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We introduce a new method for implicit structural modeling. The main developments in this paper are the new regularization operators we propose by extending inherent properties of the classic one-dimensional discrete second derivative operator to higher dimensions. The proposed regularization operators discretize naturally on the Cartesian grid using finite differences, owing to the highly symmetric nature of the Cartesian grid. Furthermore, the proposed regularization operators do not require any special treatment on boundary nodes, and their generalization to higher dimensions is straightforward. As a result, the proposed method has the advantage of being simple to implement. Numerical examples show that the proposed method is robust and numerically efficient.

In numerous industrial processes involving fluids, viscosity is a determinant factor for reaction rates, flows, drying, mixing, etc. Its importance is even more determinant for phenomena observed at the micro- and nanoscale such as in nanopores or in micro and nanochannels, for instance. However, despite notable progress in the techniques used in microrheology in recent years, the quantification, mapping, and study of viscosity at small scales remain challenging. Fluorescent molecular rotors are molecules whose fluorescence properties are sensitive to local viscosity; thus, they allow us to obtain viscosity maps by using fluorescence microscopes. While they are well-known as contrast agents in bioimaging, their use for quantitative measurements remains scarce. This paper is devoted to the use of such molecules to perform quantitative, in situ, and local measurements of viscosity in heterogeneous microfluidic flows. The technique is first validated in a well-controlled situation of a microfluidic co-flow, where two streams mix through transverse diffusion. Then, a more complex situation of mixing in passive micromixers is considered and the mixing efficiency is characterized and quantified. The methodology developed in this study thus opens a new path for viscosity characterization in confined, heterogeneous, and complex systems.