Flow structure at floating of a single bubble in a liquid with solute surfactant

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Abstract

The results of mathematical modeling of the nonstationary problem of gas bubble surfacing in a viscous liquid with a solute dissolved in it are presented. The problem formulation is written taking into account the adsorption and desorption effects of the surfactant at the interface and the dependence of the surface tension coefficient on concentration according to Langmuir’s law. The numerical algorithm of the solution is based on the original Lagrangian-Eulerian technique, which allows us to explicitly allocate the free surface at the discrete level and realize natural boundary conditions on it. The process of establishing the stationary velocity of bubble surfacing is studied, and parametric research on the influence of the surfactant concentration and bubble size on the stationary velocity and flow structure in its vicinity is performed. The distributions of the velocity vector and surface concentration along the interface are presented, demonstrating the influence of the Marangoni effect on the surfacing process.

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About the authors

E. I. Borzenko

National Research Tomsk State University

Author for correspondence.
Email: borzenko@ftf.tsu.ru
Russian Federation, Tomsk

A. S. Usanina

National Research Tomsk State University

Email: borzenko@ftf.tsu.ru
Russian Federation, Tomsk

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Supplementary files

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2. Fig. 1. Dependence of the free surface shape for La=0.01 (a): 1 – t = 0; 2 – t = 1; 3 – t = 2; 4 – t = 3; 5 – t = 4; 6 – t = 7; 7 – t = 15; 8 – t = 30; dependence of the ellipticity coefficient on time (b): 1 – La = 1, 2 – La = 0.01.

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3. Fig. 2. Evolution of surface viscosity profiles: (a) La = 0.01; (b) La = 1: 1 – t = 0; 2 – t = 1; 3 – t = 2; 4 – t = 3; 5 – t = 4; 6 – t = 7; 7 – t = 15; 8 – t = 30.

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4. Fig. 3. Distribution of Marangoni stresses at time t = 30: 1 – La = 0.01; 2 – La = 1.

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5. Fig. 4. Evolution of the velocity component profiles: (a) La = 0.01; (b) La = 0.01; (c) La = 1: 1 – t = 0; 2 – t = 1; 3 – t = 2; 4 – t = 3; 5 – t = 4; 6 – t = 7; 7 – t = 15; 8 – t = 30.

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6. Fig. 5. Distribution of components of radial (a), axial (b) velocities and modified pressure (c) at time t = 30.

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7. Fig. 6. Distribution of volume concentration of surfactant (a) and instantaneous current lines (b) at time t = 30.

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8. Fig. 7. Dependence of the ascent velocity on time (a) and steady-state shapes of the free surface (b): 1 – La = 0; 2 – La = 0.01; 3 – La = 0.02; 4 – La = 0.05; 5 – La = 0.1; 6 – La = 0.5.

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9. Fig. 8. Distributions of the radial (a), axial (b) components of the velocity and surface concentration of surfactants (c) after establishing the ascent velocity: 1 – La = 0; 2 – La = 0.01; 3 – La = 0.02; 4 – La = 0.05; 5 – La = 0.1; 6 – La = 0.5; 7 – La = 1.

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10. Fig. 9. Steady-state bubble ascent velocity depending on the La number (Ma = 0.2): 1 – Ga = 35, Bo = 0.025, Pe = PeΣ =35355, Ha = 0.035, K = 0.1; 2 – Ga = 100, Bo = 0.1, Pe = PeΣ = 105, Ha = 0.05, K = 0.05; 3 – Ga=184, Bo = 0.225, Pe = PeΣ = 183711, Ha = 0.061, K = 0.033; 4 – Ga=300, Bo = 0.441, Pe = PeΣ = 304318, Ha = 0.072, K = 0.024.

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11. Fig. 10. Streamlines during the ascent of a single bubble: (a) Ga = 35; (b) Ga = 300.

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