LuckyTran |
September 9, 2022 03:37 |
It comes directly from the constitutive law including both first and second viscosity. See Lame's constant.
It often appears in a general derivation of Navier-Stokes because if dynamic viscosity is a thermodynamic parameter (i.e. you can look it up from a table of simply temperature and pressure) then it automatically implies a specific value for Lame's constant. However, experimental measurements yields Lame's constants that are very different than this assumed value and even with the opposite sign. It is there as a reminder that the conventional Navier-Stokes equations, as difficult as they are to solve, are still not the ultimate form governing all fluid phenomenon. But also, it is the original Stoke's hypothesis.
Frank White's book contains a detailed derivation from Hooke's law but not in vector nabla notation. Otherwise it's identical to the same one on Wikipedia which you've already seen.
The way to derive it yourself is to take your unit cube and apply taylor series to the forces on each face and write down the face normal and face parallel (i.e. the shear) forces, example is given here in Cauchy equation. Note that when you go to derive Boussinesq's eddy viscosity closure, you will need to do the exact same thing for the Reynolds stresses.
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