Free-surface laminar flow of a Herschel–Bulkley fluid over an inclined porous bed

•A theoretical solution for the velocity profile is brought to free-surface flow of Herschel--Bulkley fluids over porous bed.•Extended formula of the velocity profile retrieves simpler cases (no porosity and other rheological models).•Analysis of fluid rheological properties and porous medium properties into velocity profile, mean flow velocity, and discharge gain, and flow parameters were performed.•Pseudoplastic fluids seem to have greater influence on the sensibility of the model than dilatant ones. The current work presents a mathematical solution for the flow of a Herschel--Bulkley fluid over a porous bed. To obtain the velocity profile, a dimensional analysis is firstly conducted for the Darcian-like flow, showing that the ratio between pore and flow scales impose the domain of validity for the Beaver-like kinematic boundary condition. A friction velocity dependent only on fluid and porous medium properties was found to identify the scale of the Darcian velocity. After applying a modified kinematic boundary condition between free flow and Darcian-like flow, free-surface flow velocity profile was obtained, which can effectively predicts non-porous solutions for less rheological complex fluids. Dimensional analysis was then performed for the velocity profile solution, which allowed to identify the effect of the porous bed on flow properties. Experimental results from the literature were employed to identify the necessary conditions for yield stress fluids to have Darcian-like flow in natural open-channel flows. Sensitivity analysis pointed out that the porous medium permeability and non-Newtonian fluid parameters have greater influence on the velocity profile for pseudoplastic fluids than for dilatant ones.