In a previous paper, a new gyrokinetic quasilinear model was proposed to study turbulent momentum transport in tokamaks with up-down asymmetric plasma shaping and toroidal flow shear (Sun et al 2024 Nuclear Fusion 64 036026). In this work, we start by applying this model to study the Prandtl number dependence on flux surface shaping, electron-ion temperature ratio and pressure gradient. It is then extended to study momentum transport in the presence of finite toroidal rotation as well as electron temperature gradient-driven turbulence. This shows that, if ion-scale turbulence can be suppressed, the Prandtl number will decrease by more than an order of magnitude, enabling fast rotation when external momentum injection is present. These results are benchmarked with high-fidelity nonlinear GENE simulations, showing the reliability of our quasilinear model in estimating momentum transport.