## 1 Introduction  
Knowledge of the rheological properties of biological sludge (BS) is essential for optimizing the management of a high-strength wastewater treatment plant for several aspects: (i) membrane performance [1], energy consumption (e.g., in sludge pumping), hydrodynamics of bioreactors [2], oxygen transfer by aeration systems [3], and sludge sedimentation [4]. The hydrodynamic behaviour of the BS is closely related to its characteristics, which include the relative concentration of solids, the nature of the wastewaters and the treatment process to which it is subjected. At low concentrations of total solids (TS), the sludge behaviour can be reasonably approximated in most cases as a Newtonian fluid with a linear relationship between shear stress and strain rate and negligible yield stress [5]. Therefore, measured viscosity is rather independent of the shear rate at given temperature and pressure values, and the flow curve is represented by a straight line. As the concentration of TS increases, the sludge deviates from the Newtonian behaviour and can assume shear-thinning or shear-thickening behaviour. For such a non-Newtonian sludge the measured viscosity becomes dependent on the shear rate and the ratio of shear stress to shear rate, so-called apparent viscosity, is introduced [6]. The rheological characterization of non-Newtonian sludge leads to practical difficulties in interpreting the behaviour of mixed liquor. The Ostwald-de Waele model, the Bingham model, the Herschel and Buckley model,and the Casson model are most commonly used to express non-Newtonian relationships between shear stress and shear rate for activated sludge. They can provide a suitable description of the sludge properties for practical applications [7]. Several studies have shown that temperature significantly influences rheological characteristics [6; 8]. For instance, [PERSON] et al. [6] studied the dependence of the rheological properties of a BS on temperature in a range from 25 \({}^{\circ}\)C to 55 \({}^{\circ}\)C, finding a reduction of the viscosity value with increasing temperature. As already described in the literature, the rheological parameters of a sludge depend upon other process parameters such as pH [9; 10], TS concentration [11; 12], and volatile solids (VS) concentration [13]. Treatments to which the sludge is subjected also influence the rheological properties. Several differences have been reported in the literature between non-digested mesophilic sludge [9; 14], digested mesophilic sludge [14; 15], and thermal hydrolys sludge [14; 16]. For instance, digestion influences the rheological properties reducing viscosity of BS due to VS degradation [9; 14; 15]. Also thermal hydrolysis processes can lead to a reduction of the consistency coefficient of BS thanks to the breaking of the long chains of fatty acids inside the sludge [14; 16]. These differences highlight how those processes to which the BS is subjected can strongly influence its rheological behaviour. In spite of how the knowledge that the rheological behaviour of BS is increasing, most of the studies were conducted in mesophilic conditions--though thermophilic sludge can also be used to biologically treat high strength wastewaters. With high temperature conditions (T: 45-48 \({}^{\circ}\)C), a biomass was selected that had several advantages such as a higher degradation rate of the organic substrate, lower excess sludge production and rapid inactivation of pathogens [17]. However, as a drawback, thermophilic biological sluges (TBS) have poor settling properties which make it necessary to couple membranes in case of operation with high biomass concentration [18]. Data on the rheological properties of TBS are limited to the influence of temperature and TS, without focusing, for example, on non-volatile solids (NVS). Furthermore, the modelling of the thixotropic behaviour of a TBS represents a new topic in the international scientific panorama.