Method for the conversion of hydrocarbon feedstocks using a vessel (1, 100, 200, 300) for the downflow of a hydrocarbon liquid containing solid particles, comprising: - a bottom comprising a cylindrical upper part of diameter D1, a lower part (112, 212, 312) of decreasing cross section and variable angle of inclination α with respect to the axis of revolution of the said cylindrical upper part, the angle α not being constant over the entire height of the lower part (112, 212, 312), excluding a conical or frustaconical lower part, and an outlet pipe of diameter D2 by means of which the hydrocarbon liquid leaves, and comprising an opening positioned in the bottom of the said lower part, the said lower part being a convex portion, preferably of elliptical cross section, extending over the entire height of the said lower part, the said convex portion being of height L1 and having a ratio L1/D1 comprised between 0.01 and 20, or said lower part is formed by a single convex portion (b), comprising said outlet pipe and surmounted by a single frustaconical portion, or said lower part is formed by a single convex portion (b) comprising said outlet pipe and surmounted by a succession of frustaconical portions, or said lower part is a succession of frustaconical portions (s1, s2, s3), - at least one injection of recirculated and/or makeup liquid into the said lower part (112, 212, 312) of decreasing cross section and variable angle of inclination; - at least one injection of recirculated and/or makeup liquid into the said cylindrical upper part; the said injection(s) situated in the said lower part (112, 212, 312) being inclined with respect to the tangent (T in) to the wall of the lower part at the injection point by an angle β1 in the vertical plane (xz) and by an angle β2 in the horizontal plane (xy); the said injection(s) situated in the said cylindrical upper part being inclined with respect to the wall of the cylindrical upper part by an angle θ1 in the vertical plane (xz) and by an angle θ2 in the horizontal plane (xy), the angles β1 and θ1 being comprised between 5° and 175°, the angles β2 and θ2 being comprised between 0° and 180°, and wherein the velocity V of the liquid injected in the said injections is comprised between 0.05 m.s -1 and 40 m.s -1. - a bottom comprising a cylindrical upper part of diameter D1, a lower part (112, 212, 312) of decreasing cross section and variable angle of inclination α with respect to the axis of revolution of the said cylindrical upper part, the angle α not being constant over the entire height of the lower part (112, 212, 312), excluding a conical or frustaconical lower part, and an outlet pipe of diameter D2 by means of which the hydrocarbon liquid leaves, and comprising an opening positioned in the bottom of the said lower part, the said lower part being a convex portion, preferably of elliptical cross section, extending over the entire height of the said lower part, the said convex portion being of height L1 and having a ratio L1/D1 comprised between 0.01 and 20, or said lower part is formed by a single convex portion (b), comprising said outlet pipe and surmounted by a single frustaconical portion, or said lower part is formed by a single convex portion (b) comprising said outlet pipe and surmounted by a succession of frustaconical portions, or said lower part is a succession of frustaconical portions (s1, s2, s3), - at least one injection of recirculated and/or makeup liquid into the said lower part (112, 212, 312) of decreasing cross section and variable angle of inclination; - at least one injection of recirculated and/or makeup liquid into the said cylindrical upper part; the said injection(s) situated in the said lower part (112, 212, 312) being inclined with respect to the tangent (T in) to the wall of the lower part at the injection point by an angle β1 in the vertical plane (xz) and by an angle β2 in the horizontal plane (xy); the said injection(s) situated in the said cylindrical upper part being inclined with respect to the wall of the cylindrical upper part by an angle θ1 in the vertical plane (xz) and by an angle θ2 in the horizontal plane (xy), the angles β1 and θ1 being comprised between 5° and 175°, the angles β2 and θ2 being comprised between 0° and 180°, and wherein the velocity V of the liquid injected in the said injections is comprised between 0.05 m.s -1 and 40 m.s -1.