  The Greer, Pujo-Menjouet andWebb model [Greer et al., J. Theoret. Biol., 242
(2006), 598-606] for prion dynamics was found to be in good agreement with
experimental observations under no-flow conditions. The objective of this work
is to generalize the problem to the framework of general
polymerization-fragmentation under flow motion, motivated by the fact that
laboratory work often involves prion dynamics under flow conditions in order to
observe faster processes. Moreover, understanding and modelling the
microstructure influence of macroscopically monitored non-Newtonian behaviour
is crucial for sensor design, with the goal to provide practical information
about ongoing molecular evolution. This paper's results can then be considered
as one step in the mathematical understanding of such models, namely the proof
of positivity and existence of solutions in suitable functional spaces. To that
purpose, we introduce a new model based on the rigid-rod polymer theory to
account for the polymer dynamics under flow conditions. As expected, when
applied to the prion problem, in the absence of motion it reduces to that in
Greer et al. (2006). At the heart of any polymer kinetical theory there is a
configurational probability diffusion partial differential equation (PDE) of
Fokker-Planck-Smoluchowski type. The main mathematical result of this paper is
the proof of existence of positive solutions to the aforementioned PDE for a
class of flows of practical interest, taking into account the flow induced
splitting/lengthening of polymers in general, and prions in particular.
