Coupled constitutive relations: a second lawbased higher-order closure for hydrodynamics

Abstract

In theclassicalframework,theNavier–Stokes–Fourier equations areobtainedthroughthelinearuncoupled thermodynamic force-fluxrelationswhichguarantee the non-negativityoftheentropyproduction. However,theconventionalthermodynamicdescrip- tion isonlyvalidwhentheKnudsennumberis sufficientlysmall.Here,itisshownthattherangeof validity oftheNavier–Stokes–Fourierequationscan be extendedbyincorporatingthenonlinearcoupling among thethermodynamicforcesandfluxes.The resultingsystemofconservationlawsclosedwith the coupledconstitutiverelationsisabletodescribe many interestingrarefactioneffects,suchasKnudsen paradox, transpirationflows,thermalstress,heat flux withouttemperaturegradients,etc.,which cannot bepredictedbytheclassicalNavier–Stokes– Fourier equations.Forthissystemofequations, a setofphenomenologicalboundaryconditions, which respectthesecondlawofthermodynamics, is alsoderived.Someofthebenchmarkproblems in fluidmechanicsarestudiedtoshowthe applicability ofthederivedequationsandboundary conditions.