Parallel plate pressure flow

Stresses Generated by CEF Fluids in Various Viscometric Flows What stresses are necessary to maintain a CEF fluid flowing in the following flows (a) parallel-plate drag flow (b) Couette flow with the inner cylinder rotating and (c) parallel-plate pressure flow. Assume the same type of velocity fields that would be expected... 

The lubrication approximation was discussed in terms of Newtonian fluids. Considering a nearly parallel plate pressure flow (H = Ho — Az), where A is the taper, what additional considerations would have to be made to consider using the lubrication approximation for (a) a shear-thinning fluid flow, and (b) a CEF fluid ... 

Before closing this chapter, we feel that it is useful to list in tabular form some isothermal pressure-flow relationships commonly used in die flow simulations. Tables 12.1 and 12.2 deal with flow relationships for the parallel-plate and circular tube channels using Newtonian (N), Power Law (P), and Ellis (E) model fluids. Table 12.3 covers concentric annular channels using Newtonian and Power Law model fluids. Table 12.4 contains volumetric flow rate-pressure drop (die characteristic) relationships only, which are arrived at by numerical solutions, for Newtonian fluid flow in eccentric annular, elliptical, equilateral, isosceles triangular, semicircular, and circular sector and conical channels. In addition, Q versus AP relationships for rectangular and square channels for Newtonian model fluids are given. Finally, Fig. 12.51 presents shape factors for Newtonian fluids flowing in various common shape channels. The shape factor Mq is based on parallel-plate pressure flow, namely,... 

For fluids with constant density, this equation is reduced to the incompressibility condition, V m = 0. In classical fluid dynamics problems, e.g., flow between parallel plates and flow in a cylindrical tube, Eqs. (2.4) and (2.5) lead to the familiar parabolic pressure-driven velocity profile, which serves as a useful baseline for microfluidics. 

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. 

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

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... 

The baffle cut determines the fluid velocity between the baffle and the shell wall, and the baffle spacing determines the parallel and cross-flow velocities that affect heat transfer and pressure drop. Often the shell side of an exchanger is subject to low-pressure drop limitations, and the baffle patterns must be arranged to meet these specified conditions and at the same time provide maximum effectiveness for heat transfer. The plate material used for these supports and baffles should not be too thin and is usually minimum thick-... 

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... 

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. 

The radio-frequency glow-discharge method [30-34] has been the most used method in the study of a-C H films. In this chapter, it is referred to as RFPECVD (radio frequency plasma enhanced chemical vapor deposition). Film deposition by RFPECVD is usually performed in a parallel-plate reactor, as shown in Figure 1. The plasma discharge is established between an RF-powered electrode and the other one, which is maintained at ground potential. The hydrocarbon gas or vapor is fed at a controlled flow to the reactor, which is previously evacuated to background pressures below lO"" Torr. The RF power is fed to the substrate electrode... 

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... 

Here tw represents the combined stress on the lower and upper walls (i.e., double the stress on a single wall). The pressure gradient for flow between parallel plate follows as... 

It is interesting to note several special cases of the system. First, it may be recognized that system approaches the parallel-plate Hagen-Poiseuille flow in the limit that V - 0 and A V -> 0. In this case both Reynolds number terms are eliminated from the momentum equation. Furthermore, since W — C/, only the U velocity remains in the pressure-gradient eigenvalue K. The momentum equation is simplified to... 

The flow at high Rep approaches the planar, finite-gap, stagnation flow between parallel plates. In this case, the injection velocity V dominates over the initial velocity U that enters the channel. The system of equations developed here are essentially the same as those for finite-gap planar stagnation flow. Indeed, it is only the relationship between K and the axial pressure gradient that distinguishes the two flows. 

Show that the limiting case of flow in a planar wedge for a < C 1 (Section 5.2.4) degenerates into the parallel-plate Hagen-Poiseuille flow. Show that the parabolic velocity profile is recovered and that the pressure gradient approaches a constant. 

Vane eiiminators force gas flow to undergo directional changes as it passes between parallel plates. Droplets impinge on plate surfaces, coalesce and fail to a liquid collecting spot for routing to the liquid-collection section of the vessel. Vane-type eliminators are sized by their manufacturers to assure a certain minimum pressure drop. 

Silicon wafers were oxygen plasma-cleaned in a parallel plate reactor at 400 W power, 300 mTorr pressure for 5 min. The gas flow was set to 100 seem total,... 

Fig. 1 a, b. Schematic diagram of a flow of fluid under combined shear conditions a — between flatly parallel plates under the action of pressure difference AP = P -P2 (the upper plane moves in the direction transverse to the main flow) b — between two coaxial cylinders rotating towards one another at angular velocities flj and fi2... 

Flow in a circular channel with a significant relative length 1/H (here 1 — is the length of circular head H = R2 — Rt is the width of clearance, i.e., the difference between the inner radius of the outer cylinder, the tip, and the outer radius of the inner cylinder, the core) was simulated by the flow of a polymer between two parallel plates removed from one another to a distance H (see Fig. la). The resultant flow occurs due to the pressure difference AP = P, — P2 and motion of the upper plate with velocity U0 in the direction transverse to the axial flow. In this case boundary conditions in the Cartesian system of coordinates are ... 

The field of transport phenomena is the basis of modeling in polymer processing. This chapter presents the derivation of the balance equations and combines them with constitutive models to allow modeling of polymer processes. The chapter also presents ways to simplify the complex equations in order to model basic systems such as flow in a tube or Hagen-Poiseulle flow, pressure flow between parallel plates, flow between two rotating concentric cylinders or Couette flow, and many more. These simple systems, or combinations of them, can be used to model actual systems in order to gain insight into the processes, and predict pressures, flow rates, rates of deformation, etc. 

One of the most common flows in polymer processing is the pressure driven flow between two parallel plates. When deriving the equations that govern slit flow we use the notation presented in Fig. 5.12 and consider a steady fully developed flow a flow where the entrance effects are ignored. 

Pressure flow of two immiscible fluids with different viscosities that flow as separate layers between parallel plates are often encountered inside dies during co-extrusion when producing multi-layer films. Such a system is schematically depicted in Fig. 6.16, which presents two layers of thickness h/2 and viscosities m and //,2, respectively. 

For this non-isothermal flow consider a Newtonian fluid between two parallel plates separated by a distance h. Again we consider the notation presented in Fig. 6.58, however, with both upper and lower plates being fixed. We choose the same exponential viscosity model used in the previous section. We are to solve for the velocity profile between the two plates with an imposed pressure gradient in the x-direction and a temperature gradient in the y-direction. 

Solve a coupled heat transfer and flow system of a temperature dependent viscosity melt that is driven by a pressure between two parallel plates. The viscosity is given by v = rioe (T To Use the dimensions given in the previous problem, where rjn = 1000 Pa-s, a = 0.04 K 1, the bottom plate temperature, To. of 140°C and the upper plate temperature of 160°C. 

---