Fluid plate model

Only a simplified version of the fluid plate model is presented here to establish a crude relationship between the stretching modulus and the bending rigidity of fluid monolayers. The resulting formulas, frequently used in the literature, permit some estimates and serve to illustrate the role of the reference surface. 

The emphasis of this review of membrane elasticity is on the fluid plate model of the amphiphilic monolayer. The homogeneous fluid plate is used for some elementary considerations and estimates. Plates modeling actual monolayers will possess a nonvanishing stress profile even when undeformed, i.e. in the flat state at zero lateral tension. 

Traditionally the fluid mechanics of the extrusion process are summarized by the simple plate model illustrated in Fig. A7.1 and as described in Section 7.4. The motion of the screw is unchanged, but the reference frame has been moved to transform the problem to a fixed boundary problem for the observer. The flow in the rectangular channel is reduced into the x-direction flow across the channel and the z-direction flow down the channel. 

Figure 13.7 The fluid mosaic model of biological cells, (image courtesy of www.wikipedia.org). See plate section for colour version of this image.

We have seen how the screw extruder pump is synthesized from a simple building block of two parallel plates in relative motion. We have also seen how the analysis of the screw extruder leads in first approximation back to the shallow channel parallel plate model. We carried out the analysis for isothermal flow of a Newtonian fluid, reaching a model (Eq. 6.3-27) that is satisfactory for gaining a deeper insight into the pressurization and flow mechanisms in the screw extruder, and also for first-order approximations of the pumping performance of screw extruders. 

Kaplan and Tadmor (74) proposed a mathematical model for isothermal pumping in this extruder for Newtonian fluids, which they termed the three plate model. We follow that... 

Obtaining equivalent expressions for AP for non-Newtonian fluids is difficult and for most rheological models it is not possible to obtain analytical expressions. Good approximate solutions, however, can be obtained by replacing the narrow annulus geometry by a parallel plate model (108, 109), that is, annular flow is represented by slot flow. The... 

In the case of viscoefastic fluids, the modeling of hydrodynamic in agitated tanks remains difficult, but Anne-Archard has given preliminary results in the case of a plate agitator [14]. In the laminar flow range, it appears that the effect of elasticity... 

Failure analysis of the failed MEAs indicated the fuel inlet region as the failure area. Links between the failure and specific features of the flow channel design were investigated through computational fluid dynamics modeling techniques to determine the most likely contributing factors, such as reactant and coolant channel locations, plate conductivity and interactions with the MEA thermal properties. The predicted maximum membrane-electrode interface temperature rose from 87 °C to 99.5 °C due to air bleed introduction, resulting in a hot spot consistent with the observed failnre. 

The basic principle of operation of the metering section of the single-screw extruder is illustrated by the simple plate model shown in Figure 8.16. The fluid between the two plates is considered to be Newtonian and under isothermal and steady flow conditions. Because of a restriction at the end of the channel (which is not shown) the pressure increases along the z direction, is assumed to depend only on y, since the aspect ratio of the plates is large (i.e., W/H > 10). The equation of motion becomes, after substituting in the expression for the shear stress for a Newtonian fluid. 

Petera, J. and Nassehi, V., 1995. Use of the finite element modelling technique for the improvement of viscometry results obtained by cone-and-plate rheometers. J. Non-Newtonian Fluid Mech. 58, 1-24. 

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... 

Two-dimensional compressible momentum and energy equations were solved by Asako and Toriyama (2005) to obtain the heat transfer characteristics of gaseous flows in parallel-plate micro-channels. The problem is modeled as a parallel-plate channel, as shown in Fig. 4.19, with a chamber at the stagnation temperature Tstg and the stagnation pressure T stg attached to its upstream section. The flow is assumed to be steady, two-dimensional, and laminar. The fluid is assumed to be an ideal gas. The computations were performed to obtain the adiabatic wall temperature and also to obtain the total temperature of channels with the isothermal walls. The governing equations can be expressed as... 

Molecularly motivated empiricisms, such as the solubility parameter concept, have been valuable in dealing with mixtures of weakly interacting small molecules where surface forces are small. However, they are completely inadequate for mixtures that involve macromolecules, associating entities like surfactants, and rod-like or plate-like species that can form ordered phases. New theories and models are needed to describe and understand these systems. This is an active research area where advances could lead to better understanding of the dynamics of polymers and colloids in solution, the rheological and mechanical properties of these solutions, and, more generally, the fluid mechaiucs of non-Newtonian liquids. 

Lin, J. R., Squeeze Film Characteristics Between a Sphere and [40] a Flat Plate Couple Stress Fluid Model, Camputers Structures, Vol. 75,2000, pp. 73-80. 

In the ASTER reactor deposition experiments were performed in order to compare with the 2D model results. Normalized deposition rates are plotted in Figure 22 as a function of radial position for data taken at 25 and 18 Pa. The deposition takes place on a square glass plate. For each pressure two profile measurements were performed, each profile perpendicular to the other (a and b in Fig. 22). A clear discrepancy is present. The use of the simplified deposition model is an explanation for this. Another recent 2D fluid model also shows discrepancies between the measured and calculated deposition rate [257], which are attributed to the relative simplicity of the deposition model. 

Gallego-Juarez JA, Rodriguez Corral G, Riera E et al. (2001) Development of industrial models of high power stepped-plate sonic and ultrasonic transducer for use in fluids. IEEE Ultrasonic Sypoos. Proceedings, pp 571-578... 

This model is a modification of the model developed by Kumar and Kuloor (K18) for bubble formation in inviscid fluids in the absence of surface-tension effects. The need for modification arises because the bubble forming nozzles actually used to collect data on bubble formation in fluidized beds differ from the orifice plates in that they do not have a flat base. Under such conditions the bubble must be assumed to be moving in an infinite medium and the value of 1/2 is more justified than the value 11/16. 

---