Between parallel plates

Derive the equation for the capillary rise between parallel plates, including the correction term for meniscus weight. Assume zero contact angle, a cylindrical meniscus, and neglect end effects. 

Figure C2.2.9. Polygonal domains of focal conics in a smectic A phase confined between parallel plates.

B. Spacing between wire and plate, or between rod and curtain, or between parallel plates in electrical precipitators m ft ... 

The field strength is uniform between parallel plates, whereas it varies in the space between concentric cylinders, being highest at the surface of the central cylinder. After corona sets in, the current flow will become appreciable. The field strength near the center electrode will be less than given by Eq. (17-18) and that in the major portion of the clearance space will be greater and more uniform [see Eqs. (17-23) and (17-24)]. 

In both cases, AB =dz, element width = dx and channel width = T Fig. 4.7 Melt Flow between parallel plates... 

If the clearance between the rolls is small in relation to their radius then at any section x the problem may be analysed as the flow between parallel plates at a distance h apart. The velocity profile at any section is thus made up of a drag flow component and a pressure flow component. 

Consider an element of fluid between parallel plates, T wide and spaced a distance H apart. For unit width of element the forces acting on it are ... 

It is worth noting that the equations for flow between parallel plates may also be used with acceptable accuracy for flow along a circular annular slot. The relevant terms are illustrated in Fig. 5.6. 

It is now necessary to derive an expression for the pressure loss in the cavity. Since the mould fills very quickly it may be assumed that effects due to freezing-off of the melt may be ignored. In Section 3.4(b) it was shown that for the flow of a power law fluid between parallel plates... 

Any real sample of a colloidal suspension has boundaries. These may stem from the walls of the container holding the suspension or from a free interface towards the surroundings. One is faced with surface effects that are small compared to volume effects. But there are also situations where surface effects are comparable to bulk effects because of strong confinement of the suspension. Examples are cylindrical pores (Fig. 8), porous media filled with suspension (Fig. 9), and thin colloidal films squeezed between parallel plates (Fig. 10). Confined systems show physical effects absent in the bulk behavior of the system and absent in the limit of extreme confinement, e.g., a onedimensional system is built up by shrinking the size of a cylindrical pore to the particle diameter. 

The dimensionahty of a system is one of its major features. Despite the fact that our surrounding space is three-dimensional, one can prepare situations that lead to an effective lowered dimension. A typical example regarding colloids is the surface between the solvent and air. One can prepare the particles to be trapped at that interface, so that they float on top of the solvent, building up a two-dimensional (2d) system. Another realization is strong confinement between parallel plates that leads to an effective 2d system. Concerning simulations, it is very convenient to simulate 2d systems, as one has fewer degrees of freedom to deal with e.g., plotting snapshots is easier in 2d than it is in 3d. So, besides their experimental realizations, 2d systems are also important from a conceptual point of view. 

Figure 3.12. Streamline flow between parallel plates...

The problem of axial conduction in the wall was considered by Petukhov (1967). The parameter used to characterize the effect of axial conduction is P = (l - dyd k2/k ). The numerical calculations performed for q = const, and neglecting the wall thermal resistance in radial direction, showed that axial thermal conduction in the wall does not affect the Nusselt number Nuco. Davis and Gill (1970) considered the problem of axial conduction in the wall with reference to laminar flow between parallel plates with finite conductivity. It was found that the Peclet number, the ratio of thickness of the plates to their length are important dimensionless groups that determine the process of heat transfer. 

One particular characteristic of conduction heat transfer in micro-channel heat sinks is the strong three-dimensional character of the phenomenon. The smaller the hydraulic diameter, the more important the coupling between wall and bulk fluid temperatures, because the heat transfer coefficient becomes high. Even though the thermal wall boundary conditions at the inlet and outlet of the solid wall are adiabatic, for small Reynolds numbers the heat flux can become strongly non-uniform most of the flux is transferred to the fluid at the entrance of the micro-channel. Maranzana et al. (2004) analyzed this type of problem and proposed the model of channel flow heat transfer between parallel plates. The geometry shown in Fig. 4.15 corresponds to a flow between parallel plates, the uniform heat flux is imposed on the upper face of block 1 the lower face of block 0 and the side faces of both blocks... 

FIGURE 8.4 Pressure driven flow between parallel plates with both plates stationary. 

FIGURE 8.5 Drag flow between parallel plates with the upper plate in motion and no axial pressure drop. 

To examine the details of the structure of flames in channels under quenching conditions, numerical methods were used. Two-dimensional CFD simulation of a propane flame approaching a channel between parallel plates was carried out using the FLUENT code [25]. The model reproduced the geometry of the real channels investigated experimentally. Close to the quenching limit, the burning velocity, dead space, and radius of curvature of the flames were all close to the experimental values. 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

Table IV includes theoretical transition times (C2, R14, SI7c) in laminar flow between parallel plates, following a concentration step at the wall (Leveque mass transfer). Clearly, in laminar flow (Re 100 or lower), transition times are comparable to those in laminar free convection. Here, however, the dependence on concentration (through the diffusivity) is weak. The dimensionless time variable in unsteady-state mass transfer of the Leveque type is...

Flow between parallel plates corresponds to the other limiting case of Eq. (27) for k -> 1 (r, = r0,de = 2h, where h = channel height) ... 

In structural terms, aggregation of porphyrin rings usually results from 7r-7r and it a electronic interactions between parallel plates 69 however, if a coordinated metal is present, and it has axial ligands, then the aromatic systems cannot come close together, and so aggregation is inhibited and PDT activity is expected to be enhanced.72... 

Similarly, from Eq. (24), we find the Casimir energy and pressure for the Dirac field confined between parallel plates, with anti-periodic boundary conditions, as ... 

The corresponding velocity profiles between parallel plates are in Figure 4.3. 

Figure 4.3. Velocity profiles of a power law fluid flowing between parallel plates, u is the mean velocity.

Method involves placing a specimen between parallel plate capacitors and applying a sinusoidal voltage (frequencies ranging from 1 mHz to 1 MHz) to one of the plates to establish an electric field in the specimen. In response to this field, a specimen becomes electrically polarized and can conduct a small charge from one plate to the other. Through measurement of the resultant current, the dielectric constant and dielectric loss constant for a specimen can be measured. The sharp increases in both the dielectric constant and the dielectric loss constant during a temperature scan are correlated with the occurrence of Tg... 

If we place an ionic conductor between parallel-plate blocking electrodes that produce an electric field E parallel to the x-axis, the electrostatic potential varies as — xE on passing from one electrode at x = 0 to the other. At equilibrium, the mobile-ion concentration Cj(x) is proportional to exp(qEx/kT), and the ionic drift-current density (7(E in the field is balanced by a diffusion current due to the concentration gradient (Fick s law) ... 

Note 4 Some experimental methods, such as capillary flow and flow between parallel plates, employ a range of shear rates. The value of tj evaluated at some nominal average value of Y is termed the apparent viscosity and given the symbol /app. It should be noted that this is an imprecisely defined quantity. 

For large bubbles where inertia effects are dominant, enclosed vertical tubes lead to bubble elongation and increased terminal velocities (G7). The bubble shape tends towards that of a prolate spheroid and the terminal velocity may be predicted using the Davies and Taylor assumptions discussed in Chapter 8, but with the shape at the nose ellipsoidal rather than spherical. The maximum increase in terminal velocity is about 16% for the case where 2 is small (G6) and 25% for a bubble confined between parallel plates (G6, G7) and occurs for the enclosed tube relatively close to the bubble axis. 

The physical characteristics of a discharge and the manner in which it is sustained can have a profound effect on the kinetics of plasma polymerization . Therefore, we shall review these topics here, with specific emphasis on the characteristics of plasmas sustained between parallel plate electrodes. This constraint is imposed because virtually all efforts to theoretically model the kinetics of plasma polymerization have been directed towards plasmas of this type. Readers interested in broader and more detailed discussions of plasma characteristics can find such in referen-... 

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