  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.
