Flow of “stress power-law” fluids between parallel rotating discs with distinct axes

• Flow of “stress power-law” fluid in an orthogonal rheometer geometry is considered.
• The symmetric part of the velocity gradient is given by a power-law of fluid stress.
• Solved by seeking semi-inverse form for stress and velocity fields.
• Can enforce no-slip at the boundary or specify stresses.
• Can get non-unique solutions or even unsolvability in Stokes flow.

The problem of flow between parallel rotating discs with distinct axes corresponds to the case of flow in an orthogonal rheometer and has been studied extensively for different fluids since the instrument׳s inception. All the prior studies presume a constitutive prescription of the fluid stress in terms of the kinematical variables. In this paper, we approach the problem from a different perspective, i.e., a constitutive specification of the symmetric part of the velocity gradient in terms of the Cauchy stress. Such an approach ensures that the boundary conditions can be incorporated in a manner quite faithful to real world experiments with the instrument. Interestingly, the choice of the boundary condition is critical to the solvability of the problem for the case of creeping/Stokes flow. When the no-slip condition is enforced at the boundaries, depending on the model parameters and axes offset, the fluid response can show non-uniqueness or unsolvability, features which are absent in a conventional constitutive specification. Moreover, in case of creeping/Stokes flow with prescribed values of the stress, the fluid response is indeterminate. We also record the response of a particular case of the given “stress power-law” fluid; one that cannot be attained by the conventional power-law fluids.