Generalized Lorentz reciprocal theorem in complex fluids and in non-isothermal systems

Abstract

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 nonequilibrium 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.

Isothermal transport of a near-critical binary fluid mixture through a capillary tube with the preferential adsorption

“…Note that this derivation of the reciprocal relations is valid even when ρ (0) is inhomogeneous, unlike the one given in Ref. [26]. We assume that Eq.…”

“…The tube length is assumed to be so large that effects of the flow near the tube edges are negligible. The inhomogeneity of the correlation length is taken into account, unlike in the previous studies on similar situations [24][25][26].…”

Designing nonlinear thermal devices and metamaterials under the Fourier law: A route to nonlinear thermotics

“…What's more, the wave isolator should break the Lorentz reciprocity (or reciprocal theorem) [113][114][115][116], meaning the coupling relationship between the source and the observed field keeps the same when exchanging the positions of source and observation point. Though we can see the definition of a thermal diode requires the existence of two sources with different temperatures, which is quite different from wave isolators, the reciprocal theorem and its breaking mechanism can also be studied in heat conduction, a diffusive system [71,119,120]. It's revealed that the absence of global steady-state reciprocity can generate the diode effect in a two-port system (which can be a more general case than the one-dimensional two-terminal cases we have focused on in this review), while intrinsic directional heat generation can also make a diode in a reciprocal system when only the heat flux flowing out of the system is focused on [71].…”