The $\vec\omega-\vec{D}$ fluid: general theory with special emphasis on stationary two dimensional flows

In this paper a theory is presented in which the extra stress tensor $\vec\tau$ is allowed to depend not only on the rate of strain tensor $\vec{D}$ but also on the relative vorticity $\vec\omega$ of the fluid, i.e. on the vorticity relative to the rate of rotation of the principal straining directions. This theory has its origin in an expansion of $\vec\tau$ in terms of kinematic tensors in the limit of stationarity in a material sense (constant stretch history flows). For two dimensional flows of an incompressible fluid three tensors suffice to completely specify $\vec\tau$. The three material functions which appear can depend only on two invariants, namely the second invariant of $\vec{D}$ and on $\omega^2$. Using the predictions of an Oldroyd 8 constant fluid in a homogeneous planar flow of constant stretch history, the three material functions are studied in detail. For the special case of a quasi-Newtonian fluid shear thinning and extension thickening can directly be accounted for in the “viscosity” function.