For a Newtonian fluid, the stress and strain rate of a fluid phase k are linearly related and are expressible as in which all quantities have been previously defined. When the viscosity coefficients λk and μk are dependent on the strain rate, they are decomposed in accordance with Eq. (5.1.8): It is straightforward to demonstrate that Subsequent time averaging gives When λk and μk are independent of velocity gradients, 3i〈λk〉LF = λk, 3i〈μk〉LF = μk, $\tilde {\lambda} _{kLF} = \lambda ^\prime _k = 0$, and $\tilde{\mu}_{kLF} = \mu ^\prime _k = 0$. In addition, ${}^{3i}\langle {\nabla \cdot \underline{\skew3\tilde U}_{ kLF} } \rangle = 0$, ${}^{3i}\langle {\nabla,{\underline{\skew3\tilde U}}_{ kLF} } \rangle = 0$, and ${}^{3i}\langle {( {\nabla,{\underline{\skew3\tilde U}}_{ kLF} } )_c } \rangle = 0$. Consequently, for Newtonian fluids, Eq. (C.4) simplifies to which is precisely the result given in Eq. (5.5.4c).