This assembly includes a measuring device (50) including a container (60) and an insert (62) arranged in the container (60). The measuring device (50) includes a means (68) for driving the container (60) in rotation around a central axis, the container (60) and the insert (62) defining an intermediate space (64) between them for measuring the drilling fluid. The assembly includes a control and measurement unit (58), capable of recording information representative of the force applied by the drilling fluid on the insert (62) during the rotation of the container (60). The assembly includes an intake pipe (84) for bringing drilling fluid to be measured into the intermediate space (94) and a pump (82), comprising an inlet connected to a sampling head (80) and an outlet connected to the intake pipe (84) to pump the drilling fluid towards the intermediate space (64).