# The simplest model for the constitutive equation of a

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The simplest model for the constitutive equation of a real fluid is the Newtonian fluid, which is a special case of the so-called St kifl idStokesian fluid. The Stokesian fluid satisfiesA Newtonian fluid is a special case of a Stokesian fluid in which),,(TpDklijijττ=A Newtonian fluid is a special case of a Stokesian fluid in which (1) τis a linear function of the components of D, and (2) there are no preferred direction properties (i.e., isotropy).The most general linear form is klijklijDβτ=hifthdthi81thlddβwhere is a fourth order tensor having 81 components, whose values depend on the two chosen independent thermodynamic variables, say, for example, the pressure, p, and the temperature, T. ijklβFluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (李雨)
Newtonian fluid (2)ijklijklikjliljkβαδ δβδ δγδ δ=++Theory of isotropic tensor ijklijklikjliljkwhere α, β, and γare functions of thermodynamic state. kljkiljlikklijijD)(δγδδβδδαδτ++=ThenDDDγβαδ++=jiijkkijDDDγβαδ++=ijkkijDD)(γβαδ++=ijjiDD=(since )2ijijDλδμ∇ ⋅+uλ: second viscosity , μ: dynamic viscosity (to be determined experimentally) DuIτμλ2+=Vector form Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (李雨)
Newtonian fluid (3)Recall tensor decomposition : τIT+=pWe haveτIT+=pijijijijDpTμλδδ2++=uDuIITμλ2++=pFluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (李雨)
Newtonian fluid (4)Proposed that qis linearly proportional to the gradient of the temperature field according to the experimental observationijixTKq=temperature field according to the experimental observation, (Fourier’s law)For isotropic fluid,jxijijkKδ=(k: thermal conductivity, a function of the thermodynamic state)TheniixTkq=Tk=qorFluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (李雨)
Newtonian fluid (5)Check against the 2nd law of thermodynamicsOn substituting the constitutive laws into the entropy equation, we have02Φ+TTTTkwe haveDD)uIDττD:+=:=:Φμλ2(where the dissipation function :DD)uIDττD:+=:=:Φμλ2((2(2λμλμ=∇ ⋅:+:=∇ ⋅+:2u)IDDDu)DD2222222222()2()DDDDDDDDDDDDλμ=+++++++++++112233111213212223313233()2()DDDDDDDDDDDDλμ=+++++++++++222222211223311222233331112233122()()[()()() ]4()33DDDDDDDDDDDDλμμμ=+++++++++20,k0,μ032+μλκ(bulk viscosity)Fluid Mechanics (Spring 2019) – Chapter 2 - U. Lei (李雨)
Example 6 : Interpretation of μ,the viscosity coefficientConsider simple shear flow :(),uu y=0,v=0=wThe flow is incompressible since 0=uThe stress tensor :duduThe traction (force) on a surface element with normal j isThe stress tensor :++=jiijITdydudydupμThe traction (force) on a surface element with normal j is ( )pdudμ==+jt jj Tidynormal part shearing partIn general :2(2)(2)xyxyyyzyzyyyTTTDpDDμμλμ=+∇ ⋅+=++=++t jj TjkijuikIn general :