  The classical Lorentz reciprocal theorem (LRT) was originally derived for
slow viscous flows of incompressible Newtonian fluids under the isothermal
condition. In the present work, we extend the LRT from simple to complex fluids
with open or moving boundaries that maintain non-equilibrium stationary states.
In complex fluids, the hydrodynamic flow is coupled with the evolution of
internal degrees of freedom such as the solute concentration in two-phase
binary fluids and the spin in micropolar fluids. The dynamics of complex fluids
can be described by local conservation laws supplemented with local
constitutive equations satisfying Onsager's reciprocal relations (ORR). We
consider systems in quasi-stationary states close to equilibrium, controlled by
the boundary variables whose evolution is much slower than the relaxation in
the system. For these quasi-stationary states, we derive the generalized
Lorentz reciprocal theorem (GLRT) and global Onsager's reciprocal relations
(GORR) for the slow variables at boundaries. This establishes the connection
between ORR for local constitutive equations and GORR for constitutive
equations at boundaries. Finally, we show that the LRT can be further extended
to non-isothermal systems by considering as an example the thermal conduction
in solids and still fluids.
