On the sufficient conditions for the S-shaped Buckley-Leverett function

The flux function in the Buckley-Leverett equation, that is, the function characterizing the ratio of the relative mobility functions of the two phases, is considered. The common conjecture stating that any convex mobilities result in an S-shaped Buckley-Leverett function is analyzed and disproved by a counterexample. Additionally, sufficient conditions for the S-shaped Buckley-Leverett function are given. The class of functions satisfying those conditions is proven to be closed under multiplication. Some functions from known relative mobility models are confirmed to be in that class

On uniqueness in Steiner problem

We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff dimension of $n$-points configurations on which some locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them

On the spectrum of the Sturm-Liouville problem with arithmetically self-similar weight

Spectral asymptotics of the Sturm-Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity property, which imposes significant restrictions on the self-similarity parameters of the weight. This work introduces a new method to estimate the eigenvalue counting function. This allows to consider a much wider class of self-similar measures

Relation between size of mixing zone and intermediate concentration in miscible displacement

We investigate the miscible displacement of a viscous liquid by a less viscous one in a porous medium, which frequently leads to the formation of a mixing zone characterized by thin fingers. The mixing zone grows in time due to the difference in speed between the leading and trailing edges. The transverse flow equilibrium (TFE) model provides estimates of these speeds. We propose an enhancement for the TFE estimates. It is based on the assumption that an intermediate concentration exists near the tip of the finger, which allows to reduce the integration interval in the speed estimate. Numerical simulations of the computational fluid dynamics model were conducted to validate the new estimates. The refined estimates offer greater accuracy than those provided by the original TFE model.Comment: 16 pages, 11 figure