Mathematical model lateral forces

Here the pair-force fj (r, r -) is unknown, so a model pair-force fij(r , rj, p, P2 pm) is chosen, which depends linearly upon m unknown parameters p, p2 - Pm- Consequently, the set of Eq. (8-2) is a system of linear equations with m unknowns p, P2 - - Pm- The system (8-2) can be solved using the singular value decomposition (SVD) method if n > m (over-determined system), and the resulting solution will be unique in a least squares sense. If m > n, more equations from later snapshots along the MD trajectory should be added to the current set so that the number of equations is greater than the number of unknowns. Mathematically, n = qN > m where q is the number of MD snapshots used to generate the system of equations. 

In Chapter 1 of this book we presented Eqs. fl-5a and b) that relate the rate of mass transfer/volume to a mass-transfer coefficient, the area/volume, and a driving force. This mathematical model is an attenpt to quantify a complicated situation. Although useful, this model and the other models presented later in this chapter can also be misleading. The term driving force inplies purpose or desire to transfer, and as noted, there is no purpose—the molecules are just moving randomly. With this caveat, let s look at the various models used to analyze mass transfer, starting with the Fickian diffusion model. 

We consider two spheres of fixed radii to have collided to form a film between them of thickness, say, /z- (less than some value to be defined presently). We further assume a random force such as that arising due to turbulent pressure fluctuations that produces a random film drainage process. A positive force is assumed to drain the film while a negative force causes it to thicken by inflow. Although the process is strictly three-dimensional, we shall assume a one-dimensional model, letting the force be always normal to the film. Further, we stipulate that if the film drains to some critical thickness, say h, the film snaps to allow aggregation between the particles. We shall see later how such a model can be formulated mathematically. The instantaneous film thickness H will serve to describe the position of one of the particles relative to the other. 

This basic model seems contrived, in that no direct experimental justification is given for it by Pelton and Goddard [47], Indeed these workers argue [47] that groups in this treatment are a mathematical convenience to facilitate the derivation and not a physically observable cluster of bubbles. However, in a later paper, which uses a similar model, Pelton [48] claims to observe formation of secondary bubbles in the foam column. It is claimed that they expand in size by coalescence until the buoyancy force exceeds the yield stress in the foam whereupon they rise rapidly to the top of the foam column and rupture. 

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