Shear flow isothermal

Hence, when solving a non-isothermal problem the question arises -is this a problem where the equations of motion and energy are coupled To address this question we can go back to Example 6.1, a simple shear flow system was analyzed to decide whether it can be addressed as an isothermal problem or not. In a simple shear flow, the maximum temperature will occur at the center of the melt. By substituting y = h/2 into eqn. (6.5), we get an equation that will help us estimate the temperature rise... 

Table 3.2 Summary of the relations for Newtonian and shear thinning/shear thickening fluids in the case of a simple fully developed pipe flow (isothermal) [61, [14]...

Figure 3.19 Profiles of shear stress, shear rate and speed through the cross-section in the case of an isothermal, simple shear flow (Couette flow) with Newtonian, shear thinning, and shearthickening fluids...

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

Birefringence setups can be designed to characterize molten materials undergoing isothermal homogeneous flow. The ranges of strains and strain rates also often coincide with those of rheometers, and consequently may be limited relative to those used in fabrication. Similarly, time-temperature superposition approaches may be used to expand the rate window. State-of-the-art setups suitable for rapid screening of new materials with research-scale quantities (5-20 g) are available for shear flow [72] and startup of uniaxial extensional flow [73,74]. 

Isothermal steady time tests are used to determine the gel point of a thermoset system as the point at which the shear viscosity tends towards infinity. In these tests the viscosity is measured as a function of time at a constant shear rate. This method has the following major disadvantages. Firstly, the infinite viscosity can never be measured due to equipment limitations and thus the gel time must be obtained by extrapolation. Secondly, shear flow may destroy or delay network formation. Finally, gelation may be confused with vitrification or phase separation since both these processes lead to an infinite viscosity (St John et al., 1993). However, some work by Matejka (1991) and Halley et al. (1994) has shown that extrapolation to zero values of reciprocal viscosity or normal stress (i.e. extrapolation to infinite viscosity and normal stress) can be used with some success. 

In fact, this is nothing more than the simple, isothermal shear-flow solution obtained earlier, and this is consistent with the fact that uq is the limiting form of (4-37) and (4-38) for Br -> 0. Next we turn to the solution of (4-51) and (4-52) for 6 and u. At this level of approximation, we see the first appearance of nonlinear terms in the equations, as well as... 

Before concluding the discussion of high-Peclet-number heat transfer in low-Reynolds-number flows across regions of closed streamlines (or stream surfaces), let us return briefly to the problem of heat transfer from a sphere in simple shear flow. This problem is qualitatively similar to the 2D problem that we have just analyzed, and the physical phenomena are essentially identical. However, the details are much more complicated. The problem has been solved by Acrivos,24 and the interested reader may wish to refer to his paper for a complete description of the analysis. Here, only the solution and a few comments are offered. The primary difficulty is that an integral condition, similar to (9-320), which can be derived for the net heat transfer across an arbitrary isothermal stream surface, does not lead to any useful quantitative results for the temperature distribution because, in contrast with the 2D case in which the isotherms correspond to streamlines, the location of these stream surfaces is a priori unknown. To resolve this problem, Acrivos shows that the more general steady-state condition,... 

In Fig. 2a, from data in Shaw (2004), the test conditions are at room temperature and low strain rate. Conditions are isothermal, and strain softening is due to the nucleation and growth of voids. The different curves result from applying compressive pressure p to the shear plane. The larger the ratio p/k, where k is the peak shear flow stress from the test, the larger is the strain y at which dx/dy = 0. 

Figure 2b considers the effect of heating on the stress-strain curve. The curve marked isothermal is that for p/k = 0.6 from Fig. 2a. That marked adiabatic is obtained from the isothermal curve assuming all of the plastic work is converted to heat and that the shear flow stress reduces with temperature rise at the rate of 30 MPa per 100 °C (a reasonable value). The two curves marked intermediate suppose one third and two thirds of the heat to be conducted away. In these cases, softening is the result of heating. The strain at which dx/dy = 0 increases from the adiabatic to the isothermal condition.

Huilgol RR, You Z (2006) On the importance of the pressure dependence of viscosity in steady non-isothermal shearing flows of compressible and incompressible fluids and in the isothermal fountain flow. J Non-Newtonian Fluid Mech 136 106-117 Hulsen MA, Van Heel APG, Van den Brule BHAA (1997) Simulation of viscoelastic flows using Brownian configuration fields. J Non-Newtonian Fluid Mech 70 79-101 Ingber MS, Mondy LA (1994) A numerical study of three-dimensional Jeffery orbits in shear flow. J Rheol 38 1829-1843... 

Fig. 3.4 Representative TEM morphology seen in quenched linear PE crystallized isothermally at 115°C. Sample had been sheared at 25 s. The arrow indicates the shear flow direction in the melt. Y. An et al. (2006) Polymer 47 5643-5656...

For the validation of the newly implemented hybrid thermal LBM, first a simple test case with a shear flow and coupled heat transfer in a micro-channel induced by a stationary bottom wall and an upper wall moving at a pre-scribed velocity (i.e. here 0.1 m/s, yielding Re = 2) was considered. The upper wall is isothermal with To = 300 K and the lower wall is heated to 7h = 600 K. The in- and outflow boundaries are treated by periodic boundary conditions (Fig. 10.25). As the problem is two-dimensional, the channel with 128 pm x 640 pm was discretized with... 

Velocity field for simple shear flow (a) isothermal and (b) viscous dissipation. 

Figure 14.4 gives plots of viscosity (log ri) versus cure time for different shear rates (y) when a general-purpose unsaturated polyester (Aropol 7030, Ashland Chemical Company) was subjected to steady-state shear flow in a cone-and-plate rheometer under an isothermal condition at 60 C. The resin had been prepared by the reaction of propylene glycol with a mixture of maleic anhydride and isophthalic anhydride. The resin system used in Figure 14.4 was cured with BPO as initiator and a solution of 5 wt % A,A-dimethylaniline, diluted in styrene, as accelerator. The following observations are worth noting in Figure 14.4. At an early stage of curing, the rj increased slowly, but then...

In References [6,114,115] shear-induced crystallization of iPP-based nanocomposites with o-MMT was examined. In Reference [114] early stages of flow-induced isothermal spherulitic crystallization of intercalated iPP/PP-g-MA/o-MMT nanocomposites with 2 wt% of the clay were studied after application of shear flow for a short time of up to 30 seconds. The flow induced enhancement of crystallization was stronger in the nanocomposite than in neat iPP. However, acceleration of crystallization in iPP/PP-g-MA blend was also observed. 

In practice there are a number of other factors to be taken into account. For example, the above analysis assumes that this plastic is Newtonian, ie that it has a constant viscosity, r). In reality the plastic melt is non-Newtonian so that the viscosity will change with the different shear rates in each of the three runner sections analysed. In addition, the melt flow into the mould will not be isothermal - the plastic melt immediately in contact with the mould will solidify. This will continuously reduce the effective runner cross-section for the melt coming along behind. The effects of non-Newtonian and non-isothermal behaviour are dealt with in Chapter 5. 

Derive expressions for the velocity profile, shear stress, shear rate and volume flow rale during the isothermal flow of a power law fluid in a rectangular section slit of width W, depth H and length L. During tests on such a section the following data was obtained. 

Polyethylene is injected into a mould at a temperature of 170°C and a pressure of 100 MN/m. If the mould cavity has the form of a long channel with a rectangular cross-section 6 mm X 1 mm deep, estimate the length of the flow path after 1 second. The flow may be assumed to be isothermal and over the range of shear rates experienced (10 -10 s ) the material may be considered to be a power law fluid. 

Any rheometric technique involves the simultaneous assessment of force, and deformation and/or rate as a function of temperature. Through the appropriate rheometrical equations, such basic measurements are converted into quantities of rheological interest, for instance, shear or extensional stress and rate in isothermal condition. The rheometrical equations are established by considering the test geometry and type of flow involved, with respect to several hypotheses dealing with the nature of the fluid and the boundary conditions the fluid is generally assumed to be homogeneous and incompressible, and ideal boundaries are considered, for instance, no wall slip. 

A first model of the calender nip flow has been presented by ArdichviUi. Further on Gaskefl presented a more precise and well-known model. Both models are very simplified, which yields that the flow is Newtonian and isothermal, and they predict that the nip force is inversely proportional with the clearance. Since mbber materials show a shear thinning behavior Ardichvilli s model seems not to be very realistic. The purpose of this section is to present a calender nip flow model based on the power law. The model is stiU being considered isothermal. Such a model was first presented by McKelvey. ... 

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