| Summary: | In this thesis we consider the propagation of an air bubble in a cylindrical column filled with
a viscoplastic fluid. Because of the yield stress of the fluid, it is possible that a bubble will
remain trapped in the fluid indefinitely. We restrict our focus to the case of slow moving or
near-stopped bubbles.
Using the Herschel-Bulkley constitutive equation to model our viscoplastic fluid, we develop a
general variational inequality for our problem. This inequality leads to a stress minimization
principle for the solution velocity field. We are also able to prove a stress maximization principle
for the solution stress field. Using these two principles we develop three stopping conditions.
For a given bubble we can calculate, from our stopping conditions, a critical Bingham number
above which the bubble will not move. The first stopping condition is applicable to arbitrary
axisymmetric bubbles. It is strongly dependent on the bubble length as well as the general
shape of the bubble. The second stopping condition allows us to use existing solutions of
simpler problems to calculate additional stopping conditions. We illustrate this second stopping
condition using the example of a spherical bubble. The third stopping condition applies to long
cylindrical bubbles and is dependent on the radius of the bubble. In addition to our stopping
conditions, we determine how the physical parameters of the problem affect the rise velocity of
the bubble.
We also conduct a set of experiments using a series of six different Carbopol solutions. From
the experiments we examine the dependence of the bubble propagation velocity on the fluid
parameters and compare this to our analytic results. We find that there is an interesting
discrepancy for low modified Reynolds number flows wherein the bubble velocity increases with
a decrease in the modified Reynolds numbers. We also compare our three stopping conditions
with the data. It appears that all the stopping conditions seem to be valid for the range of
bubbles examined despite the fact that when applying the second and third stopping conditions
most bubble shapes are not well approximated by a sphere or a cylinder.
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