When two miscible fluids, such as glycerol (glycerin) and water,
are brought in contact, they immediately diffuse in each other.
However if the diffusion is sufficiently slow, large concentration gradients exist
during some time. They can lead to the appearance of an
“effective interfacial tension”. To study these phenomena we
use the mathematical model
consisting of the diffusion equation with convective terms and of
the Navier-Stokes equations with the Korteweg stress.
We prove the global existence and uniqueness of the solution for the
associated initial-boundary value problem in a two-dimensional bounded domain.
We study the longtime behavior of the solution and show that it converges
to the uniform composition distribution with zero velocity field.
We also present numerical simulations of miscible drops and show how
transient interfacial phenomena can change their shape.