  Binary-fluid flows can be modeled using the Navier-Stokes-Cahn-Hilliard
equations, which represent the boundary between the fluid constituents by a
diffuse interface. The diffuse-interface model allows for complex geometries
and topological changes of the binary-fluid interface. In this work, we propose
an immersed isogeometric analysis framework to solve the
Navier-Stokes-Cahn-Hilliard equations on domains with geometrically complex
external binary-fluid boundaries. The use of optimal-regularity B-splines
results in a computationally efficient higher-order method. The key features of
the proposed framework are a generalized Navier-slip boundary condition for the
tangential velocity components, Nitsche's method for the convective
impermeability boundary condition, and skeleton- and ghost-penalties to
guarantee stability. A binary-fluid Taylor-Couette flow is considered for
benchmarking. Porous medium simulations demonstrate the ability of the immersed
isogeometric analysis framework to model complex binary-fluid flow phenomena
such as break-up and coalescence in complex geometries.
