The topic originally from 50's Saffman-Taylor instability, which occurs in Hele-Shaw cell research. Now, we focus on a variant way of conventional Hele-Shaw set-up where viscous fingering patterns were produced, namely, lifting plate problem.
Think about that model in a simple way: 2 flat plates, with oil stay evenly between them. Lifting the top plate by a given speed while leaving the bottom plate with natural rest status. In that way, oil between these 2 plates will be gathering to the center, and its fluid boundary forms a fingering pattern.
Notes that although the experimental results of lifting plate problem is visually similar to radical Hele-Shaw problem, its physical driving is different: this problem is extensional free surface instability. Even though, lifting plate problem can be observed as an alternative sense from Hele-Shaw problem.
One thing we interested in this problem is the number of fingers. Theoretical studies over 10 years proved the number of fingers depends on surface tensor (dimensionless surface tensor), while the preceding experimental results show that there is another important parameter need to be considered: confinement, i.e. aspect ratio of fluid, which is the ratio of initial radius and initial plate gap width. Roughly speaking, the number of fingers increasing with the increasing of this ratio. However, this happens under a certain kind of good condition, e.g. const moving rate of top plate, short time slot, etc. (that's why we said it is in an instability status). Bad conditions were not considered too mush in the previsous studies.
The most recent studies focus on the rapid increasing gap, and this rate even can be exponentially large. Those all pointed at non-circular vanishing limiting shapes (at the level of linear theory), and non-linear dynamics and emergent pattern formation are hence too hard to predict. Again, not so many works so far on that direction.
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snowflakes pattern formation, bacterial colony growth.