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Normalizing flows are objects used for modeling complicated probability density functions, and have attracted considerable interest in recent years. Many flexible families of normalizing flows have been developed. However, the focus to date has largely been on normalizing flows on Euclidean domains; while normalizing flows have been developed for spherical and other non-Euclidean domains, these are generally less flexible than their Euclidean counterparts. To address this shortcoming, in this work we introduce a mixture-of-normalizing-flows model to construct complicated probability density functions on the sphere. This model provides a flexible alternative to existing parametric, semiparametric, and nonparametric, finite mixture models. Model estimation is performed using the expectation maximization algorithm and a variant thereof. The model is applied to simulated data, where the benefit over the conventional (single component) normalizing flow is verified. The model is then applied ....
Abstract In this study, a numerical investigation of fluid flow and heat transfer of Al O water nanofluids in a pillow plate heat exchanger with a two-phase model is investigated. The flow regime studied in this study is in the laminar flow regime such that the Reynolds numbers are 250, 500, 750, and 1000 and the volume fraction of the nanoparticles remained so that the fluid stayed in Newtonian. The volume fraction of nanoparticles is (Formula presented.) 0, 1, 2, and 3 %. In this study, the effect of the shape of the welding points used in the pillow plate has also been investigated in such a way that three modes of circular welding point, large-diameter elliptic welding point along with the flow and large-diameter elliptic welding point perpendicular to the stream have been investigated. Numerical simulation results show that using nanofluid and increasing Reynolds number improves heat transfer, as well as circular welding point, shows the best heat transfer rate. The shape ....