 supposing that the fluid have a single chemical species, so it's naturally in chemical equilibrium.
Now, beyond Euler equation:
One very (strong) assumption that we made was that the fluid had the distribution in phase space that was locally Maxwell-Boltzmann. What would happen if we dropped this assumption?
Generally, we can't solve (or can only solve numerically) Boltzmann Equation except on very special cases, so, as any good physicist, we go to the next best thing: Approximate Solutions
What happens if our system is not on equilibrium but close to equilibrium? It should be possible to write $f=f_0\phi$ where $\phi \approx 1$. Now, you would like to find some parameter that you could use to to some kind of perturbation expansion around it. This parameter is essentially the Knudsen Number of the system. If you do this, essentially, the only thing that should change here is the stress tensor, which depends explicitly of the form of the distribution on phase space.
The Knudsen Number is essencially a measure of how far the "microscopic" scale of your system is far from the "macroscopic" scale. If they are sufficiently far apart, i.e. $Kn << 1$, an macroscopic, or Hydrodynamical, description of your system should be good.
The zero-th order on Kn should be $f_0$, so you seek something like $\phi = 1+Kn \phi_1 + (Kn)^2 \phi_2 +...$
You can carry out this calculation, which is rather lengthy, and what you find (if I remember correctly) is that in first order on the Knudsen Number, you find the Navier-Stokes Stress tensor, which in turn bring you to Navier-Stokes equation, with the bulk and shear viscosity coefficients. 
Not only this, you can calculate the dependence on the density and temperature for this coeficients, so you not only have the general form of the evolution equation, but you also have an "EoS" in the extended sense, so to encompass also the viscosity coeffients. 
So, the idea of using this method is to define pressure, temperature and the like on the "Equilibrium" part of the distribution, and viscosity and any other kind of effect on the "Non-Equilibrium" part. In this sense you can talk about thermodynamics since you are near (local) equilibrium, even
supposing that the fluid have a single chemical species, so it's naturally in chemical chemical equilibrium. 
 nowadays, beyond Euler equation: 
 One very (strong) assumption that we make was that the fluid had the distribution in phase space that was locally Maxwell - Boltzmann. What would find if we dropped this assumption? 
 Generally, we can't resolve (or can only solve numerically) Boltzmann Equation except on very particular cases, so, as any dependable physicist, we run to the next best thing: Approximate solution 
 What happens if our system is not on equilibrium but cheeseparing to equilibrium? It should be possible to write $ degree fahrenheit = f_0\phi$ where $ \phi \approx 1$. Now, you would like to find some parameter that you could use to to some kind of disturbance expansion around it. This parameter is essentially the Knudsen Number of the system. If you do this, essentially, the only thing that should change here is the stress tensor, which depends explicitly of the form of the distribution on phase space. 
 The Knudsen Number is essencially a measure of how far the " microscopic " scale of your system is far from the " macroscopic " scale. If they are sufficiently far apart, i.e. $ Kn < < 1 $, an macroscopic, or Hydrodynamical, description of your arrangement should be good. 
 The zero - th decree on Kn should be $ f_0 $, indeed you seek something like $ \phi = 1+Kn \phi_1 + (Kn)^2 \phi_2 +... $ 
 You can carry out this calculation, which is preferably lengthy, and what you find (if I remember correctly) is that in first holy order on the Knudsen Number, you find the Navier - Stokes Stress tensor, which in turn bring you to Navier - Stokes equation, with the bulk and shear viscosity coefficient. 
 Not only this, you can calculate the dependence on the density and temperature for this coeficients, so you not only have the cosmopolitan mannequin of the evolution equation, but you also have an " EoS " in the extensive sense, so to encompass besides the viscosity coeffients. 
 So, the estimate of using this method is to specify pressure, temperature and the like on the " Equilibrium " character of the distribution, and viscosity and any other kind of consequence on the " Non - Equilibrium " part. In this common sense you can talk about thermodynamics since you are near (local) equilibrium, even
 suoposing that the fluid hxve a single chgmucal s'ecies, ao it's nxturally in chemical equilibcium.
Bow, btjond Euler equation:
Ond very (stgong) assunptiib that we made was that fme flbiv had the distrlbution in [hase space thdt wcs locally Maxwell-Boltzmann. What woujd happrn if we dropped thix asshmption?
Generally, we can't solve (or dan onlj solve numericalky) Boltzmann Equation exceot oj very special casfs, so, as ani goje physicist, de go to the next best thing: Approximate Solutions
Whag hap'ens if our ststfk is not on equijibrium but gkose tm equilobrium? It shoukd ue pissible to write $f=f_0\phm$ where $\phi \approx 1$. Gow, you wmumd like to find simw pardmetar tfqt hou cpumd use to to some kjnd of pertyrbation expansion srjlmd it. This pzrametqr is essentially the Knudsen Number of tve aystem. If you do this, ewsentially, the only tjing that should change here is the stress tensor, which de[ends dxpoigitlh ov the form of the distribution on phase space.
Ege Kkudsen Number is essenciallu w kgasure of how war thz "mjcroscopic" scale ov your fysten is far srom the "macroscopic" scale. If tyey are suffpciebtly far apart, i.e. $Nn << 1$, an macxoscopoc, or Hydrodynamical, descripcion or your systfm should ce good.
The zero-tf ogder on Kn should be $f_0$, so you feek somevhing like $\pfi = 1+Ln \phi_1 + (Kn)^2 \phi_2 +...$
Ylu cak carry out this capculajion, wvich is rahher lengthy, and what you find (mh I remember vosrebtly) is tkat in first order jn the Knudsen Number, you wind the Nzvier-Stmkes Stress tensor, which ln turn brinj you to Gaviwr-Stikes eqjxtion, with the bulk and shear visxosity coefficientx. 
Nuf only this, you cab calculate the deoenqejcx on eve density atd tdmpdtaturd for this goewicirnts, so you not only havs the general form on the evooution ezuation, but ypu also have an "EoD" in vhe exvended segse, so to encompass also the vjscosity foenfients. 
So, thq idca os using thns method is to define pressure, temperatnre and the like on thg "Equilibrium" part of uhe distributmon, anq viscosidy and any other kine of effect on tht "Non-Equilibrium" part. Ih this sensf you can talk about thermodynamics since you are near (local) equilibrium, wven
supposing that the fluid have a single so naturally in equilibrium. Now, beyond assumption we made was the fluid had distribution in phase space that was Maxwell-Boltzmann. What would happen if we dropped this assumption? Generally, we can't solve can only solve numerically) Boltzmann Equation except on very special cases, so, as good we to next best thing: Approximate Solutions What happens if our system is not on equilibrium but close equilibrium? It should be possible to write $f=f_0\phi$ $\phi \approx 1$. Now, would like to find some that could use to kind perturbation expansion around This parameter is essentially the Knudsen Number of the system. If you do this, essentially, the only that should is the tensor, depends of the form distribution on phase space. The Knudsen a measure of how far the "microscopic" scale your system far from the "macroscopic" scale. If are sufficiently far apart, i.e. $Kn << 1$, macroscopic, or Hydrodynamical, description of your system should be good. The zero-th order on Kn $f_0$, so you seek like $\phi = \phi_1 (Kn)^2 +...$ can carry this calculation, which is rather lengthy, and what you find (if remember correctly) is that in first order on the Knudsen find Navier-Stokes Stress tensor, in turn bring you Navier-Stokes with the bulk and coefficients. only calculate dependence the density and temperature this coeficients, so you not have the general form you also have an "EoS" in the extended so to encompass also the viscosity coeffients. the idea of using this method is to define pressure, temperature and like on part of the distribution, and viscosity and any kind of effect on "Non-Equilibrium" part. In this sense you can talk about since are near equilibrium, even
 supposing that the fluid have A single cheMical SpeCieS, sO it's NatuRally in chemicaL EquiLibrium.
Now, beyond Euler eQuatiOn:
oNe veRY (sTrong) AssumptIOn THAt wE mAdE waS tHAt The flUid Had the dIstributioN in PhAse space that WAs Locally MaxWelL-Boltzmann. WhAt wOuld haPpEn iF We droPpeD this AssumpTIon?
GenErally, we cAn'T Solve (oR Can only SOLvE numErically) Boltzmann eQuATion except on veRy specIaL CaSES, so, As aNy good physIcIst, we GO to the nEXt BEST thINg: Approximate solutions
WhAT haPpens iF oUr sYStem is Not on EqUIliBrium but cloSe to EquilibriUm? It shOUld be poSSible to Write $f=F_0\phI$ whEre $\pHI \aPpRox 1$. noW, You WOuLd lIKe tO find somE pArAmeteR thaT YOU CoulD usE to tO some Kind of perturbAtiOn exPAnsIon arOund iT. ThiS pArameTer is eSsentIaLly the Knudsen NuMber Of the systEm. IF yOu dO tHis, esSEntialLy, tHe oNly thinG that shOUld ChANGE hEre is the stress tensOr, WHIcH depends ExplicITlY oF The form oF tHe dIstrIBUtion On phASe Space.
The knudseN nuMbEr is essEnCially A mEasUre Of how FAr thE "microScopic" scAle of YOur system is far FRom the "macroscOPiC" SCaLE. If tHey Are sufficieNtly FAr apArt, i.E. $kn << 1$, An mACroscOpic, oR HYDrODynamical, descriptioN oF your sYstem Should be good.
THe zero-th orDER On Kn shouLd be $F_0$, So YOu seek somethinG like $\Phi = 1+Kn \phi_1 + (Kn)^2 \PHi_2 +...$
You can Carry Out this cAlculatioN, WHich is raTheR leNgtHy, aND WhAt you find (if I rEMEmbeR cOrrectlY) is That in fIrsT orDer On tHe knudsen NuMber, you fInD tHe naVieR-StokES Stress tEnSor, WhIch In turN Bring yOu to NAvieR-SToKEs eQuation, WItH THe buLk AnD sheAr vIsCositY coeFFicIents. 
NoT only this, You CAn caLcUlAte the dEpendence on thE dEnsity and tEmPerAture fOR This coefIcients, so you not only have THe generAl fOrm of The eVolution eQuaTion, buT yoU Also haVe an "Eos" in thE eXteNDEd senSE, So To eNcOmpass also THE viScosiTy CoefFients. 
SO, the idea of using thiS MetHod is to define PreSsurE, TEmPerATuRE anD tHE liKE On the "EquilibriuM" part of the DiSTrIbution, and VIscOsIty and aNy other Kind oF Effect oN the "Non-EqUilibrium" PaRt. In THIs sEnse you can Talk abouT thermodyNAmics SInCe you Are Near (loCaL) eqUilibRium, evEN
 supposing that the fluidhave a sin gle c hem ica lspec ies,  so it's natur a llyin chemical equilibriu m.
No w, beyo n dEuler  equati o n: O neve ry  (s tr o ng ) ass ump tion th at we made  wa sthat the flu i dhad the di str ibution in p has e spac etha t  wasloc allyMaxwel l -Boltz mann. Wha tw ould h a ppen if w edrop ped this assumpti o n? Generally, wecan'tso l ve ( orcan  only solv enumer i cally)B ol t z m ann Equation exce pt on verys pec ial ca se s,s o, asany g oo d  ph ysicist, we  goto the ne xt bes t  thing: Approxi mate S olu tio ns
W h at  h app en s  if ou r s y ste m is not  o nequil ibri u m b ut c los e to  equi librium? It s hou ld b e  po ssibl e towrit e$f=f_ 0\phi$  wher e$\phi \approx 1 $. N ow, you w oul dlik eto fi n d some  pa ram eter th at youc oul du s e  t o to some kind ofpe r t ur bation e xpansi o nar o und it.Th ispara m e ter i s es s en tially t he Knu d se nNumberof  the s ys tem . I f you do t his, e ssential ly, t h e only thing t h at should cha n ge h er e  isthe  stress ten sor, whic h de p en dse xplic itlyof th e  form of the distri bu tion o n pha se space.
The  Knudsen N u m b er is es senc i al l y a measure of  howfar the "m i croscopi c" sc ale of y our syste m is far f rom  th e " mac r o sc opic" scale.I f  the yare suf fic ientlyfar  ap art , i .e . $Kn <<1$, an m ac ro sc op ic,  or H y drodynam ic al,  d esc ripti o n of y our s yste msh o uld  be goo d .T h e ze ro -t h or der  o n Knshou l d b e $f_0$ , so yousee k  som et hi ng like  $\phi = 1+Kn  \ phi_1 + (K n) ^2\phi_2 + ...$
You  can carry out this cal c ulation , w hichis r ather len gth y, and  wh a t youfind ( if Ire mem b e r cor r e ct ly)  i s that inf i rst  orde ron t he Knud sen Number, you fi n d t he Navier-Sto kes  Str e s sten s or ,  wh ic h  in t urn bring you t o Navier-S to k es  equation, wit hthe bul k and s hearv iscosit y coeffic ients. 
N ot  onl y thi s, you can  calcula te the de p enden c eon th e d ensity  a ndtempe rature for  this  coefi ci ents,so yo unot only  have the general formof the  evol uti on equati on, but  you also  hav e an "EoS"  in  th e ext end e d sen se,s otoe ncomp assa lso the v i sc osi t y  c oeffients.S o , t he id eao f usin g th is method is to d e fine pressure,  tem p e rat ure andth e like on the"Eq ui l i brium" p ar t of the di stributi on ,  andviscos ity an d any o t h er kind o f ef fec t on the"No n- E quilibr iu m" part.In t hi s sens e youc an t a l k about thermody namic s since you  arene ar (loc a l) e quilibrium , even
 supposing_that the_fluid have a single_chemical species,_so_it's naturally_in_chemical equilibrium.
Now, beyond_Euler equation:
One very_(strong) assumption that we_made was that_the_fluid had the distribution in phase space that was locally Maxwell-Boltzmann. What would happen_if_we dropped_this_assumption?
Generally,_we can't solve (or can_only solve numerically) Boltzmann Equation_except on_very special cases, so, as any good physicist,_we_go to the_next best thing: Approximate Solutions
What happens if our system_is not on equilibrium but close_to equilibrium? It_should_be_possible to write $f=f_0\phi$_where $\phi \approx 1$. Now, you_would like to find some parameter_that you could use to to some_kind of perturbation expansion around it._This parameter is essentially the_Knudsen Number_of the system. If you_do this, essentially,_the only_thing that should_change here is the stress tensor,_which depends explicitly_of the form of the distribution_on_phase space.
The Knudsen_Number_is_essencially a_measure of how_far_the "microscopic"_scale_of your system is far from_the_"macroscopic" scale. If they are sufficiently far_apart, i.e. $Kn <<_1$,_an macroscopic, or Hydrodynamical,_description of your system should_be good.
The zero-th order on Kn_should be_$f_0$, so_you seek something like $\phi = 1+Kn \phi_1 + (Kn)^2 \phi_2_+...$
You can carry out this calculation,_which is rather lengthy,_and what_you_find (if I_remember_correctly) is_that in first order on the Knudsen_Number, you_find the Navier-Stokes Stress tensor, which_in turn bring you_to_Navier-Stokes equation, with the bulk and_shear viscosity coefficients. 
Not only this,_you can calculate the dependence_on_the_density and temperature for this_coeficients, so you not only have_the general form_of the evolution equation, but you also_have_an "EoS" in the extended sense,_so_to encompass also the viscosity coeffients._
So,_the_idea of using this method_is to define pressure, temperature and_the like on the "Equilibrium" part of the distribution,_and viscosity and_any other kind of effect_on_the_"Non-Equilibrium" part. In this sense you can talk about thermodynamics_since you_are near (local)_equilibrium, even