  The aim of this short paper is threefold. First, we develop an implicit
generalization of a constitutive relation introduced by Korteweg [10] that can
describe the phenomenon of capillarity. Second, using a sub-class of the
constitutive relations (implicit Euler equations), we show that even in that
simple situation more than one of the members of the sub-class may be able to
describe one or a set of experiments one is interested in describing, and we
must determine which amongst these constitutive relations is the best by
culling the class by systematically comparing against an increasing set of
observations. (The implicit generalization developed in this paper is not a
sub-class of the implicit generalization of the Navier-Stokes fluid developed
by Rajagopal [20, 21] or the generalization due to Prusa and Rajagopal [19], as
spatial gradients of the density appear in the constitutive relation developed
by Korteweg [10].) Third, we introduce a challenging set of partial
differential equations that would lead to new techniques in both analysis and
numerical analysis to study such equations.
