A theory of the dynamics of a sliver or roving under roller draft is presented in a general form based on the scheme of the fluid mechanics. The motion of a fiber bundle being drafted is described by three differential equations; one relates the local elongation of the bundle to the velocity distribution along the bundle, and the other two govern the conservation of mass and momentum. The momentum flux is composed of three terms due to mass flow, tension, and inter-fiber slip. The last term, the momentum transfer by interaction among fibers, has no trivial analogue in any of traditional theories proposed earlier, though indirectly it might connect with the force for pulling out fibers of the bundle. A hypothetical system simulated by a viscoelastic fluid flow is introduced as a standard model and its stationary motion is examined for illustrating how the basic scheme works, how the velocity and tension are formed along the bundle, how the drafting force depends on the system parameters, etc.