The Saffman–Taylor problem and several sets of remarkable integral identities

The methodology based on the so-called global relation, introduced by the first author, has recently led to the derivation of a novel nonlinear integral-differential equation characterizing the classical problem of the Saffman–Taylor fingers with nonzero surface tension. In the particular case of zero surface tension, this equation is satisfied by the explicit solution obtained by Saffman and Taylor. Here, first, for the case of zero surface tension, we present a new nonlinear integrodifferential equation characterizing the Saffman–Taylor fingers. Then, by using the explicit Saffman–Taylor solution valid for the particular case of zero surface tension, we show that the above equations give rise to sets of remarkable integral trigonometric identities.