Flow past a sphere: Numerical predictions of thixo-viscoelastoplastic wormlike micellar solutions

Abstract

In this work, we present numerical solutions to the benchmark flow past a sphere problem considering the rheology of semi-diluted and concentrated wormlike micellar viscoelastic solutions using the most recent model-variant of the Bautista–Manero–Puig (BMP) family of fluids, i.e. the BMP+ _ τ p model (López-Aguilar et al., [1] ). This study is performed using our in-house finite volume/element algorithm, stabilised to attain highly-elastic non-linear solutions with two novel constitutive and boundary-condition implementations, i.e. the ABS- f (ABSolute f -functional) and the VGR (Velocity GRadient) corrections (López-Aguilar et al., [2] ). The structuring level of various wormlike micellar fluids with different solute concentrations (from semi-diluted to extremely concentrated solutions) and degrees of strain-hardening is studied with a dynamic dimensionless fluidity that connects the internal structure construction-destruction dynamics with viscoelasticity. In complex flow, a viscoelastic flow-regime unveils the prediction of a flow transition from a stable steady flow (related to a steady sphere settling) to an unsteady quasi-periodic flow regime, in which transient cycles of vortex formation-disappearance behind the sphere are present, resembling experimental findings on negative-wake instabilities and oscillations in the settling velocity for the descend of particles in wormlike micellar solutions. This instability is predicted here for semi-diluted to concentrated wormlike micellar solutions and it is located in flow-rate ranges corresponding to extensional Deborah numbers in the interval of around 26 . 7 ≤ D e e x t ≤ 40 . 3 , as reported experimentally. This behaviour is correlated with a chaotic flow-structure behind the obstacle, for which strong fluctuation in the fluid’s internal-structure is recorded. In the steady viscoelastic flow regime, correlation between the BMP+ _ τ p fluidity structure-variable and its normal-stress response around and downstream of the sphere (in the form of localised extrema) is detected, where interaction between the rheological response of the material and mixed shear-to-extensional deformation provide insights into the forces the fluid elements experiment in the sphere wake. Here, a monotonic decline in the drag-correction factor is observed, this being one of the main experimental manifestations of the steady sphere-settling phase in these materials. These findings may serve to further understand flow transitions and instabilities related to negative wakes in this kind of shear-thinning extensional-hardening fluids (Rothstein and Mohammadigoushki, [3] ). In a plastic regime, complex thixo-viscoelastoplastic BMP+ _ τ p features manifest through the prediction of fore-aft asymmetries promoted by the thixotropic structure construction-destruction and viscoelastic properties of the model under extremely low solute-concentrations. Predicted yield-fronts resemble those reported experimentally by Holenberg et al. [4] for carbopol solutions. • Negative-wake instability predictions in the De -range recorded experimentally. • Correlation between extensional viscosity and normal stresses in the sphere wake. • Localised N 1 and internal-structure f -functional in the sphere wake. • First normal-stress activity in the sphere wake follows extensional viscosity response. • Asymmetric yield-fronts under highly-concentrated gel-like wormlike micellar solutions.