  The paper presents numerical methods for unsteady flows of a viscous
incompressible fluid in internal domains with many inlet/outlet sections. The
novel variants of dissipative boundary conditions augmented by the inertia
terms are used at the inlets/outlets of the flow domain. Volumetric flow rates
or inlet/outlet average pressure are imposed as additional constrains imposed
on a fluid motion. The variational formulations of the Stokes problem with such
conditions and constrains are presented and the solution methods are proposed.
These methods are based on superposition of appropriately defined auxiliary
Stokes problems. Extension of the proposed methodology to the Navier-Stokes
flows, based on the Operator-Integration-Factor-Splitting (OIFS) technique, is
also described. Next, a nonlinear extension of the inertial-dissipative
conditions is formulated and incorporated in the solution framework for the
Navier-Stokes flows. Finally, an alternative approach to flow problems with
such conditions, based on optimal control theory and adjoint evaluation of the
goal functional gradient, is presented. A few remarks concerning numerical
implementation issues are formulated.
