Purpose: This paper sets out to present a fully explicit smoothed particle hydrodynamics (SPH) method to solve non‐Newtonian fluid flow problems.Design/methodology/approach: The governing equations are momentum equations along with the continuity equation which are described in a Lagrangian framework. A new treatment similar to that used in Eulerian formulations is applied to viscous terms, which facilitates the implementation of various inelastic non‐Newtonian models. This approach utilizes the exact forms of the shear strain rate tensor and its second principal invariant to calculate the shear stress tensor. Three constitutive laws including power‐law, Bingham‐plastic and Herschel‐Bulkley models are studied in this work. The imposition of the incompressibility is fulfilled using a penalty‐like formulation which creates a trade‐off between the pressure and density variations. Solid walls are simulated by the boundary particles whose positions are fixed but contribute to the field variables in the same way as the fluid particles in flow field.Findings: The performance of the proposed algorithm is assessed by solving three test cases including a non‐Newtonian dam‐break problem, flow in an annular viscometer using the aforementioned models and a mud fluid flow on a sloping bed under an overlying water. The results obtained by the proposed SPH algorithm are in close agreement with the available experimental and/or numerical data.Research limitations/implications: In this work, only inelastic non‐Newtonian models are studied. This paper deals with 2D problems, although extension of the proposed scheme to 3D is straightforward.Practical implications: This study shows that various types of flow problems involving fluid‐solid and fluid‐fluid interfaces can be solved using the proposed SPH method.Originality/value: Using the proposed numerical treatment of viscous terms, a unified and consistent approach was devised to study various non‐Newtonian flow models.