## 2 Modeling of Lubrication and Viscous Drag Forces  
With dense particle suspensions, the forces between particle pairs are dominated by the lubrication limit, which holds when the gap width between particles, h, is much smaller than their mean diameter, \(\dot{h}\). The interaction needs to be modified to take account of the roughness of the particle surfaces (no matter how small this may be) to avoid a singularity at h = 0. The interaction needs further augmentation to indicate what happens at h = 0, and for this eventuality a simple collision rheology is introduced as follows. When two perfectly smooth particles in a fluid in slow flow come close together, the fluid mechanics may be approximated by the lubrication limit. In this limit, the interactive force for approach or departure normal to the smooth surfaces at the surface-to-surface distance h is proportional to \(\dot{h}/\)h. It is therefore never possible for perfectly smooth particles to have a solid contact (h = 0) at a finite velocity. In nature, physical surfaces are never perfectly smooth, and the interaction deviates from the ideal. A roughness dimension has to be introduced. In a paper by [PERSON] and [PERSON] (2005) [6], it is argued that the rough surface effectively behaves as a permeable medium, so that the fluid in squeeze flow can escape by flowing parallel to the surface through the asperities on the surface. In this case a finite relative velocity is possible, and the interactive force behaves more like \(\dot{h}/(\)h + h\({}_{0})\), where h\({}_{0}\) is a measure of the surface asperity size. Now it is possible to reach h = 0; the particles can touch. For the purposes of the investigation, suspension flow is here approximated by the motion of an aggregate of neutrally buoyant particles of more or less equal diameter. Then the lubrication force F\({}_{12}\) is calculated by using the following equation;  
\[\mathrm{F}_{12}\ =\ -3\pi\eta D^{2}/8\ (\mathrm{h}+\mathrm{h}_{0})\.((\mathrm{v}_{1} -\ \mathrm{v}_{2}).\dot{n}_{12})\ \hat{n}_{12} \tag{1}\]  
where \(\eta\) is the viscosity, \(\hat{n}_{12}\) is the unit normal vector pointing from the surface of particle 1 to the surface of particle 2, \(D\) is the diameter of the particles, \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) are the velocities of particles 1 and 2. The force mediated by the fluid in this way works purely normal to the surfaces, and a detailed discussion and derivation is in [PERSON] (2008) [7]. The tangential interaction is of an order less than the leading term in Equation (1) behaving as \(\dot{h}\)ln (h/\(D\)), see [PERSON] and [PERSON] (1984) [8]. When particles touch at a finite velocity, a collision mechanism is invoked. The normal velocity is updated using the coefficient of restitution, e using the unit normal vector  
\[\left({\rm v_{2}\,-\,v_{1}}\right)^{\prime}.\,\hat{n}_{12}=\,-(1\,-\,{\rm e}) \,\left({\rm v_{2}\,-\,v_{1}}\right).\,\hat{n}_{12} \tag{2}\]  
\(\overline{n}_{12}\) is introduced as the tangential vector such that \(\overline{n}.\overline{n}=0\). The tangential velocity is unaffected by the collision, so  
\[\left({\rm v_{2}\,-\,v_{1}}\right)^{\prime}.\,\overline{n}_{12}=\,\left({\rm v _{2}\,-\,v_{1}}\right).\,\overline{n}_{12} \tag{3}\]  
and momentum is conserved,  
\[\left({\rm m_{1}v_{1}+m_{2}v_{2}}\right)={\rm m_{1}v_{2}+m_{2}v_{2}} \tag{4}\]  
These equations have been solved (with a symbolic manipulation program) for \({\rm v_{1}}^{\prime}\) and \({\rm v_{2}}^{\prime}\) and then implemented in the computer program. In order to characterize the viscous drag forces in a particle suspension flow, two alternative approaches may be introduced: The \"ensemble-averaged\" option, based on a continuum consideration of the interstitial fluid, is to choose the width of the channel and the \"suspension viscosity\" that is the viscosity of the particle-fluid mixture. In general, the mixture viscosity is a solid fraction dependent factor greater than that of the interstitial fluid, see for example [PERSON] (1965) [9].