For slit flow

Determine the velocity profile and traction profiles in a pressure driven slit flow of a Newtonian fluid. Use Ap =1000 Pa, //, =1000 Pa-s, h 1 mm and a distance from entrance to exit of 1000 mm. Solve the problem using isoparametric 2D quadratic elements and different gauss points, compare your solutions with the analytical solution for slit flow. 

In addition in round-hole capillaries, it is possible to use a slit die to perform viscosity measurements. Theoretically such a die would have no side walls (be infinitely wide), but in practice it has been found that wall effects are negligible if the slit die is at least 10 times wider than it is high. The equations for slit flow are similar to those for round-hole capillaries ... 

This same expression could be found by using the expression for slit flow in Table 2.5 and replacing the width, W, by %R(1 -I- k) and the height, H, by / (1 - k) (see Problem 2B.4). In essence, the annular region is opened up and treated as plane slit flow. One will find this a useful approach throughout the design of polymer processes. 

To complete the equation, we must specify G and dL/ dl. G is determined by assuming that developed flow occurs in the land and dL/dl, which is a geometric variable, is then determined. We first solve for G assuming that the rheological properties are described by the power-law model. For slit flow the volumetric flow rate per unit width, q, is given as (see Table 2.5)... 

The foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

A number of instruments are based on the extmsion principle, including slit flow and normal capillary flow (Table 6). These instruments are useful when large numbers of quality control or other melt viscosity test measurements are needed for batches of a single material or similar materials. When melt viscosities of a wide range of materials must be measured, rotational viscometers are preferable. Extmsion rheometers have been applied to other materials with some success with adhesives and coatings (10,161). 

A first necessary condition for the existence of the ID analog of Eq. (25) is the existence of a self-similar (asymptotic) velocity profile (itself equivalent to the existence of a ID equation for the flow field). This self-similar profile depends only on the wall Reynolds (Rew) number and has the following form (planar slit geometry) ... 

Derive the equation for the steady state temperature profile in pressure driven slit flow with viscous dissipation. Assume a Newtonian viscosity model. 

Hermite interpolation for the velocity and velocity gradient of a power law fluid in a narow slit. Consider the analytical expressions for the velocity and the velocity gradient for the flow of an incompressible power law fluid through a narrow slit due to apressure gradient to represent velocity measurements. With eqns. (7.17) and (7.18), the velocity and its gradient are evaluated in 10 equally spaced points through the thickness. An expression to interpolate both, the velocity and its gradient, is required for a point i + 1/2 located between point i and + 1. 

Figures 11.15 and 11.16 compare the temperature and velocity profiles, respectively, for the steady-state, fully developed flow of the coupled flow-heat transfer pressure driven slit flow problem, using RFM and FDM. The agreement between the two solutions is excellent.

Upon exiting the die, the sheet extrudate will swell to a level determined by the polymer, the melt temperature, the die length-to-opening ratio, and the shear stress at the die walls. Additionally, flow instabilities will occur at values of the corrected shear stress at the wall, of the order of, but higher than 105 N/m2, as found by Vlachopoulos and Chan (58), who also concluded that, for PS, HDPE, and LDPE, the critical Sr in slits is 1.4 times higher than in tubes of circular cross section. Aside from these differences, the information presented in Section 12.1 and 12.2 applies to slit flow. 

The Reynolds number Re is here defined by setting L in (13.4) equal to the tube diameter, 2R. As for the channel, Re must not exceed a magnitude of 1000 for the flow to be laminar. The velocity gradient at the wall is very similar to that for the slit (13.6), with y replaced by r and h by R. 

Flow between two parallel stationary walls as a result of an applied pressure gradient is known as slit flow. It can be considered as the two-dimensional analog of capillary flow. This technique is used when studying practical industrial processes such as injection molding of plastics, for example, for bottle-pack lines. The shear stress is here calculated from ... 

Determine Oout/Oin as a function of for the slit flow reactor in Figure 16.6. The... 

PoiseuiUe (capillary or slit), large yes functions of spatial functions of spatial coor- for laminar flows small ... 

The deformability of the viscoelastic drops in Newtonian matrix was studied in the convergent slit flow. Both, the experimental observations and the boundary element method computations were carried out. It was reported that deformation of the Boger fluid drop, was quite low —about 1/3 of that recorded for the deformability of a strongly shear-thinning, viscoelastic solution. The latter drops showed deformability similar to these observed for Newtonian drops of similar viscosities. 

The reactive vapours leave the nozzle in a laminar gas flow and mix only by diffusion. The optimum distance between the glass surface and the slit nozzle depends on the total gas flow rate and is for a flow rate of 1 m3 h 1 per 1 m nozzle length in the range of 3 mm. The chemical reaction to form Sn02 takes place mainly on the hot... 

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