Particle-Surface Interactions Low Speeds

As two surfaces approach each other, the fluid between them must be displaced. First, we consider the ca,se of two plane parallel circuUtr disks of radius a approaching each other along llieir common axis (Fig. 4.1). The disks are immersed in a fluid in which the pressure is p(). Without loss of generality, it is possible to assume that one of the disks is fixed and that the other is in relative motion. The motion is sufficiently slow to neglect the inertial and unsteady terms in the equations of fluid motion. 

The flow is axisymmetric, and for low velocities the pressure gradient across the gap in the 3 direction, dp/5z, can be neglected. Hence the pressure is a function only of r. The applicable equations are the r component of the equation of motion 

We now make the following educated guesses for the forms of the velocity and pressure distributions  

When we integrate with the boundary conditions (4.2a) to obtain Z(z)t the following expression is found for the radial velocity  

If the continuity relation (4.2) is integrated with respect to z across the gap between the disks, making use of the boundary conditions on U , the result is 

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