Polymer-induced drag reduction (DR) in fluids was studied using a rotational rheometer with double-gap concentric cylinder geometry. Although both polymers (polyacrylamide (PAM) and 2-acrylamido-2methylpropane sulfonic acid (SPAM)) had molecular weights of several MDa, the contrasting polymer charge, nonionic and anionic, led to different polymer overlap concentrations (c*), PAM ≫ SPAM, and fluid rheology, with PAM fluids mostly Newtonian and SPAM fluids non-Newtonian (shearthinning). Based on these differences, it was important to account for the infinite shear viscosity and normalize the polymer concentration by the intrinsic concentration (c int ) so that the DR performance of the two polymer fluids could be accurately compared. Both polymers induced DR, and the maximum DR by SPAM (DR% = 28) was slightly higher than that by PAM (DR% = 22) when Re p ∼ 1700. For PAM, the loss of DR with time diminished at higher polymer concentrations (≥100 ppm, at Re p = 3149) but was found to be sensitive to high Re p , with polymer chain scission the likely cause of the reduced performance. For the semi-dilute SPAM fluids, the shear stability contrasted that of PAM, showing negligible dependence on the polymer concentration and Re p . The apparent rapid loss of DR was predominantly attributed to a time-dependent effect and not polymer degradation. In pipe flow, the maximum DR for SPAM was higher than that measured by rheometry and was attributed to differences in the flow conditions. However, changes in the normalized DR/c with polymer concentration were found to be consistent between the two flow geometries. Furthermore, the high fluid stresses in pipe flow (at high Re p ) led to drag reduction losses consistent with PAM, as the time-dependent effect was not seen.

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In industrial applications such as hydraulic water fracking, polymers are added at dilute concentrations to the flow causing significant drag reduction (DR) and leading to a massive reduction to pump energy cost. In recent years, an alternative approach that is based on rotational flow has shown capabilities of measuring DR based purely on rheological testing. Nonetheless, there are some limiting assumptions in this approach that can lead to inaccurate interpretation of the data, especially for non-Newtonian polymer solutions, where the Reynolds number (Re) is evaluated at the infinite-shear or solvent viscosities. However, it is well known that the apparent viscosity of the polymer solutions is higher than that of a solvent at a stable region and lower than infinite shear viscosity at an unstable one. In this study, we propose a theoretical form of the DR expression that is based on the Re, which is estimated at the apparent local viscosity measure. The work establishes a promising approach for screening DR agents using rheological measurements. Moreover, the study presents new theoretical findings and analyses for estimating DR and extrapolating the results to high Re. Two polymer solutions of xanthan gum (XG) and partially hydrolyzed polyacrylamide (HPAM) in distilled water are tested at concentrations between 5 and 150 ppm using a concentric double-gap cylinder. The proposed approach is found to be more consistent with the theory of linear flow (i.e., flow loop), where the DR in the stable region is found to be identically zero. The transition from stable to unstable regions is also consistent with the existing linear flow theory. This enhances the role of rheological testing for DR measurements and DR agent screening, which provides a platform for the application of simple and cost-effective rheometry in the DR industry.

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e., the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; b) a Taylor expansion of the fibre constitutive function in the structure tensor; c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument.Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e., different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.