The unsteady Kármán problem for a dilute particle suspension

We consider the unsteady three-dimensional Kármán flow induced by the impulsive rotation of an infinite rotating plane immersed in an incompressible viscous fluid with a dilute suspension of small solid monodisperse spherical particles. The flow is described in terms of a ‘dusty gas’ model, which treats the discrete phase (particles) and the continuous phase (fluid) as two continua occupying the same space and interacting through a Stokes drag mechanism. The model is extended to allow for a local gravitational acceleration in a direction parallel to the axis of rotation, and is valid for cases in which gravity acts either in the same direction as or in the opposite direction to the Ekman axial flow induced by the rotation of the plane. Analysis based on the theory of characteristics shows that the role of gravity is crucial to the treatment of the discrete-phase equations, particularly in regard to the appropriate boundary conditions to be applied at the solid surface. Other notable features include the presence of an essential singularity in the solution when gravity is absent; indeed this phenomenon may help to explain some of the difficulties encountered in previous studies of this type. If the gravitational force is directed away from the rotating surface, a number of other interesting features arise, including the development of discontinuities in the particle distribution profiles, with corresponding particle-free regions contained between the interface and the rotating boundary. These ‘shock’ features can be associated with a critical axial location in the boundary layer at which a balance is achieved between Ekman suction induced by the rotating boundary and the influence of gravitational effects acting to move particles away from the boundary.