  The paper is devoted to homogenization of two-phase incompressible
viscoelastic flows with disordered microstructure. We study two cases. In the
first case, both phases are modeled as Kelvin-Voight viscoelastic materials. In
the second case, one phase is a Kelvin-Voight material, and the other is a
viscous Newtonian fluid. The microscale system contains the conservation of
mass and balance of momentum equations. The inertial terms in the momentum
equation incorporate the actual interface advected by the flow. In the
constitutive equations, a frozen interface is employed. The interface geometry
is arbitrary: we do not assume periodicity, statistical homogeneity or scale
separation. The problem is homogenized using G-convergence and oscillating test
functions. Since the microscale system is not parabolic, previously known
constructions of the test functions do not work here. The test functions
developed in the paper are non-local in time and satisfy divergence-free
constraint exactly. The latter feature enables us to avoid working with
pressure directly. We show that the effective medium is a single phase
viscoelastic material that is not necessarily of Kelvin-Voight type. The
effective constitutive equation contains a long memory viscoelastic term, as
well as instantaneous elastic and viscous terms.
