A review of the second normal-stress difference; its importance in various flows, measurement techniques, results for various complex fluids and theoretical predictions

Abstract

Abstract Shear flow is ubiquitous. Not only is it arguably the most widely-used deformation type to characterise complex fluids in rheological studies but also, in practice, the deformation most likely to occur in the great majority of flows, e.g. involving fluid transport through pipes or conduits. In steady simple shear flow the rheological properties of a complex fluid are completely characterised in just three material functions; the variation with shear rate of the shear viscosity and the so-called first and second normal-stress differences. Despite requiring only three material functions to be completely characterised, most shear-flow rheological characterisations are usually restricted simply to the shear viscosity and, at best, the variation of the first normal-stress difference N1 with shear rate. The second normal-stress difference N2 remains very much neglected. For dilute polymer solutions where this quantity may be negligibly small in comparison to the first normal-stress difference, such neglect is justified but for a whole range of complex fluids – indeed even polymer solutions outside of the dilute regime and especially melts – it is not clear that N2 may be safely disregarded. Indeed, in this review article we spotlight a number of flows where second normal-stress differences are of importance and potentially major consequence. Following this attention, we review the many experimental techniques which have been proposed for its measurement and survey the available literature for measurements of this quantity for various complex fluids including the aforementioned polymeric solutions, melts, liquid crystals, dense non-Brownian suspensions (both with Newtonian and complex fluid bases), semi-dilute wormlike micellar fluids and magnetorheological fluids. Theoretical predictions for N2 from various commonly-used continuum constitutive equations – primarily from the polymer literature – are also given and their asymptotic predictions at low and high shear rates compared. Finally, we end with a brief summary and outlook.