The interplay between gravity and centrifugal forces in the continuous circular hydraulic jump

The flow of a thin film over a rotating disk is crucial in many industrial applications, such as spin coating and jet cooling. Accurate prediction of hydrodynamic features is essential for optimizing these processes. In this study, we examine the vertical downward impingement of a circular Newtonian liquid jet on a horizontally rotating disk, focusing on two distinct regimes based on the rotation level. For a stationary disk and low rotation speed, a continuous hydraulic jump is formed, while increased rotation speed leads to the transformation of the jump into a hump. A composite mean-field thin-film approach is utilized to analyze the flow dynamics in different regions of the domain. The effects of gravity and rotation are considered by developing a model to capture the continuous jump and vortex structure. The influence of rotation on the laminar boundary layer near impingement is found to be negligible. Specific conditions for gravity and rotation are established to differentiate between the jump and hump regimes. The model is validated for both regimes against existing experiment. In the jump regime, the flow transitions from predominantly azimuthal near the disk to predominantly radial toward the free surface, while in the hump regime, the flow maintains an azimuthal character around the hump. The vortex associated with the jump diminishes with increasing rotation speed, indicating the occurrence of a type-0 jump on a rotating disk. For small gravity, the vortex does not form in conjunction with the jump at any rotation level. In the case of small rotation, large gravity, and large disk size, the film exhibits a hydraulic jump near impingement followed by a sharp rise in thickness near the edge of the disk.