Developments in the Flow of Complex Fluids in Tubes

The focus of this monograph is on the tube and channel flow of non-linear viscoelastic fluids and non-Brownian suspensions in straight tubes of arbitrary but longitudinally constant cross section. The state of the science in predicting the longitudinal field of viscoelastic fluids closely related to the secondary field and the prediction of the friction factors is thoroughly covered. Laminar and turbulent longitudinal flow fields and related phenomena including the all-important drag reduction, the early developments in the history of transversal flows, similarities with secondary transversal deformations associated with the simple shearing of solid materials in non-linear solid mechanics, as well as the analogy between the laminar flow of non-linear fluids and the turbulent flow of linear fluids in noncircular cross-sectional tubes together with constitutive criteria for the existence of secondary flows are commented on as well as the relatively recent research in secondary flows of dilute solutions in rotating pipes and channels and the related drag reduction together with the fundamental aspects of transversal flows and industrial applications. The importance and implications of secondary flows of non-linear viscoelastic fluids and the analytical methods used to investigate secondary flows are discussed. Archival literature concerning the study of secondary flows of non-Newtonian fluids is rather thin when compared to the flurry of research output in other aspects of the behavior of constitutively non-linear liquids in spite of the occurrence and the importance of secondary flows in almost every industrial operation of significance involving constitutively non-linear fluids such as extrusion processes, for instance. A comprehensive and detailed survey of secondary flows of incompressible viscoelastic liquids in straight stationary and rotating tubes is given. Only hydrodynamically and thermally developed flows are considered. Developing flows and the more complex phenomena of secondary flows of complex fluids in curved circular as well as non-circular curved tubes are not included and will be discussed elsewhere.

The state of the science in predicting the hydrodynamically developed longitudinal field of non-linear viscoelastic fluids in straight tubes of arbitrary but longitudinally constant cross section and in predicting the friction factors is summarized. Laminar and turbulent longitudinal flow fields and related phenomena including recent progress concerning the all-important drag reduction phenomena are addressed.

Early developments in the history of transversal flows, similarities with secondary transversal deformations associated with the simple shearing of solid materials in non-linear solid mechanics, as well as the analogy between the laminar flow of non-linear fluids and the turbulent flow of linear fluids in non-circular cross-sectional tubes together with constitutive criteria for the existence of secondary flows are commented on as well as the relatively recent research in secondary flows of dilute solutions in spanwise rotating pipes and channels and the related drag reduction in laminar flow. Recent findings on the fundamental aspects of transversal steady flows in straight tubes and the impact on industrial applications are reviewed emphasizing the importance of secondary flows in almost every industrial operation of significance involving constitutively non-linear fluids such as extrusion processes and viscous encapsulation. Only hydrodynamically and thermally developed flows are considered.

The partitioning of the flow domain of fluids of instantaneous elasticity into two distinct regimes, a relatively high-speed supercritical region away from the boundaries where the viscoelastic Mach number M ≫ 1 and where the vorticity is governed by a hyperbolic equation and a low-speed region near the boundaries where it is governed by an elliptic equation, is discussed. The smallness of the Reynolds Re number is not a sufficient reason for neglecting the inertial non-linearity in the flow of viscoelastic fluids with instantaneous elasticity, those which do not have a Newtonian base viscosity. The pivotal parameter is the product of the Reynolds number and the Weissenberg number M ² = Re We, where M is the viscoelastic Mach number, the ratio of a characteristic velocity of the fluid to that of the speed of shear waves. The neglect of the inertial terms may only be justified for small values of the Mach number M.

Mean secondary flows in straight tubes of non-circular cross section turbulent driven of Newtonian fluids by constant pressure gradients are discussed in their historical context as well as in terms of the most recent findings. The fundamental issues and their impact on industrial processes, in particular on processes involving particle laden flows are reviewed. Similarities with the driving mechanism of secondary laminar flows of viscoelastic fluids, criteria for the existence of secondary flows, and general classification and closure approximations for homogeneous and wall-bounded flows are discussed.

The rheology of dilute, semi-dilute, and concentrated non-Brownian suspensions is reviewed. Computing shear viscosity in different concentration regimes and recent progress in determining the normal stress functions of semi-dilute and concentrated non-colloidal suspensions are summarized. Macroscopic constitutive models for suspension flow, shear-induced and stress-induced particle migration, applications to Stokesian dynamics simulations (SDS), and efforts to improve the predictions through SDS both in unbounded and bounded flows are discussed together with challenges in shear-driven migration of non-colloidal concentrated suspensions. The complex nature and sometimes contradictory behavior reported in the literature make it challenging to construct a theoretical model. Efforts to understand the motion of particles in viscoelastic suspending media are summarized and recent research on secondary field in Poiseuille flow of shear-driven migration of suspensions is discussed together with secondary field in single-phase and multiphase turbulent flow of suspensions in tubes.