Parallel plate flow Newtonian fluids, isothermal

Parallel-Plate Flow of Newtonian Fluids A Newtonian polymeric melt with viscosity 0.21b(S/in2 and density 481b/ft3, is pumped in a parallel-plate pump at steady state and isothermal conditions. The plates are 2 in wide, 20 in long, and 0.2 in apart. It is required to maintain a flow rate of 50 lb/h. (a) Calculate the velocity of the moving plate for a total pressure rise of 100 psi. (b) Calculate the optimum gap size for the maximum pressure rise, (c) Evaluate the power input for the parts (a) and (b). (d) What can you say about the isothermal assumption ... 

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. 

In this example, we consider the viscous, isothermal, incompressible flow of a Newtonian fluid between two infinite parallel plates in relative motion, as shown in Fig. E2.5a. As is evident from the figure, we have already chosen the most appropriate coordinate system for the problem at hand, namely, the rectangular coordinate system with spatial variables x, y, z. 

According to the lubrication approximation, we can quite accurately assume that locally the flow takes place between two parallel plates at H x,z) apart in relative motion. The assumptions on which the theory of lubrication rests are as follows (a) the flow is laminar, (b) the flow is steady in time, (c) the flow is isothermal, (d) the fluid is incompressible, (e) the fluid is Newtonian, (f) there is no slip at the wall, (g) the inertial forces due to fluid acceleration are negligible compared to the viscous shear forces, and (h) any motion of fluid in a direction normal to the surfaces can be neglected in comparison with motion parallel to them. 

Consider an incompressible Newtonian fluid in isothermal flow between two non-parallel plates in relative motion, as shown in Fig. E2.8, where the upper plate is moving at constant velocity V0 in the z direction. The gap varies linearly from an initial value of H0 to H over length L, and the pressure at the entrance is P0 and at the exit P. Using the lubrication approximation, derive the pressure profile. 

Note that the shape factors plotted in Fig. 6.13 are a function of only the II/W ratio. The effect of the flight on the pressure flow is stronger than that on drag flow. When the ratio Il/W diminishes, both approach unity. In this case, Eq. 6.3-19 reduces to the simplest possible model for pumping in screw extruders, that is, isothermal flow of a Newtonian fluid between two parallel plates. 

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. 

Now, for convenience, we assume that the barrel surface is stationary and that the upper and lower plates representing the screws move in the opposite direction, as shown in Fig. 6.57, but for flow rate calculations, it is the material retained on the barrel rather than that dragged by the screw that leaves the extruder. We assume laminar, isothermal, steady, fully developed flow without slip on the walls of an incompressible Newtonian fluid. We distinguish two flow regions marked in Fig. 6.57 as Zone I and Zone II. In the former, the flow is between two parallel plates with one plate moving at constant velocity relative to... 

Non-Newtonian Flow between Jointly Moving Parallel Plates (JMP) Configuration Derive the velocity profile for isothermal Power Law model fluid in JMP configuration. 

The Superposition Correction Factor Combined drag and pressure flow between parallel plates (or concentric cylinders17) of a Newtonian fluid at isothermal conditions leads to a flow-rate expression that is the linear sum of two independent terms, one for drag flow and another for pressure flow ... 

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,... 

Squeeze flow between parallel plates was analyzed in Section 6.3 as an elementary model of compression molding. In that treatment we were able to obtain an analytical solution to the creeping flow equations for isothermal Newtonian fluids by making the kinematical assumption that the axial velocity is independent of radial position (or, equivalently, that material surfaces that are initially parallel to the plates remain parallel). In this section we show a finite element solution for non-isothermal squeeze flow of a Newtonian hquid. The geometry is shown schematically in Figure 8.16. We retain the inertial terms in the Navier-Stokes equations, thus including the velocity transient, and we solve the full transient equation for the temperature, including the viscous dissipation terms. The computational details. 

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