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Analytical solutions for shear stress in large-amplitude oscillatory shear flow (LAOS), for continuum or molecular models, often take the form of the first few terms of a power series in the shear rate amplitude. For corotational models, we get this truncated series using the Goddard-Miller integral expansion (GIE). Our previous work shows that the best Padé approximants for this truncated series, and specifically for the corotational Maxwell model in LAOS, can agree closely with the corresponding exact solution. We observe this close agreement, even for the Padé approximant for the series truncated after the fifth shear stress harmonic. In this paper, we begin with the extension of the GIE truncated after the next, seventh, order in the shear rate amplitude [Phys. Fluids 29, 043101 (2017)], and we then explore its Padé approximants. We uncover its best approximant, the [2,4], and compare it with both the GIE and the exact solution [Macromol. Theory Simul. 24, 352 (2015)]. We use Ewoldt grids to show the stunning accuracy of the [2,4] approximant in LAOS. We quantify this accuracy with an objective function and then map this function onto Pipkin space. We find the [2,4] approximant (from the GIE truncated after the seventh order in the shear rate amplitude) to be a simple accurate expression for the shear stress in LAOS. Our worked examples illustrate how researchers can use our new approximant reliably. For this, we use the Spriggs relations to extend the Padé approximant to multimode.

This review sheds light on the catalytic valorisation of agroforestry biomass through levulinic acid and formic acid towards γ-valerolactone and other higher-value chemicals.

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Background: Hydrothermal carbonization (HTC) of dairy processing waste was performed to investigate the effect of temperature and initial pH on the yield and composition of the solid (hydrochar) and liquor produced. All hydrochars met the EU requirements of organo-mineral solid fertilizers defined in the Fertilizing Products Regulation in terms of phosphorus (P) and mineral content. Methods: Laboratory scale HTC was performed using pressurized reactors, and the products (solid and liquid) were collected, stored and analyzed for elemental composition and nutrient content using Inductively coupled plasma optical emission spectroscopy (ICP-OES), ultraviolet-visible spectrophotometry (UV-Vis) and other analytic techniques. Results: Maximum hydrochar yield (60.67%) was observed at T=180℃ and pH=2.25, whereas the maximum P-recovery was 80.38% at T=220℃ and pH=4.6. The heavy metal content of the hydrochars was mostly compliant with EU limitations, except for Ni at T=220℃ and pH=8.32. Meanwhile, further study of Chromium (Cr) species is essential to assess the fertilizer quality of the hydrochars. For the liquid product, the increase in temperature beyond 200℃, coupled with an increase in initial acidity (pH=2.25) drove P into the liquor. Simultaneously, increasing HTC temperature and acidity increased the concentration of NO3- and NH4+ in the liquid products to a maximum of 278 and 148 mg/L, respectively, at T=180℃ and pH=4.6. Furthermore, no direct relation between final pH of liquor and NH4+ concentration was observed. Conclusions: HTC improved the nutrient content of dairy waste, allowing for the production of potential solid organo-mineral fertilizers requiring additional treatment to ensure safe fertilizer application.