Two Parallel Plates

Derive, from simple considerations, the capillary rise between two parallel plates of infinite length inclined at an angle of d to each other, and meeting at the liquid surface, as illustrated in Fig. 11-23. Assume zero contact angle and a circular cross section for the meniscus. Remember that the area of the liquid surface changes with its position. 

The measurement of the streaming potential developed when a solution flows through two parallel plates [74-76] allows the characterization of macroscopic surfaces such as mica. 

Wlien an electrical coimection is made between two metal surfaces, a contact potential difference arises from the transfer of electrons from the metal of lower work function to the second metal until their Femii levels line up. The difference in contact potential between the two metals is just equal to the difference in their respective work fiinctions. In the absence of an applied emf, there is electric field between two parallel metal plates arranged as a capacitor. If a potential is applied, the field can be eliminated and at this point tire potential equals the contact potential difference of tlie two metal plates. If one plate of known work fiinction is used as a reference electrode, the work function of the second plate can be detennined by measuring tliis applied potential between the plates [ ]. One can detemiine the zero-electric-field condition between the two parallel plates by measuring directly the tendency for charge to flow through the external circuit. This is called the static capacitor method [59]. 

Two parallel plates of conducting material separated by an insulation material, called the dielectric, constitutes an electrical condenser. The two plates may be electrically charged by connecting them to a source of direct current potential. The amount of electrical energy that can be stored in this manner is called the capacitance of the condenser, and is a function of the voltage, area of the plates, thickness of the dielectric, and the characteristic property of the dielectric material called dielectric constant. 

In spite of these problems, polymer melts have been sufficiently studied for a number of useful generalisations to be made. However, before discussing these it is necessary to define some terms. This is best accomplished by reference to Figure 8.2, which schematically illustrates two parallel plates of very large area A separated by a distance r with the space in between filled with a liquid. The lower plate is fixed and a shear force F applied to the top plate so that there is a shear stress (t = F/A) which causes the plate to move at a uniform velocity u in a direction parallel to the plane of the plate. 

Figure 8.2. Velocity distribution of a fluid between two parallel plates, one stationary, the other...

For a fluid between two parallel plates, each moving at a velocity Vj, the drag flow velocity is equal to Vd- In the case of a calender with rolls of radius, R, rotating at a speed, N, the drag velocity will thus be given by InRN. 

The velocity component due to pressure flow between two parallel plates has already been determined in Section 4.2.3(b). 

The crossover 2d 2d behavior can be described in a similar manner to the case of a tube confinement. For the chain, trapped between two parallel plates a distance D apart, one again has N/g blobs but they arrange to a two-dimensional random coil configuration ... 

FIG. 10 A colloidal suspension between two parallel plates. There is strong confinement perpendicular to the plates, but an infinite system in the lateral orientations. 

Liquids are able to flow. Complicated stream patterns arise, dependent on geometric shape of the surrounding of the liquid and of the initial conditions. Physicists tend to simplify things by considering well-defined situations. What could be the simplest configurations where flow occurs Suppose we had two parallel plates and a liquid drop squeezed in between. Let us keep the lower plate at rest and move the upper plate at constant velocity in a parallel direction, so that the plate separation distance keeps constant. Near each of the plates, the velocities of the liquid and the plate are equal due to the friction between plate and liquid. Hence a velocity field that describes the stream builds up, (Fig. 15). In the simplest case the velocity is linear in the spatial coordinate perpendicular to the plates. It is a shear flow, as different planes of liquid slide over each other. This is true for a simple as well as for a complex fluid. But what will happen to the mesoscopic structure of a complex fluid How is it affected Is it destroyed or can it even be built up For a review of theories and experiments, see Ref. 122. Let us look into some recent works. 

Quenching Distance The distance between two parallel plates (flat walls) that will jnst qnench (extingnish) the flame front of a particnlar fnel-oxidant mixtnre. It is smaller than the qnenching diameter in cylindrical tnhes. 

The simple Helmholz model, in which the charge on the model is regarded as the plate of a capacitor that attracts a counter layer of ions of opposite charge and results in two parallel plates of the same charge density, is inconsistent with the shapes of the electrocapillary curves obtained in practice. It can be shownthat if the Helmholz model applied, the electrocapillary curve would conform to the relationship... 

Our experimental techniques have been described extensively in earlier papers (2, 13). The gamma ray irradiations were carried out in a 50,000-curie source located at the bottom of a pool. The photoionization experiments were carried out by krypton and argon resonance lamps of high purity. The krypton resonance lamp was provided with a CaF2 window which transmits only the 1236 A. (10 e.v.) line while the radiation from the argon resonance lamp passed through a thin ( 0.3 mm.) LiF window. In the latter case, the resonance lines at 1067 and 1048 A. are transmitted. The intensity of 1048-A. line was about 75% of that of the 1067-A. line. The number of ions produced in both the radiolysis and photoionization experiments was determined by measuring the saturation current across two electrodes. In the radiolysis, the outer wall of a cylindrical stainless steel reaction vessel served as a cathode while a centrally located rod was used as anode. The photoionization apparatus was provided with two parallel plate nickel electrodes which were located at equal distances from the window of the resonance lamp. 

Fig. 4.15 Effect of axial conduction, channel between two parallel plates (Maranzana et al. 2004) (schematic view) 1 cover plate, 2 micro-channel, 3 silicon block 1,...

Mala GM, Li D, Werner C (1997b) Flow characteristics of water through a micro-channel between two parallel plates with electro kinetic effects. Int J Heat Fluid Flow 18 491 96 Male van P, Croon de MHJM, Tiggelaar RM, Derg van den A, Schouten JC (2004) Heat and mass transfer in a square micro-channel with asymmetric heating. Int J Heat Mass Transfer 47 87-99 Maranzana G, Perry I, Maillet D (2004) Mini- and micro-channels influence of axial conduction in the walls. Int J Heat Mass Transfer 47 3993 004 Maynes D, Webb BW (2003) Full developed electro-osmotic heat transfer in microchannels. Int J Heat Mass Transfer 46 1359-1369... 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The van der Waals force between two parallel plates on unit area is given as follows ... 

In the laser flash method, a melt of interest is placed between two parallel plates. The upper plate is heated stepwise and the thermal diffusiv-ity is measured from the rise in temperature. The specific design for molten materials and especially slags employed by Ohta et al. is based on the differential three-layer technique utihzing a special cell that can be accommodated in the system. A schematic diagram of the principle of the measurement section is shown in Fig. 31. A laser pulse irradiates the upper (platinum) crucible and the temperature response of the surface of the lower platinum crucible is observed, a liquid specimen being sandwiched between the two. 

In order to sustain a discharge between two parallel plate electrodes, the product of pressure p and interelectrode distance L has to satisfy the Paschen law for the... 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

Other researchers used flow between two parallel plates as the experimental and theoretical system to incorporate diffusion plus convection into their dissolution modeling and avoid film model approximations [10]. Though they did not consider adding reactions to their model, these workers did show that convection was an important phenomenon to consider in the mass transfer process associated with solid dissolution. In fact, the dissolution rate was found to correlate with flow as... 

Figure 14 illustrates a simple ionization circuit consisting of two parallel plates of metal with an air space between them. The plates are connected to a battery which is connected in series with a highly sensitive ammeter. 

Imagine a Maxwell liquid placed between two parallel plates and sheared by moving the upper plate in its own plane. However, instead of moving the plate at a constant velocity as discussed in Chapter 1, let the displacement of the plate vary sinusoidally with time, ie the plate undergoes simple harmonic motion. If the maximum displacement of the upper plate is X and the distance between the plates is h, then the amplitude A of the shear strain in the liquid is given by... 

The relationship between charge and potential are derived by assuming that the planes can be treated as plates of two parallel plate capacitors in series (18) with... 

Penney and Lynch (P2, Method No. 2) passed an aerosol between two parallel plates with an adjustable potential across them. The collection efficiency achieved is a measure of the average electrostatic mobility of the particles. 

Couette flow is shear-driven flow, as opposed to pressure-driven. In this instance, two parallel plates, separated by a distances h, are sheared relative to one another. The motion induces shear in the interstitial fluid, generating a linear velocity profile that depends on the motion of the moving surface. If we assume a linear shear rate, the shear stress is given simply by... 

Rowell and Finlayson described leakage flow through the flight clearance as flow between two parallel plates [7]. The rate of energy dissipation between the flight lands and the barrel wall is given by Eq. 7.82. This dissipation is considered to be the same as that for barrel rotation, and it is provided in the literature [9]. 

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