Extensional flow defined

Affine Deformation of a Liquid Droplet in Extensional Flow The shape of a droplet in extensional flow defined by vv = dx/dt = Gx and vv = dy/dt = —Gy is shown in the accompanying figure. 

The Giesekus criterion for local flow character, defined as

extensional flow, 0 in simple shear flow and — 1 in solid body rotation [126]. The mapping of J> across the flow domain provides probably the best description of flow field homogeneity current calculations in that direction are being performed in the authors laboratory. 

In the context of the preceding model, a drop is said to break when it undergoes infinite extension and surface tension forces are unable to balance the viscous stresses. Consider breakup in flows with D mm constant in time (for example, an axisymmetric extensional flow with the drop axis initially coincident with the maximum direction of stretching). Rearranging Eq. (26) and defining a characteristic length Rip113, we obtain the condition, for a drop in equilibrium,... 

We first derive the kinematics of the deformation. The flow situation is shown in Fig. 14.14. Coordinate z is the vertical distance in the center of the axisymmetric bubble with the film emerging from the die at z = 0. The radius of the bubble R and its thickness 8 are a function of z. We chose a coordinate system C, embedded in the inner surface of the bubble. We discussed extensional flows in Section 3.1 where we defined the velocity field of extensional flows as... 

Extensional flow (also called elongational flow) is defined as a flow where the velocity changes in the direction of the flow dvi/ dxy in contradistinction with shear flow where the velocity changes normal to the direction of flow (dv1/dx2). In uniaxial flow in the x1 direction the extensional rate of strain is defined as ... 

Wall thickness uniformity is often compromised when fabricating near the material break point. Thinner walls are expected in the most stretched regions, but only if the melt is not allowed to recoil after cessation of flow. Often fabrication conditions are selected to be well away from the break point to minimize these issues. Key break point metrics for the startup of data illustra-trated in Figure 13.5 are the time /b and tensile stress coefficient rj (e,ti,) at break. Both quantities may be multiplied by the strain rate e to estimate the Hencky break strain (sb = tb s) and the break stress (tTb = kr (s, t)). Similar metrics can be defined for other startup extensional flows. 

The most general extensional flow is defined as a flow with a velocity gradient of the form... 

It is convenient to express the capillarity number in its reduced form K = K / K, where the critical capillary number, K., is defined as the minimum capillarity number sufficient to cause breakup of the deformed drop. Many experimental studies have been carried out to establish dependency of K on X. For simple shear and uniaxial extensional flow, De Bruijn [1989] found that droplets break most easily when 0.1 4 ... 

Another axisymmetric extensional flow is biaxial extension. In this deformation, there is a compression along the axis of symmetry that stretches in the radial direction, as shown in Fig. 9. The principal strain rate is defined as... 

A cylindrical fiber is subjected to elongational flow along the fiber axis such that the z-component of the velocity vector is = Az, where A is a positive constant that defines the rate of elongational flow. The fiber is isotropic with a Poisson ratio of 0.5, which means that there is no volume change during extensional flow. Newton s law of viscosity is valid to describe this phenomenon. 

It is now possible to compare the changes in interfacial area produced in unidirectional shear, equation (11.7), uniaxial elongation, equation (11.19) and planar extension, equation (11.20). However, as pointed out be Cheng, this comparision needs to be done on a rational basis. For example, it is possible to examine the area ratios at equal strain defined by y, = Te = 7pe and it is clear from this viewpoint that effectiveness in generating new surface area increases in the order simple shear, uniaxial extoision to planar extension. It is clear that at large strains the advantage is heavily in favour of the extensional flows. If a value of strain equal to 10 is considered (Mohr states that most shear mixers exceed this value) then the area ratios SISn are given in Table 11.1. However,... 

In extensional flows, the velocity increases (fibre melt spinning) or decreases (radial flow from the sprue in an injection mould) along the streamlines, but there is no velocity gradient in the perpendicular direction. Figure 5.5 shows fibre melt spinning where the velocity increases with distance x from the spinneret, as the result of a tensile stress along the fibre. The tensile strain rate 6x is defined by... 

In marked contrast to measurements of shear rheological properties, such as apparent viscosity in steady shear, or of complex viscosity in small amplitude oscillatory shear, extensional viscosity measurements are far from straightforward. This is particularly so in the case of mobile elastic liquids whose rheology can mitigate against the generation of well-defined extensional flow fields. 

As developed in Section 10.3, the stability of extensional flow in visco-plastic solids is governed by intrinsic properties of the solid, such as its plastic resistance, its strain-hardening rate and its strain-rate-hardening rate, through the sensitivity of the plastic resistance to the strain rate. In many instances, however, the deforming bar or fiber contains imperfections that can affect or hasten localization in necks and subsequent rupture. Such perturbations of flow by imperfections and their effect on material stability in extensional flow have been of great interest. A well-defined scenario of this was conceived by Hutchinson and Obrecht (1977) and further developed by Hutchinson and Neale (1977). 

In order to characterize polymeric fluids and to test rheological equations of state it is customary to use simple, well defined flows. The two main flows are simple shear and simple elongational. These are shown schematically in Figure 1. In shear flow, material planes (see Figure 1) move relative to each other without being stretched, whereas in extensional flow the material elements are stretched. These two different flow histories generate different responses in not only flexible chain polymers but in liquid crystalline polymers. When these flows are carried... 

During shear or uniaxial extensional flow, the initially spherical drop deforms into a prolate ellipsoid with the long axis, ai, and two orthogonal short axes, a2. It is convenient to define the drop deformability parameter, D, as... 

Another important class of fluid flow is the extensional flow (or, elongational flow), which refers to flow where the rate of deformation tensor is diagonal. For an incompressible fluid, one can define the following form (Dealy 1994) ... 

For the uniaxial extensional flow the extensional viscosity is defined as (Dealy 1994)... 

In extensional flow, the diagonal components of are non-zero (i.e. T,y = 0 for i j). In the case of uniaxial extension, Th is the primary stress that can be measured, while T22 and T33 are generally equal to the pressiure of the environment. Thus, the uniaxial extensional viscosity rj is defined by. 

Following Newton (1640), the viscosity is defined as the ratio of the stress over the deformation rate. Whether a shear or a simple extensional flow is considered, one has then the shear or the extensional viscosity, and if such quantities are rate dependent, one deals with shear or extensional viscosity functions. Experimentally,... 

Figure 10.3.7 gives the intrinsic viscosity of prolate spheroids in uniaxial extensional flow. We define the intrinsic viscosity in extensional flow as... 

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