^{1}, Utkan Demirci

^{2}and Metin Muradoglu

^{1}

### Abstract

The effect of solublesurfactants on the unsteady motion and deformation of a bubble rising in an otherwise quiescent liquid contained in an axisymmetric tube is computationally studied by using a finite-difference/front-tracking method. The unsteady incompressible flow equations are solved fully coupled with the evolution equations of bulk and interfacial surfactant concentrations. The surface tension is related to the interfacial surfactant concentration by a nonlinear equation of state. The nearly spherical, ellipsoidal, and dimpled ellipsoidal-cap regimes of bubble motion are examined. It is found that the surfactant generally reduces the terminal velocity of the bubble but this reduction is most pronounced in the nearly spherical regime in which the bubble behaves similar to a solid sphere and its terminal velocity approaches that of an equivalent solid sphere. Effects of the elasticity number and the bulk and interfacial Peclet numbers are examined in the spherical and ellipsoidal regimes. It is found that the surface flow and interfacial surfactant concentration profiles exhibit the formation of a stagnant cap at the trailing end of the bubble in the ellipsoidal regime at low elasticity and high interfacial Peclet numbers. Bubble deformation is first reduced due to rigidifying effect of the surfactant but is then amplified when the elasticity number exceeds a critical value due to overall reduction in the surface tension.

This work is supported by the Scientific and Technical Research Council of Turkey (TUBITAK), Grant No. 105M043. The computations are performed by using the high performance computing center at Koc University.

I. INTRODUCTION

II. FORMULATION AND NUMERICAL METHOD

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Surfactants
- 143.0
- Surfactant effects
- 46.0
- Elasticity
- 28.0
- Reynolds stress modeling
- 19.0
- Lagrangian mechanics
- 17.0

## Figures

Schematic illustration of the computational setup for a buoyancy-driven bubble rising in an axisymmetrical channel with soluble surfactant.

Schematic illustration of the computational setup for a buoyancy-driven bubble rising in an axisymmetrical channel with soluble surfactant.

Spherical bubble. (a) Reynolds number vs nondimensional time for , 2.5, 5, 7.5, 10, and 15, and (b) steady Reynolds number vs nondimensional channel diameter for clean (solid lines) and contaminated (dashed lines) bubbles. ( and .)

Spherical bubble. (a) Reynolds number vs nondimensional time for , 2.5, 5, 7.5, 10, and 15, and (b) steady Reynolds number vs nondimensional channel diameter for clean (solid lines) and contaminated (dashed lines) bubbles. ( and .)

Spherical bubble. The streamlines and the velocity vectors at steady-state in a coordinate system moving with the bubble centroid for (a) a clean bubble and (b) a contaminated bubble. Every third grid points are used in the velocity vector plots. ( and .)

Spherical bubble. The streamlines and the velocity vectors at steady-state in a coordinate system moving with the bubble centroid for (a) a clean bubble and (b) a contaminated bubble. Every third grid points are used in the velocity vector plots. ( and .)

Spherical bubble. (a) Surface velocity profiles of a clean (solid lines) and a contaminated (dashed lines) bubble, and (b) the interfacial surfactant concentration profiles for the channel diameters , 2.5, 5, and 15 at . ( and .)

Spherical bubble. (a) Surface velocity profiles of a clean (solid lines) and a contaminated (dashed lines) bubble, and (b) the interfacial surfactant concentration profiles for the channel diameters , 2.5, 5, and 15 at . ( and .)

Spherical bubble. The contour plots of constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (left plot) and 1 (right plot) at . Contour levels are the same in both cases. (, , and .)

Spherical bubble. The contour plots of constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (left plot) and 1 (right plot) at . Contour levels are the same in both cases. (, , and .)

Spherical bubble. (a) The surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 0.5, and 1 at .

Spherical bubble. (a) The surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 0.5, and 1 at .

Spherical bubble. The contour plots of constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (left plot) and 1000 (right plot) at . Contour levels are the same in both cases. (, , and .)

Spherical bubble. The contour plots of constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (left plot) and 1000 (right plot) at . Contour levels are the same in both cases. (, , and .)

Spherical bubble. (a) The surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 100, 500, and 1000 at . ( and .)

Spherical bubble. (a) The surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 100, 500, and 1000 at . ( and .)

Spherical bubble. Reynolds number vs time for , , 2.5, and 1.25. ( and .)

Spherical bubble. Reynolds number vs time for , , 2.5, and 1.25. ( and .)

Ellipsoidal bubble. Reynolds number vs time for , 0.1, 0.5, and 1.0. ( and .)

Ellipsoidal bubble. Reynolds number vs time for , 0.1, 0.5, and 1.0. ( and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration vs arc length measured from the centerline in the counter-clockwise direction for (solid lines) and 1.0 (dashed lines) at times , 28.4, 48.4, and 67.8. ( and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration vs arc length measured from the centerline in the counter-clockwise direction for (solid lines) and 1.0 (dashed lines) at times , 28.4, 48.4, and 67.8. ( and .)

Ellipsoidal bubble. (Top row) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (from left to right) , 0.25, 0.5, and 1.0. (Bottom row) The streamlines and the velocity vectors in a coordinate system moving with the bubble centroid. Every third grid point is used in the vector plots. (, , , , and .)

Ellipsoidal bubble. (Top row) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (from left to right) , 0.25, 0.5, and 1.0. (Bottom row) The streamlines and the velocity vectors in a coordinate system moving with the bubble centroid. Every third grid point is used in the vector plots. (, , , , and .)

Ellipsoidal bubble. Effect of elasticity number on bubble deformation. (, , , and .)

Ellipsoidal bubble. Effect of elasticity number on bubble deformation. (, , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of nondimensional arc length measured from the centerline in the counterclockwise direction with , 0.25, 0.5, and 1.0. (, , , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of nondimensional arc length measured from the centerline in the counterclockwise direction with , 0.25, 0.5, and 1.0. (, , , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for (solid lines) and 1000 (dashed lines) at times , 28.4, 48.4, and 67.8. (, , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for (solid lines) and 1000 (dashed lines) at times , 28.4, 48.4, and 67.8. (, , , and .)

Ellipsoidal bubble. (Top row) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (from left to right) , 100, 500, and 1000. (Bottom row) The velocity vectors and the streamlines in a coordinate system moving with the bubble centroid. Every third grid point is used in the vector plots. (, , , , and .)

Ellipsoidal bubble. (Top row) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (from left to right) , 100, 500, and 1000. (Bottom row) The velocity vectors and the streamlines in a coordinate system moving with the bubble centroid. Every third grid point is used in the vector plots. (, , , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 100, 500, and 1000. (, , , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 100, 500, and 1000. (, , , , and .)

Ellipsoidal bubble. (Top row) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (from left to right) , 1000, and . (Bottom row) The velocity vectors and the streamlines in a coordinate system moving with the bubble centroid. Every third grid point is used in the vector plots. (, , , , and .)

Ellipsoidal bubble. (Top row) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side) with (from left to right) , 1000, and . (Bottom row) The velocity vectors and the streamlines in a coordinate system moving with the bubble centroid. Every third grid point is used in the vector plots. (, , , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 1000, and . (, , , , and .)

Ellipsoidal bubble. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction for , 1000, and . (, , , , and .)

Dimpled ellipsoidal cap. (a) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side). The streamlines and the velocity vectors in a frame of reference moving with the bubble centroid for (b) a contaminated and (c) a clean bubble at . Every eighth grid point is used in the vector plots. (, , , , and .)

Dimpled ellipsoidal cap. (a) The contour plots of the constant surfactant concentration in the bulk fluid (left side) and the distribution of the surfactant concentration at the interface (right side). The streamlines and the velocity vectors in a frame of reference moving with the bubble centroid for (b) a contaminated and (c) a clean bubble at . Every eighth grid point is used in the vector plots. (, , , , and .)

Dimpled ellipsoidal cap. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction at times , 28.9, 46.2, and 63.5. (, , , , and .)

Dimpled ellipsoidal cap. (a) Surface velocity and (b) interfacial surfactant concentration as a function of arc length measured from the centerline in the counterclockwise direction at times , 28.9, 46.2, and 63.5. (, , , , and .)

Effects of the Eötvös number on the motion of the clean and contaminated bubbles. (a) The steady Reynolds number vs the Eötvös number. (b) The drag coefficient vs the Reynolds number.

Effects of the Eötvös number on the motion of the clean and contaminated bubbles. (a) The steady Reynolds number vs the Eötvös number. (b) The drag coefficient vs the Reynolds number.

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