Local and global existence of solutions to equations for flows of fibre suspensions

We establish the existence and uniqueness, locally and globally in time, of solutions to the governing equations for fibre suspension flows, for sufficiently small data. The linear and quadratic closure rules are considered, and the rotary diffusivity is assumed to be constant. The existence of a unique classical solution, local in time, is proven for the cases of both linear and quadratic closure rules. By restricting consideration to the physically significant case of constant rotary diffusivity it is possible to obtain the first results on global existence for this problem. In particular, the existence and uniqueness of a classical solution, globally in time, is established for sufficiently small data. The solution is shown to be stable in the absence of a body force. Existence and uniqueness of solutions to the steady problem are also established.