Three-dimensional fluid phases

For a three-dimensional fluid phase like, for example, a gas enclosed in a volume, the free enthalpy so-called Gibbs energy depends on the temperature, the pressure and the amount of gas = f(T, p, n ). The differential Gibbs energy is ... 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

R. Yamamoto and K Nakanishi, Computer Simulation of Vapor-liquid Phase Separation in two- and three-dimensional Fluids Growth Law of Domain Size, Phys. Rev. B 49 (1994) 14958-14966 II. Domain Structure, Phys. Rev. B 51... 

A more detailed representation of phase equilibrium in a pure fluid, including the presence of a single solid phase, js given in the three-dimensional PVT phase diagram of Fig. 7.3-5. Such complete phase diagrams are rarely available, although data may be available in the form of Fig. 7.3-4, which is a projection of the more complete diagram onto the P-V plane, and Fig. 7.3-6, which is the projection onto the P-T plane. 

As described above, the kink in the molecular shape and the requirement to fill the space as effectively as possible are not compatible with a three-dimensional fluid order. In other words, when translating a bent-core molecule in the melt of the neighbouring bent-core molecules, it experiences a periodic potential with its periodicity determined by the length I of the molecules. To allow for fluidity at the macroscopic level, one needs to frustrate the bent-core structure so that they do not lock into smectic layers easily. Such a frustration can be introduced by some steric or electrostatic disturbance of the bare bent-core (or peeled banana ) shape, which has been seen in some modulated smectic phases.As discussed by Bailey and Jdkli,a steric or electrostatic inclusion in the core of the molecules leads to layer modulation, an SmCc structure and broken smectic layers that effectively correspond to a columnar phase, as the inclusions increase. Following this picture, here we postulate that bent-core nematics are probably more frustrated than the electrically unswitchable B7 (columnar) phases, in which the broken smectic ribbons are separated by melted fluid nematic regions. Such over-frustrated B7 materials are characterized... 

However, two-phase dynamics are commonly treated in a very simple way (or even ignored by some modeling systems) by both fairly simple integral models and more complex three-dimensional fluid models. Practically all of these models rely on the homogeneous equilibrium assumptions, implying that the liquid is uniformly distributed in the cloud and that the liquid and the gas are at a uniform temperature and in thermodynamical equilibrium. 

The simplest smectic phase is the smectic A(S ) phase. This phase has traditionally been described as a system that is a solid in the direction along the director and a fluid normal to the director, or equivalently, as stacked two-dimensional fluids it is more properly described as a one-dimensional density wave in a three-dimensional fluid with the density wave along the nematic director. The phase is similar except the density wave vector makes a finite angle with the director. In both and phases there is complete translational symmetry perpendicular to the density wave vector. 

Stabilization of the Cellular State. The increase in surface area corresponding to the formation of many ceUs in the plastic phase is accompanied by an increase in the free energy of the system hence the foamed state is inherently unstable. Methods of stabilizing this foamed state can be classified as chemical, eg, the polymerization of a fluid resin into a three-dimensional thermoset polymer, or physical, eg, the cooling of an expanded thermoplastic polymer to a temperature below its second-order transition temperature or its crystalline melting point to prevent polymer flow. 

Figure 7.2 A three-dimensional phase diagram for a Type I binary mixture (here, CO2 and methanol). The shaded volume is the two-phase liquid-vapor region. This is shown ti uncated at 25 °C for illustration purposes. The volume surrounding the two-phase region is the continuum of fluid behavior.

Lelea D, Nishio S, Takano K (2004) The experimental research on micro-tube heat transfer and fluid flow of distilled water. Int J Heat Mass Transfer 47 2817-2830 Li J, Peterson GP, Cheng P (2004) Three-dimensional analysis of heat transfer in a micro-heat sink with single phase flow. Int J Heat Mass Transfer 47 4215-4231 Lin TY, Yang CY (2007) An experimental investigation by method of fluid crystal thermography. Int. J. Heat Mass Transfer 50(23-24) 4736-4742... 

Pore shape is a characteristic of pore geometry, which is important for fluid flow and especially multi-phase flow. It can be studied by analyzing three-dimensional images of the pore space [2, 3]. Also, long time diffusion coefficient measurements on rocks have been used to argue that the shapes of pores in many rocks are sheetlike and tube-like [16]. It has been shown in a recent study [57] that a combination of DDIF, mercury intrusion porosimetry and a simple analysis of two-dimensional thin-section images provides a characterization of pore shape (described below) from just the geometric properties. 

To be semisolid, a system must have a three-dimensional structure that is sufficient to impart solidlike character to the undistributed system that is easily broken down and realigned under an applied force. The semisolid systems used pharmaceutically include ointments and solidified w/o emulsion variants thereof, pastes, o/w creams with solidified internal phases, o/w creams with fluid internal phases, gels, and rigid foams. The natures of the underlying structures differ remarkably across all these systems, but all share the property that their structures are easily broken down, rearranged, and reformed. Only to the extent that one understands the structural sources of these systems does one understand them at all. 

Computational fluid dynamics (CFD) is rapidly becoming a standard tool for the analysis of chemically reacting flows. For single-phase reactors, such as stirred tanks and empty tubes, it is already well-established. For multiphase reactors such as fixed beds, bubble columns, trickle beds and fluidized beds, its use is relatively new, and methods are still under development. The aim of this chapter is to present the application of CFD to the simulation of three-dimensional interstitial flow in packed tubes, with and without catalytic reaction. Although the use of... 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

A conglomerate in real liquid crystalline phases was first observed in the smectic phase of a rod-shaped mesogen with two stereogenic centers in its tail [42], We used a racemic mixture which was supposed not to electrically switch. Evidence for conglomerate formation was provided by clear electro-optic switching and texture observation under a polarizing microscope domains with stripes, which themselves display fine stripes. These stripes are tilted in two different directions with respect to the primary stripes. This is a still very rare example now that fluid soft matter is known to resolve spontaneously into a three-dimensional conglomerate. 

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