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.