Tensor Maxwellian

Most fluids in engineering are Newtonian. However, fluids such as petroleum and blood are Maxwellian, that is, the stress tensor is not only a function of the strain rate but also a function of strain itself. The Maxwellian relaxation is the ratio between the fluid viscosity and stiffness. 

Both the absolute- and local Maxwellians are termed equilibrium distributions. This result relates to the local and instantaneous equilibrium assumption in continuum mechanics as discussed in chap. 1, showing that the assumption has a probabilistic fundament. It also follows directly from the local equilibrium assumption that the pressure tensor is related to the thermodynamic pressure, as mentioned in sect. 2.3.3. 

Applying the local Maxwellian distribution function (2.225), explicit expressions can be obtained for the heat flux, q and the pressure tensor P. 

It is important to note that this distribution function (2.244), defined so that it resembles (2.243) but with the constant values of n, v and T in (2.243) replaced by the corresponding functions of r and t, remains a solution to (2.242). This distribution function, which is called the local Maxwellian, makes the kinetic theory much more general and practically relevant. Both the absolute- and local Maxwellians are termed equilibrium distributions. This result relates to the local and instantaneous equilibrium assumption in continuum mechanics as discussed in Chap. 1, showing that the assumption has a probabilistic fundament. It also follows directly from the local equilibrium assumption that the pressure tensor is related to the thermodynamic pressure, as mentioned in Sect. 2.5. 

The relaxation equation for a Maxwellian incompressible isotropic liquid is obtained from the law of anisotropic relaxation in the following partial form of the relaxation time Xj.ju and viscosity tensors ... 

If one allows for anisotropic frictional forces by retaining the friction tensor fin equation (51), and allowing for anisotropic Brownian motion by allowing the Maxwellian velocity distribution to be skewed (so that = — (kT/ F)[(5/5ry) f F]), then the diffusion equation and stress tensor expressions become... 

If hydrodynamic interaction is neglected, if a Maxwellian velocity distribution is assumed for the beads and if the friction tensor is taken to be isotropic, then the diffusion equation for the distribution function t) is... 

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