Tanveer later found a wider class of possible bubble shapes in this channel geometry by relaxing the centreline symmetry constraint thereby introducing a doubly connected flow domain and writing the solutions in terms of elliptic functions. Here, in the case of a single bubble in a Hele-Shaw channel which is reflectionally symmetric about the channel centreline, Taylor & Saffman appealed to hodograph methods in simply connected domains to exactly determine the shape of the propagating bubble. There is a similar history for zero-surface-tension problems in a channel geometry (where the flow is confined to a channel whose parallel walls are perpendicular to the Hele-Shaw plates). As with the single bubble problem, solutions exist for a continuum of bubble speeds U. For multiple bubbles, exact solutions have been constructed by Crowdy by exploiting the function theory of the Schottky–Klein prime function , an approach which is crucial for solving problems like this with arbitrary finite connectivity. In the case of a single bubble in an unbounded Hele-Shaw cell, it is relatively simple to show that the bubbles are elliptic in shape with aspect ratio ( U− V)/ V. The standard zero-surface-tension model for Hele-Shaw flow predicts that any number of bubbles can translate steadily with a continuum of possible bubble speeds U for a given background fluid speed V in the same direction (where U> V). We will focus on the theoretical situation where the Hele-Shaw cell is unbounded, therefore completely removing the effect of any side walls on any of the bubbles and flow field variables. We are interested in steadily propagating bubbles in a Hele-Shaw cell, a topic which dates back to Taylor & Saffman . Many exact and numerical solutions have been found to these problems, in both steady and unsteady cases, which is particularly remarkable given the highly nonlinear nature of these problems (see, e.g. Mathematically, Hele-Shaw models with fluid interfaces give rise to many interesting free boundary problems which have been well studied over the past several decades, each of which come under the general umbrella of Laplacian growth processes. For this reason, flow in a Hele-Shaw cell can be used to produce good visualizations of the streamline patterns in potential flow fields, such as the flow around a circular cylinder or around a body with sharp corners, because the fluid is able to negotiate corners without separation (Van Dyke ).Ī particularly interesting set-up involves two immiscible fluids of differing viscosity where, if the less viscous fluid is displacing the more viscous fluid, the interface between the two fluids is unstable and can evolve to form one or more fingers . From a modelling perspective, if the fluid is assumed to be incompressible and the flow irrotational, any mathematical solutions will be necessarily harmonic. A Hele-Shaw cell is a classical experimental apparatus which sandwiches a thin layer of viscous fluid between two parallel glass plates in such a way that the fluid dynamics can be modelled as completely two dimensional.
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