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

## TL;DR

This paper extends the classical Lorentz reciprocal theorem to complex fluids with non-equilibrium stationary states and non-isothermal systems, linking local constitutive relations with boundary conditions and broadening its applicability.

## Contribution

The work derives a generalized Lorentz reciprocal theorem and global Onsager's reciprocal relations for complex fluids and non-isothermal systems in quasi-stationary states.

## Key findings

- Derived the generalized Lorentz reciprocal theorem (GLRT) for complex fluids.
- Established the connection between local Onsager's reciprocal relations and boundary conditions.
- Extended the LRT to non-isothermal systems like thermal conduction in solids and fluids.

## 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 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.

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Source: https://tomesphere.com/paper/1906.04458