Force in Normal Direction

The hydrodynamic interaction of a solid sphere moving in normal direction toward a solid plane has been calculated by Happel and Brenner [325,327, p. 330] and later by Cox [628]. We follow a derivation of Chan and Horn [629], which is valid for small distances, that is, for D Rp. The shape of the sphere, which confines the liquid film 

When a particle approaches the planar surface, liquid is squeezed out of the gap in between. In particular for small distances, the flow of the liquid is primarily in radial direction. Thus, for D Rp, we neglect the vertical component of the flow and set Vz = 0. One consequence of the last line in Eq. (6.10) is that in such a case the pressure depends only on x, y and not on the vertical z-component since for Vz = 0 also 0 P/0Z = 0. We further assume that variations in the flow velocity v, Vy in and y-directions are small compared to variations of the velocity v, Vy in z-direction. This is known as the lubrication approximation first applied by Reynolds [630]. Then, 0 v /0x, 0 v /0y, 0 Vy/0x, and 0 Vy/0y in the first two lines of Eq. (6.10) can be 

Cl is the first integration constant. The integration is easy because QP/Qx does not depend on z, since P does not depend on z. With the second integration, we obtain 

To determine the two integration constants Ci and C2, we apply the so-called no-slip boundary conditions. No-slip boundary conditions require that the liquid molecules directly in contact with the surfaces are stationary 

Inserting the first condition in Eq. (6.17) directly leads to C2 = 0. Applying the second condition to Eq. (6.17)  

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There are a number of computational models used to investigate granular media. Event-driven or hard-sphere algorithms are based on the calculation of changes from distinct collisions between single grains that are often approximated by spheres, ellipsoids, or polyhedral.MD of soft-particle models are another common way to simulate granular materials. In this approach, the repulsive contact force in normal direction is typically proportional to the particle... 

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