Past exam of the advanced fluid dynamics course of the University of Oxford / 2016 / 2 / c / i / Solution

Let $g=\dot{\gamma }$ and use $G_{yx}=g$ for the source's velocity gradient convention. Seek the homogeneous stationary stress reached under steady simple shear flow. The material derivative of that stress tensor vanishes. In the $xy$ plane,The $xx$, $yy$ and $zz$ components of the constitutive equation give, respectively,Thus, putting $D=1+\alpha (2\tau -\alpha )g^2$ and using the shear stress calculated below,The out-of-plane shear components vanish on the homogeneous branch. The two normal-stress differences areThe ratio is defined at nonzero shear when $N_1\ne 0$; at $g=0$ both differences vanish, and the displayed constant is their limiting ratio. An arbitrary isotropic pressure in the total Cauchy stress tensor cancels from both differences; it is not an arbitrary addition to the constitutively defined extra stress $\mathsf{T}$ itself.