Geodesic
On the sufficient conditions for the S-shaped Buckley--Leverett function
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 51, Tome 536 (2024), pp. 247-260

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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. 
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     author = {N. V. Rastegaev},
     title = {On the sufficient conditions for the {S-shaped} {Buckley--Leverett} function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     publisher = {mathdoc},
     volume = {536},
     year = {2024},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_536_a11/}
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N. V. Rastegaev. On the sufficient conditions for the S-shaped Buckley--Leverett function. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 51, Tome 536 (2024), pp. 247-260. http://geodesic.mathdoc.fr/item/ZNSL_2024_536_a11/