## Numerical Simulations of Interactions of Solid Particles and Deformable Gas Bubbles in Viscous Liquids

##### Abstract

Studying the interactions of solid particles and deformable gas

bubbles in viscous liquids is very important in many applications,

especially in mining and chemical industries. These interactions

involve liquid-solid-air multiphase flows and an

arbitrary-Lagrangian-Eulerican (ALE) approach is used for the direct

numerical simulations. In the system of rigid particles and

deformable gas bubbles suspended in viscous liquids, the

Navier-Stokes equations coupled with the equations of motion of the

particles and deformable bubbles are solved in a finite-element

framework. A moving, unstructured, triangular mesh tracks the

deformation of the bubble and free surface with adaptive refinement.

In this dissertation, we study four problems. In the first three

problems the flow is assumed to be axisymmetric and two dimensional

(2D) in the fourth problem.

Firstly, we study the interaction between a rising deformable bubble

and a solid wall in highly viscous liquids. The mechanism of the

bubble deformation as it interacts with the wall is described in

terms of two nondimensional groups, namely the Morton number (Mo)

and Bond number (Bo). The film drainage process is also

considered. It is found that three modes of bubble-rigid wall

interaction exist as Bo changes at a moderate Mo.

The first mode prevails at small Bo where the bubble deformation

is small. For this mode, the bubble is

hard to break up and will bounce back and eventually attach

to the rigid wall. In the second mode, the bubble may break up after

it collides with the rigid wall, which is determined by the film

drainage. In the third mode, which prevails at high Bo, the bubble

breaks up due to the bottom surface catches up the top surface

during the interaction.

Secondly, we simulate the interaction between a rigid particle and a

free surface. In order to isolate the effects of viscous drag and

particle inertia, the gravitational force is neglected and the

particle gains its impact velocity by an external accelerating

force. The process of a rigid particle impacting a free surface and

then rebounding is simulated. Simplified theoretical models are

provided to illustrate the relationship between the particle

velocity and the time variation of film thickness between the

particle and free surface. Two film thicknesses are defined. The

first is the thickness achieved when the particle reaches its

highest position. The second is the thickness when the particle

falls to its lowest position. The smaller of these two thicknesses

is termed the minimum film thickness and its variation with the

impact velocity has been determined. We find that the interactions

between the free surface and rigid particle can be divided into

three regimes according to the trend of the first film thickness.

The three regimes are viscous regime, inertial regime and jetting

regime. In viscous regime, the first film thickness decreases as the

impact velocity increases. Then it rises slightly in the inertial

regime because the effect of liquid inertia becomes larger as the

impact velocity increases. Finally, the film thickness decreases

again due to Plateau-Rayleigh instability in the jetting regime.

We also find that the minimum film thickness corresponds to an

impact velocity on the demarcation point between the viscous and

inertial regimes. This fact is caused by the balance of viscous

drag, surface deformation and liquid inertia.

Thirdly, we consider the interaction between a rigid particle and a

deformable bubble. Two typical cases are simulated: (1) Collision of

a rigid particle with a gas bubble in water in the absence of

gravity, and (2) Collision of a buoyancy-driven rising bubble with a

falling particle in highly viscous liquids. We also compare our

simulation results with available experimental data. Good agreement

is obtained for the force on the particle and the shape of the

bubble.

Finally, we investigated the collisions of groups of bubbles and

particles in two dimensions. A preliminary example of the oblique

collision between a single particle and a single bubble is conducted

by giving the particle a constant acceleration. Then, to investigate

the possibility of particles attaching to bubbles, the interactions

between a group of 22 particles and rising bubbles are studied. Due

to the fluid motion, the particles involved in central collisions

with bubbles have higher possibilities to attach to the bubble.

bubbles in viscous liquids is very important in many applications,

especially in mining and chemical industries. These interactions

involve liquid-solid-air multiphase flows and an

arbitrary-Lagrangian-Eulerican (ALE) approach is used for the direct

numerical simulations. In the system of rigid particles and

deformable gas bubbles suspended in viscous liquids, the

Navier-Stokes equations coupled with the equations of motion of the

particles and deformable bubbles are solved in a finite-element

framework. A moving, unstructured, triangular mesh tracks the

deformation of the bubble and free surface with adaptive refinement.

In this dissertation, we study four problems. In the first three

problems the flow is assumed to be axisymmetric and two dimensional

(2D) in the fourth problem.

Firstly, we study the interaction between a rising deformable bubble

and a solid wall in highly viscous liquids. The mechanism of the

bubble deformation as it interacts with the wall is described in

terms of two nondimensional groups, namely the Morton number (Mo)

and Bond number (Bo). The film drainage process is also

considered. It is found that three modes of bubble-rigid wall

interaction exist as Bo changes at a moderate Mo.

The first mode prevails at small Bo where the bubble deformation

is small. For this mode, the bubble is

hard to break up and will bounce back and eventually attach

to the rigid wall. In the second mode, the bubble may break up after

it collides with the rigid wall, which is determined by the film

drainage. In the third mode, which prevails at high Bo, the bubble

breaks up due to the bottom surface catches up the top surface

during the interaction.

Secondly, we simulate the interaction between a rigid particle and a

free surface. In order to isolate the effects of viscous drag and

particle inertia, the gravitational force is neglected and the

particle gains its impact velocity by an external accelerating

force. The process of a rigid particle impacting a free surface and

then rebounding is simulated. Simplified theoretical models are

provided to illustrate the relationship between the particle

velocity and the time variation of film thickness between the

particle and free surface. Two film thicknesses are defined. The

first is the thickness achieved when the particle reaches its

highest position. The second is the thickness when the particle

falls to its lowest position. The smaller of these two thicknesses

is termed the minimum film thickness and its variation with the

impact velocity has been determined. We find that the interactions

between the free surface and rigid particle can be divided into

three regimes according to the trend of the first film thickness.

The three regimes are viscous regime, inertial regime and jetting

regime. In viscous regime, the first film thickness decreases as the

impact velocity increases. Then it rises slightly in the inertial

regime because the effect of liquid inertia becomes larger as the

impact velocity increases. Finally, the film thickness decreases

again due to Plateau-Rayleigh instability in the jetting regime.

We also find that the minimum film thickness corresponds to an

impact velocity on the demarcation point between the viscous and

inertial regimes. This fact is caused by the balance of viscous

drag, surface deformation and liquid inertia.

Thirdly, we consider the interaction between a rigid particle and a

deformable bubble. Two typical cases are simulated: (1) Collision of

a rigid particle with a gas bubble in water in the absence of

gravity, and (2) Collision of a buoyancy-driven rising bubble with a

falling particle in highly viscous liquids. We also compare our

simulation results with available experimental data. Good agreement

is obtained for the force on the particle and the shape of the

bubble.

Finally, we investigated the collisions of groups of bubbles and

particles in two dimensions. A preliminary example of the oblique

collision between a single particle and a single bubble is conducted

by giving the particle a constant acceleration. Then, to investigate

the possibility of particles attaching to bubbles, the interactions

between a group of 22 particles and rising bubbles are studied. Due

to the fluid motion, the particles involved in central collisions

with bubbles have higher possibilities to attach to the bubble.

##### Collections

- Doctoral Dissertations [11228]