Planar walls

P. Attard, D. R. Berard, C. P. Ursenbach, G. N. Patey. Interaction free energy between planar walls in dense fluids an Omstein-Zernike approach for hard-sphere, Lennard-Jones, and dipolar systems. Phys Rev A 44 8224-8234, 1991. 

The generic inhomogeneity in liquid state theory is the planar wall (Fig. 7). It is relatively simple. First, the position of the wall does not matter, as long... 

It is evident that in many situations the reaction rate will be directly proportional to the surface area between phases whenever mass transfer hmits reaction rates. In some situations we provide a fixed area by using solid particles of a given size or by membrane reactors in which a fixed wall separates phases Ifom each other. Here we distinguish planar walls and parallel sheets of sohd membranes, tubes and tube bundles, and spherical solid or liquid membranes. These are three-, two-, and one-dimensional phase boundaries, respectively. 

The constants A and B can be determined from the solutions available for the simple cases in which either = 0 or uw = 0. The first limiting case corresponds to the dissolution of a planar wall in a semiinfinite liquid in steady flow (Blasius flow), while the second limit corresponds to steady-state diffusion with chemical reaction in a stagnant fluid. For Blasius flow at large Schmidt numbers [2],... 

There are many situations in which the spherically symmetrical case is specifically invoked, as in the Debye-Hiickel theory of electrolyte nonideality, for example. We consider situations for which this is the case in Chapter 12. For now, however, we consider the potential distribution adjacent to a planar wall that carries a positive charge. 

FIG. 12.3 Location of a volume element of solution adjacent to a planar wall. 

Further analysis requires specification of the system under study and the spatial dependence of the uniform potential, iA0(z), pertaining to that system. Below we consider the two cases studied earlier an isolated planar wall and two interacting planar walls. 

For a one-component fluid confined by a planar wall to which the z-axis is perpendicular, the inhomogeneous OZ equation can be written as [102]... 

One failing of the theory of VF is the neglect of electron overspill from a metal into the solution which we proceed to examine. The interaction V(z) of the planar wall with a particle in Eq. (15) can take various forms. If the wall is assumed to be a vacuum, then since in the nonprimitive model, the solvent molecules and ions in the electrolyte solution are also in vacuo, there are no image forces so that V(z) = 0. This is one extreme where the wall is described as inert or neutral. 

FIG. 5-1 Steady, one-dimensional conduction in a homogeneous planar wall with constant k. The thermal circuit is shown in (b) with thermal resistance Ax/(kA). 

Second, similar simulations in cylindrical pores are reported, in which non-planar wall would hinder the liquid s freezing even with favorable "excess" potential energy. Non-monotonous variation of freezing point against the pore size, which was observed for U-methane in carbon pores, can be interpreted as the result of competition between the geometrical difficulty and the compression by the excess potential energy. 

Below we present expressions for the forces and torqnes for some of the elementary motions. In all cases we assnme that the Reynolds number is small, the coordinate plane xy is parallel to the planar wall and h is the shortest surface-to-snrface distance from the particle to the wall. 

Second, we consider the case of pure rotation a solid spherical particle of radius R is situated at a surface-to-surface distance, h, from a planar wall and rotates with angular velocity, to, around the X axis in an otherwise quiescent fluid. The corresponding force and torque resultants are ... 

This consideration can be extended to include capillaries of radii comparable to k- and the cases when the potential distribntion cannot be approximated nsing the resnlts for a planar wall. - ... 

E. P. Ascoli, D. S. Dandy, and L. G. Leal, Low Reynolds number hydrodynamic interaction of a solid particle with a planar wall, Int. J. Numer. Methods Fluids 9, 651-88 (1989) E. P. Ascoli, D. S. Dandy, and L. G. Leal, Buoyancy-driven motion of a deformable drop toward a planar wall at low Reynolds number, J. Fluid Mech. 213, 287-311 (1990). 

FIG. 5-1 Steady, one-dimensional conduction in a homogeneous planar wall... 

Vossen, M., Forstmann, F. (1994). The structure of water at a planar wall An integral equation approach with the central force model, J. Chem. Phys., 101 2379. 

Fig. 2.3.1-1 Stationa heat transfer from a hot liquid (H) to cold liquid (K). a) Through three planar walls (inner coke layer, steel wall, and outer fouling layer), b) Through three cylindrical walls (inner coke layer, steel wall, and outer fouling layer).

Simulations of Water Confined Between Planar Walls. 

In the case of the phosphazene molecule, w e assume first that the six-membered ring system is planar, and that any deviations from planarity wall modify slightly the resultant picture. Of the two d,j-orbitals of the phosphorus atom is tangential to the plane of the ring and dyz is radially disposed. These are showai in Fig. 5. Of these two, the dj g-orbital will give greatest overlap with the... 

Demixing, Fig. 8 Log-log plot of the length scale C(z, t) of a thin film of a binary Lennard-Jones mixture between two flat planar walls D = 20 Lennard-Jones parameters apart plotted versus time, for two different values of z. Lennard-Jones parameters were chosen ffAA = = ft = 1, Sa4 = Egg = S = 1, S B = eI2,... 

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