Force constants, solid-fluid equilibrium

The behaviour of a fluid is different. If an ordinary liquid is placed between the plates and a constant shearing force F applied to the upper plate, the lower plate being fixed, the upper plate does not come to an equilibrium position but continues to move at a steady speed. The liquid adheres to each plate, ie there is no slip between the liquid and the solid surfaces, and at any instant the deformation of the sample is as shown in Figures 1.10 and 1.11. Thus the liquid sample is continuously sheared when subjected to a constant shear stress. The distinction between a fluid and a solid is that a fluid cannot sustain a shear stress without continuously deforming (ie flowing). 

Let us consider a one-component fluid confined in a pore of given size and shape which is itself located within a well-defined solid structure. We suppose that the pore is open and that the confined fluid is in thermodynamic equilibrium with the same fluid (gas or liquid) in the bulk state and held at die same temperature. As indicated in Chapter 2, under conditions of equilibrium a uniform chemical potential is established throughout the system. As the bulk fluid is homogeneous, its chemical potential is simply determined by the pressure and temperature. The fluid in the pore is not of constant density, however, since it is subjected to adsorption forces in the vicinity of the pore walls. This inhomogeneous fluid, which is stable only under the influence of the external field, is in effect a layerwise distribution of the adsorbate. The density distribution can be characterized in terms of a density profile, p(r), expressed as a function of distance, r, from the wall across the pore. More precisely, r is the generalized coordinate vector. 

In dynamic situations forces are not balanced at every instant one special case of dynamic situations is the steady state, in which forces are constant but they are not balanced by opposing forces. In addition, dynamic situations may include states at equilibrium. In dynamic equilibria forces fluctuate at every instant, but the forces are balanced when they are averaged over finite durations and finite parts of the system. The relevant time and length scales may or may not be sensible or important to an observer. Moreover, these scales can differ substantially for systems in different phases of aggregation for example, property fluctuations in solids are typically orders of magnitude smaller than those in fluids. 

Many researches adopted one of the aforanentioned approaches and modified it to include various aspects of the pneumatic drying process. Andrieu and Bressat [16] presented a simple model for pneumatic drying of polyvinyl chloride (PVC), particles. Their model was based on elementary momentum, heat and mass transfer between the fluid and the particles. In order to simplify their model, they assumed that the flow is unidirectional, the relative velocity is a function of the buoyancy and drag forces, solid temperature is uniform and equal to the evaporation temperature, and that evaporation of free water occurs in a constant rate period. Based on their simplifying assumptions, six balance equations were written for six unknowns, namely, relative velocity, air humidity, solid moisture content, equilibrium humidity, and both solid and fluid temperatures. The model was then solved numerically, and satisfactory agreanent with their experimental results was obtained. A similar model was presented by Tanthapanichakoon and Srivotanai [24]. Their model was solved numerically and compared with their experimental data. Their comparison between the experimental data and their model predictions showed large scattering for the gas temperature and absolute humidity. However, their comparisons for the solid temperature and the water content were failed. 

The molecular theory is similar to Cauchy s description of the elastic theory of solids [1] and utilizes additive local molecular pair interactions to describe elasticity. The latter approach was taken by Oseen [2], who was the first to establish an elastic theory of anisotropic fluids. Oseen assumed short-range intermolecular forces to be the reason for the elastic properties, and he derived eight elastic constants in the expression for the elastic free energy density of uniaxial nematic phases. Finally, he retained only five of them, which enter the Euler-Lagrange equations describing equilibrium deformation states of the nematic mesophase, and omitted the other three. 

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