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

Xinpeng Xu, Tiezheng Qian*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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 LRT 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.

Original languageEnglish
Article number475101
JournalJournal of Physics Condensed Matter
Issue number47
StatePublished - 22 Aug 2019


  • Lorentz reciprocal theorem (LRT)
  • Onsager s reciprocal relations
  • complex fluids
  • micropolar fluids
  • two-phase binary fluids


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