The quadratic rheology model considers the yield stress,viscous stress,turbulent stress and disperse stress,so it is used in this study to derive the velocity profile of debris flows.The quadratic model with the parabolic eddy viscosity was numerically solved,and an analytical solution was derived for the quadratic model with a constant eddy viscosity.These two solutions were compared with the Arai-Takahashi model that excluded the viscous stress and the yield stress.The three models were tested by using 17 experiment cases of debris flows over rigid beds.The results prove that the quadratic model with parabolic and constant eddy viscosities is applicable to muddy and granular flows,whereas the Arai-Takahashi model tends to overestimate the flow velocity near the water surface if a plug-like layer exists.In addition,the von Karman constant and the zero-velocity elevation in the three models are related to sediment concentration.The von Karman constant decreases first and then increases as the sediment concentration increases.The zero-velocity elevation is below the bed surface,likely due to the invalidity of the non-slip boundary condition for the debris flows over fixed beds.