The rheology of highly concentrated suspensions of solid particles dispersed in a viscous fluid features a number of surprising phenomena [1]. By varying the rate of deformation we can observe shear thinning, continuous or discontinuous shear thickening and, at very high concentrations, even the phenomenon of shear jamming. While the understanding of the microscopic origin of these phenomena has reached a significant depth [2, 3], the definition of suitable continuum-level models able to capture the rheology of dense suspensions is still the subject of active research efforts [4]. Building on a recently proposed tensorial model for shear-jamming suspensions [5], we develop consti- tutive relations that allow to include further effects such as shear thickening and yielding. In particular, the inclusion of rate-dependent phenomena requires the addition of evolution equations for tensorial measures of strain, akin to the conformation tensors used in viscoelastic fluid models. The presence of such evolution equations and the nonlinearities that arise in coupling them to the flow equation require appropriate computational strategies to guarantee the reliability of the numerical results [6]. Different features of the proposed continuum model are analyzed and illustrated with numerical solutions in paradigmatic examples of complex flows, in which the geometric features of the domain and the possible presence of free surfaces make it necessary to have available a fully tensorial model. In parallel, we discuss the numerical strategies employed to treat such heterogeneous flows. REFERENCES [1] Morris, J.F. Annu. Rev. Fluid Mech. (2020) 52: 121. [2] Seto, R., Mari, R., Morris, J.F., and Denn, M.M. Phys. Rev. Lett. (2013) 111: 218301. [3] Wyart, M. and Cates, M.E. Phys. Rev. Lett. (2014) 112: 098302. [4] Gillissen J.J.J.et al., Phys. Rev. Lett. (2019) 123: 214504. [5] Giusteri, G.G. and Seto, R. Phys. Rev. Lett. (2021) 127: 138001. [6] Moreno, L. et al. Comput. Methods Appl. Mech. Engrg. (2019) 354: 706–731.