocity and low viscosity, while the wake presents pseudo-plug flow. The high shear rate corresponds to the low viscosity region, while the low shear rate forms a high viscosity region.

The minimum specific viscosity decreases with the increase of the concentration of CMC solution. For mass concentrations of 0. This can be attributed to the shear thinning effect of CMC solutions, because when the concentration of CMC solutions increases, both the flow index n and the minimum of specific viscosity decrease, as shown in Figure 6 a. As shown in Figure 6 a , the viscosity distribution in 0. The bubble rose with a spiral trajectory in 0. The large relative velocity between the liquid around the vortex rope and the bulk liquid led to the occurrence of a large shear stress, which resulted in a low viscosity region in a spiral rope form.

Simultaneously, there existed a spiral high viscosity region in the center of the bubble wake. In Figure 7 a , the radial viscosity distribution decreased alternately due to the spiral rise of the bubble, but in Figure 7 b and 7 c presented a perfect symmetry with the trajectory of the bubble motion gradually tending to linear.

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The shear behavior is principally responsible for the viscosity change of a non-Newtonian fluid. Therefore, the shear behavior is directly reflected in the viscosity distribution. Figure 8 and Figure 9 show the shear stress fields and shear rate fields around a single bubble rising in solutions with different concentrations of CMC.

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