Viscosity flow geometries

The flow behavior of the polymer blends is quite complex, influenced by the equilibrium thermodynamic, dynamics of phase separation, morphology, and flow geometry [2]. The flow properties of a two phase blend of incompatible polymers are determined by the properties of the component, that is the continuous phase while adding a low-viscosity component to a high-viscosity component melt. As long as the latter forms a continuous phase, the viscosity of the blend remains high. As soon as the phase inversion [2] occurs, the viscosity of the blend falls sharply, even with a relatively low content of low-viscosity component. Therefore, the S-shaped concentration dependence of the viscosity of blend of incompatible polymers is an indication of phase inversion. The temperature dependence of the viscosity of blends is determined by the viscous flow of the dispersion medium, which is affected by the presence of a second component. 

RANS turbulence models are the workhorse of CFD applications for complex flow geometries. Moreover, due to the relatively high cost of LES, this situation is not expected to change in the near future. For turbulent reacting flows, the additional cost of dealing with complex chemistry will extend the life of RANS models even further. For this reason, the chemical-source-term closures discussed in Chapter 5 have all been formulated with RANS turbulence models in mind. The focus of this section will thus be on RANS turbulence models based on the turbulent viscosity hypothesis and on second-order models for the Reynolds stresses. 

We assume that the flow geometry, as well as the fluid density and viscosity (/x > 0, given) are not affected by the reactions and that the flow is described by the Stokes equations relating the fluid velocity q and fluid pressure p ... 

Here is a typical Leslie viscosity, AT is a Frank constant, V is a flow velocity, /i is a length scale of the flow geometry, such as the tube diameter in Poiseuille flow, and ftn is the average shear rate. The Ericksen number is the ratio of the flow-induced viscous stress 6Ye.fi = /h to the Frank stress K/h. The appropriate Leslie viscosity or Frank constant... 

The modified dilute viscosity is approximated based on the relation Mdiiute oc p%lpV0. The bulk density is proportional to a, whereas the mean free path Ip is inversely proportional to aj,. A typical value, ad.min = 1 x 10 , was used to limit the mean free path in an unconfined flow. The limit /p,max was thus either given as a characteristic dimension of the flow geometry, or calculated from min as /j — ... 

Figure 2-9. A number of simple flow geometries, such as concentric cylinder (Couette), cone-and-plate, and parallel disk, are commonly employed as rheometers to subject a liquid to shear flows for measurement of the fluid viscosity (see, e.g., Fig. 3-5). In the present discussion, we approximately represent the flow in these devices as the flow between two plane boundaries as described in the text and sketched in this figure.

Elaborate computer codes that recognize this coupling in complex flow geometries have been devised and verified. The present examples are representative of a general class of single-phase, variable-viscosity, variable-density problems yet avoid undue complications in mathematical or numerical analysis. 

Boundary Layer Concept. The transfer of heat between a solid body and a liquid or gas flow is a problem whose consideration involves the science of fluid motion. On the physical motion of the fluid there is superimposed a flow of heat, and the two fields interact. In order to determine the temperature distribution and then the heat transfer coefficient (Eq. 1.14) it is necessary to combine the equations of motion with the energy conservation equation. However, a complete solution for the flow of a viscous fluid about a body poses considerable mathematical difficulty for all but the most simple flow geometries. A great practical breakthrough was made when Prandtl discovered that for most applications the influence of viscosity is confined to an extremely thin region very close to the body and that the remainder of the flow field could to a good approximation be treated as inviscid, i.e., could be calculated by the method of potential flow theory. 

The fact that in highly turbulent flows liquid viscosity doesn t influence on main volume medium movement is interesting and important enough [3, 135]. In this case they say that flow is self-simila in relation to viscosity and the influence of the last one is displayed in a narrow enough wall layer. The value of Reynolds criterion above which the self-simila field is observed in many respects is determined by flow geometry. For example, in [3] they showed that under sphere flow the... 

A non-Newtonian fluid is one whose flow curve (shear stress versus shear rate) is non-linear or does not pass through the origin, i.e. where the apparent viscosity, shear stress divided by shear rate, is not constant at a given temperature and pressure but is dependent on flow conditions such as flow geometry, shear rate, etc. and sometimes even on the kinematic history of the fluid element under eonsideration. Such materials may be conveniently grouped into three general elasses ... 

In practice, it is advantageous to be able to decrease the viscosity (thus decreasing the resistance to flow) by increasing pressure or velocity, as well as to modify the flow geometry in order to increase shear rates. In some cases, there is an effect of the history of previous deformation. This is exemplified by thixotropic liquids that show a low viscosity after a shear history (like stirring), while a high viscosity prevails at rest. This phenomenon is much desired in the case of paints and varnishes. 

In the differential rheometers discussed above,the material property of interest was the steady or zero-shear-rate viscosity. While viscosity is an important property of materials, it can only be measured on fluids. The dynamic mechanical properties, on the other hand, can be measured equally well on solids or fluids and can be very sensitive to changes in the material structure. Generally, the low frequency properties are the most sensitive to small changes in structure (4). Thus, the objective of this work was to investigate the theoretical response of an opposed squeeze flow geometry (5,6,7) and compare it to the experimental results for representative viscoelastic materials. While the experimental confirmation of the analyses of these problems was confined to a limited number of well-characterized materials, the general purpose of the combined theoretical and experimental approach was to demonstrate the applicability of the rheometer to the study of the viscoelastic properties of any material within the instrument s force, size and speed capabilities. 

The Re value of the lower automodel area boundary is largely determined by the flow geometry. For example, [38] demonstrates that the automodel area in a sphere-around flow, where the resistance coefficient and therefore, the viscosity is independent of Re, forms at Re = 500. As higher flow turbulisation is achieved, in comparison with a cylindrical channel in a diffuser-confusor reactor at the same Re values, we can expect automodel mode formation at lower Re values. 

The root cause of melt fracture has been the subject of numerous studies, resulting in a few theories. It is clear, however, that excessive shear stress in the polymer as it passes through the die is one source of the defect. A cause of high shear stress is abrupt changes in flow streamlines. If the internal flow geometry of a die contains sharp corners, this could lead to melt fracture. More often, excessive shear stress is due to high polymer viscosity or high polymer flow rate. Therefore, action that reduces viscosity or flow rate... 

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