The effect of slip on the motion of a sphere close to a wall and of two adjacent spheres

Summary The motion of a sphere towards a plane or another sphere is opposed by the fluid between them with a force which is inversely proportional to the gap. In consequence, it is impossible for a constant force to produce contact in a finite time, unless the Stokes equations are modified. When the gap is of the same order as the mean free path of the air molecules, the Stokes theory for the motion of the air must be modified. The Maxwell slip flow approximation is used in this paper to show that, when the gap is small, the resisting force between the approaching surfaces becomes only logarithmically dependent on the gap, and contact can be achieved in a finite time. The difficulty in applying the Stokes theory to the problem of determining collision efficiencies for cloud droplets is thereby removed.The calculated values of the resistance to approach are used to determine the motion of a sphere falling towards a plane. If the motion is compared with the corresponding motion when no allowance is made for slip flow, the sphere without slip would still be at a distance of 1.3 times the mean free path from the plane, when the sphere with slip has made contact.Transverse motion must also be considered if the trajectory of a particle close to a collector is required. The forces and couples on the sphere in that situation have a logarithmic dependence on the gap without slip, but they tend to constant values when the effect of slip is included. Some calculations of collision efficiency of drops falling under gravity (Hocking and Jonas [1]) have been amended to include the effect of slip when the colliding drops are very close together, and show a significant increase in the collision efficiency.