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Center for Nonlinear Studies |
Flowing foams are used in many engineering and technical applications. For example they have been used for decades as displacing fluids for enhanced oil recovery and aquifer remediation. Another more recent application is the remediation of polluted soils: the foam is injected to carry chemical amendments to the pollutants present in the medium. In this context, apart from potential interesting physico-chemical and biochemical properties, foams have peculiar flow properties that applications might benefit of. In particular, viscous dissipation arises mostly from the contact zones between the soap films and the walls. In most experimental studies, no local information of the foam structure is available, and only global quantities such as the effective viscosity can be measured. We present an investigation of foam flows in a two-dimensional (2D) porous medium consisting of circular obstacles positioned randomly in a Hele-Shaw cell. The foam structure is recorded at regular times by a video camera and subsequently analyzed by image processing, which provides us with the velocity field and spatial distribution of bubble sizes. The flow exhibits a rich phenomenology, including flow irreversibility, preferential flow paths, local flow intermittency/non-stationarity despite the permanent global flow rate. Moreover, the medium impacts the local rheology of the flowing fluid by selecting the bubble size distribution through processes of lamella creation or destruction. In our system, lamella creation through bubble fragmentation is by far the dominant such mechanism. We investigate how preferential flow paths and intermittency depend on the imposed global flow rate and foam quality (the water content), and measure the evolution, along the mean flow direction, of the probability density of bubble sizes. We present a fragmentation model to explain that evolution, based on two statistical distributions: that for the fragmentation frequency of a bubble of given size a, f(a), and that for the size b of a bubble that results from the fragmentation of a bubble of size a, g(b|a). Under simplifying assumptions for the functions f(a) and g(b|a), an analytical resolution of the model's constitutive equation provides a behavior that is qualitatively similar to the experimental observations. Furthermore, a more realistic functional form for f(a) and g(b|a) can be directly inferred from the experimental data, in which case the corresponding model solution is obtained numerically. The results are in excellent agreement with direct measurements of the bubble size distribution. We also evidence a correlation between the spatial distribution of bubble sizes the velocity field, a finding which may have significant implications for large scale modeling.
Host: Joaquin Jimenez-Martinez