The governing equations, interphase actions, constitutive relations and turbulence closures of the twofluid model and diffusion model for the liquid-solid two-phase flow are studied in this paper. The pressure actions in and between two phases are analyzed, and the governing equations of the two-fluid model are conveniently closed by adding the interphase pressure-difference force into the pressure terms of momentum equations. The constitutive relations for the liquid-solid mixture in the diffusion model are given by the models of Newtonian fluid and Bingham fluid, and the Bingham relation of shear stress and deformation rate for the single-directional shear flow of the mixture is extended to a general form for the multi-directional shear flow. The equation for the interphase velocity difference, which is used to close the generalized diffusion model, is derived from the momentum equations of two-fluid model and then simplified to more useful forms. For the turbulent flow, the mean movement equations of the two-fluid model and diffusion model are derived by using Reynolds averaging method. The closure techniques for the mean movement equations on the levels of zero-order turbulence model and κ-ε turbulence model are suggested. This paper presents the general basis of the mathematical models, which is to be followed by their validations and applications in the near future. The governing equations, interphase actions, constitutive relations and turbulence closures of the twofluid model and diffusion model for the liquid-solid two-phase flow are studied in this paper. The pressure actions in and between two phases are analyzed, and the governing equations of the two-fluid model are conveniently closed by adding the interphase pressure-difference force into the pressure terms of momentum equations. The constitutive relations for the liquid-solid mixture in the diffusion model are given by the models of Newtonian fluid and Bingham fluid, and the Bingham relation of shear stress and deformation rate for the single-directional shear flow of the mixture is extended to a general form for the multi-directional shear flow. The equation for the interphase velocity difference, which is used to close the generalized diffusion model , is derived from the momentum equations of two-fluid model and then simplified to more useful forms. For the turbulent flow, the mean mo vement equations of the two-fluid model and diffusion model are derived by using Reynolds averaging method. The closure techniques for the mean movement equations on the levels of zero-order turbulence model and κ-ε turbulence model are suggested. This paper presents the general basis of the mathematical models, which is to be followed by their validations and applications in the near future.