Numerical simulation of Poiseuille flow for S-shaped rheology fluid: Streamwise banding and viscous sandglasses

Recent experiments on pressure-driven Poiseuille flow of cornstarch in a cylindrical tube (Talon and Salin, 2024) show a surprising behavior. The measured flow curve, i.e. the flow rate versus the applied pressure drop, is indeed non-monotonic: the flow rate increases monotonically at low pressure drops up to a maximum, after which it decreases abruptly to an almost constant flow rate regardless of further increases in pressure drop. Cornstarch is known to exhibit discontinuous shear thickening (DST) behavior (Fall et al., 2012). In addition, recent experiments (Denn et al., 2018; Darbois Texier et al., 2020; Bougouin et al., 2024) suggest that the rheology may ultimately be S-shaped, where the shear rate is a nonmonotonic function of stress, similar to the model proposed by Wyart and Cates (Wyart and Cates, 2014). To account for the observed jump- plateau behavior of the flow rate, one possibility is that Poiseuille flow for S-shaped rheology exhibits some kind of phase segregation, where the pressure gradient becomes non-uniform. The pressure gradient segregate between two types of region, with either high pressure gradient or low one. This kind of "streamwise banding"were analyzed in Talon and Salin (2024) using the lubrication approximation and assuming simple dynamical stochastic version of the nonmonotonic S-shaped rheology Wyart-Cates model. The plateau behavior is then related to an increase of the high viscous region as the pressure is increased. The mere presence of anon- monotonic rheological curve could then be sufficient to predict the occurrence of banding in the streamwise direction, even if the suspension remains homogeneous. In this paper, we aim to analyze this prediction by disregarding the lubrication approximation and directly solving the flow of a shear thickening fluid with S-shaped rheology. Using 2D TRT Lattice Boltzmann simulations, we observe that the plateau inflow rate is indeed associated with a streamwise segregation of the pressure gradient. In addition, we show that regions of high pressure gradients are due to the formation of a highly viscous structure similar to a "sandglass"shape. We then analyze the occurrence of these sandglass structures as a function of the system parameters.