  A high-order, degree-adaptive hybridizable discontinuous Galerkin (HDG)
method is presented for two-fluid incompressible Stokes flows, with boundaries
and interfaces described using NURBS. The NURBS curves are embedded in a fixed
Cartesian grid, yielding an unfitted HDG scheme capable of treating the exact
geometry of the boundaries/interfaces, circumventing the need for fitted,
high-order, curved meshes. The framework of the NURBS-enhanced finite element
method (NEFEM) is employed for accurate quadrature along immersed NURBS and in
elements cut by NURBS curves. A Nitsche's formulation is used to enforce
Dirichlet conditions on embedded surfaces, yielding unknowns only on the mesh
skeleton as in standard HDG, without introducing any additional degree of
freedom on non-matching boundaries/interfaces. The resulting unfitted HDG-NEFEM
method combines non-conforming meshes, exact NURBS geometry and high-order
approximations to provide high-fidelity results on coarse meshes, independent
of the geometric features of the domain. Numerical examples illustrate the
optimal accuracy and robustness of the method, even in the presence of badly
cut cells or faces, and its suitability to simulate microfluidic systems from
CAD geometries.
