The falling sphere problem and capturing enhanced drag with Boger fluids

In this computational study, the ability of an extensional White-Metzner construction with the FENE-CR model is considered to reflect experimental enhanced drag data of Jones et al. [1]. The numerical drag predictions for three different aspect ratios of sphere:tube radii {0.5, 0.4, 0.2} are obtained with a hybrid finite element/volume (fe/fv) algorithm. Excellent agreement is extracted for all three aspect ratios against the experimental measurements, and at any specified rate, the tighter-fitting the aspect ratio the lower the resulting drag. Moreover, as the Weissenberg number is increased, the transition between steady-state and oscillatory flow is recognised from the instantaneous pressure data, prior to numerical divergence. A main realisation in this study is that it is important to select the same procedure of Wi-continuation across experimental and computational protocols, to extract comparable levels of drag. Clearly the λ1-increase mode (common computational form), is more involved than the Q-increase mode (usual experimental form), and as such, less robust as a reliable method for accurate drag prediction and enhanced drag capture. In general, flow-rate increase (Q-increase) conditions generate larger drag enhancement, when compared to fluid-relaxation time increase (λ1-increase), at comparable levels of dissipative-factor (λD). The investigation also follows parametric variation in solvent fraction (βsolvent) in one particular geometric aspect-ratio instance. This reveals that at any specific fixed elasticity level, there is an increase in drag observed with rise in βsolvent. In addition, high solute/low-solvent fractions at low dissipative-factor, were only found to generate drag reduction, consistent with the literature. New and key facets to this fe/fv implementation are summarised, in appealing to: an improved velocity gradient boundary conditions imposed at the centreline (VGR-correction); continuity correction; absolute value of the stress-trace function (ABS-f-correction); increasing flow-rate solution continuation; alongside advanced techniques in fv-time discretisation, discrete treatment of pressure terms, and compatible stress/velocity-gradient representation.