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For very small power-law indices, n has little effect on the Froude number. The behavior and shape of the interface between phases in a multiphase flow lead to different flow patterns.
In a multiphase flow, mechanisms and competitive forces determine the flow regime. One of the most important multiphase flow patterns is slug flow [ 1 ].
The slug flow is known as a flow pattern in which the dispersed phase is surrounded by a continuous phase in the form of large bubbles or drops Taylor bubbles of Taylor drops. There are many applications of slug flow in industries such as cooling in the nuclear reactors, heaters and fermentation vessels.
Many studies have considered the slug flow between gas and liquid phases. However, there is no a comprehensive knowledge about the liquid—liquid slug flow.
Taylor and Davis [ 2 ] studied numerically and experimentally the motion of large bubbles in a liquid fluid. They carried out measurements related to the shape and ascending rate of air bubbles in nitrobenzene and water with different volumes and obtained bubble profiles. The large and long bubbles observed in this study were called Taylor bubbles. Ono and Kintner [ 3 ] investigated the rising motion of air bubbles in a quiescent liquid to determine their velocity for a wide range of fluid viscosity and surface tension in a vertical tube.
They correlated the velocity of the rising bubble relative to the velocity of a similar moving sphere in the liquid using a correction coefficient. Maida [ 4 ] performed experiments on the rising motion of Taylor bubbles and spherical ones in stagnant and flowing fluids water—air system inside the vertical cylindrical tubes.
He showed that in a stagnant fluid, there is a critical diameter for the tube that the bubble with smaller diameter than critical value cannot raise in the tube. Caro et al. They showed that an increase in the fluid viscosity or an increase in the surface tension causes the nose of the bubble to widen, and as a result, the bubble rise velocity decreases. Liao and Zhao [ 6 ] investigated a theoretical model to predict the velocity of a Taylor bubble in small vertical channels with square and triangle cross sections.
The end of channels was blocked, and they were filled with a stagnant liquid. The corresponding relations were determined to predict bubble rise velocity in the channels using dimensionless parameters.
Nogueira et al. They found that the length of the Taylor bubble depends on the inverse of the viscosity. Luo and Prosperti [ 8 ] analyzed the linear stability of large gas bubbles that lost their stability during their rising motion. It was found that the relative velocity between the bubble and the liquid decreases with increasing the flow rate.
Dirito et al. Numerical simulations were carried out using a volume of fluid method for a symmetric geometry.
They provided details of flow patterns and drop shapes under different conditions. The rising motion of a Taylor drop in stagnant liquids was investigated by Dirito et al. They reported drop velocity profile and demonstrated that axisymmetric behavior is not valid when viscosity ratio is very small or very large. Lertnuwat [ 11 ] proposed an empirical model to predict the shapes of a Taylor bubble rising in a stagnant liquid for different values of surface tension.
It was concluded that since the surface tension impact on the Froude number, it controls the fluid flow around the Taylor bubble. Present study aims to evaluate the motion of a Taylor drop in a non-Newtonian stagnant fluid in a vertical tube Fig. The main objective of the present work is to study the shear-thinning effect on the rising motion a Taylor drop.
The geometry of the Taylor drop includes two hemispheres and a cylinder in such a way that the length of the tube is 8 times the length of the drop. The present study considers the motion of a Taylor drop rising in a liquid continues phase. There are some techniques to solve the governing equations such as level set LS [ 12 , 13 ], coupled level set and volume of fluid CLSVOF [ 14 ], VOF [ 15 , 16 , 17 ] and front tracking techniques [ 18 , 19 , 20 , 21 , 22 , 23 , 24 ].
In the present study, to measure the motion of the two-phase flow, the VOF method is employed. The governing equations for unsteady, laminar and incompressible flow include continuity and Navier—Stokes equations [ 16 , 23 ]:. In this study, glycerol—water solution is selected as continuous phase fluid. Glycerol—water solution is prepared by mixing certain weights of glycerol and distilled water. This non-Newtonian fluid is modeled by the Bird—Carreau fluid model. The Bird—Carreau model is defined as follows [ 23 ]:.
Finite volume scheme is employed to discretize the governing equations. Generalized Reynolds number [ 27 ]:. The Froude number [ 27 ]:. The physical properties of the glycerol—water solution and the Bird—Carreau model constants are presented in Table 1 , and the physical properties of the Taylor drop silicon oil KF are shown in Table 2.
To select the appropriate solution grid for the present simulations, the velocity of Taylor drop is calculated for several grid resolutions with , 18, and 29, elements. Figure 2 shows the effect of the number of grid elements on the Taylor drop velocity. It is observed that the drop velocity does not change for the grid with more than 18, elements.
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