Principal elongational ratios

For example, the principal elongation ratios in simple shear can be expressed as ... 

The area Ao is now subjected to strain with principal directions in the Cartesian coordinates and with principal elongation ratios kx,ky, and A, (Erwin, 1978a). The area Ao will become A, and the vectors defining A are a, b, and d, so that... 

K Principal elongational ratio Eq. 6.37 ru Normal component of extra stress Eq. 9.1... 

That is, the interfacial area increases exponentially with time. This is attributed to Erwin [22]. Equation (6.14) forms the basis of the view that uniaxial elongational flow is important to achieve good mixing. For a simple shear flow, the principal extension ratios A, An, and Am may be shown to be [28] ... 

Where Aj, A2, and are strains expressed as the elongational ratios in the direction of the principal axes. This equation assumes that u(Aj), u(A2), and u(A ) may in general take different values this means the coordinate axes are fixed and are not free to be chosen. This is more like the usual shear and elongational measurements where the magnitude of strain is tied to a specific coordinate system. The concept of strain invariance is not used. 

In certain occasions the volume criterion is not appropriate. Fn particular when we have an ill-conditioned problem, use of the volume criterion results in an elongated ellipsoid (like a cucumber) for the joint confidence region that has a small volume however, the variance of the individual parameters can be very high. We can determine the shape of the joint confidence region by examining the cond( ) which is equal to and represents the ratio of the principal axes of... 

The quantities C and d are the second-order tensors that are an array of nine Cij(iJ = 1,2,3) and dy terms C is known as the Green deformation tensor and d is the rate of deformation tensor. It is always possible to find a principal axis coordinate system where C and d have diagonal forms, that is, where Cy and dy are zero when i is not equal toj. In this principal axis coordinate system, all deformations are representable as multi-axial elongational and may be expressed through extension ratios li, In, and Ini corresponding to these principal axes. 

When values of the capillarity number and the reduced time are within the region of drop breakup, the mechanism of breakup depends on the viscosity ratio, A. In shear, four regions have been identified for X 0.l tip spinning, for 0.1 < X < 1 drop breaks into two principal and an odd number of satellite droplets for 1 small droplets is the preferred mechanism, whereas for A >3.8 in shear drops do not break, but they do in elongation. 

Whereas the dispersed phase is predominantly present as droplets in a continuous matrix when the flow intensity is rather mild, more intense shear or extension results in the formation of elongated fibrils. For Ca-numbers that largely exceed the critical value, the droplet dynamics is completely governed by the hydrodynamic forces [103]. If the viscosity ratio is sufficienfly close to 1, the dispersed domains will follow the principal stretch of the matrix and their shape can be described by means of the affine deformation equations. Elemans et al. found that the ratio CalCa rit should exceed 2 in order to obtain affine deformation for a system withp = 0.1 [104]. For highly extended domains, the aspect ratio LIB is generally used to quantify the droplet deformation rather than the deformation parameter D. In case of affine deformation, the following expression provides the evolution of the aspect ratio LIB in shear flow [103] ... 

This alternative dramatically affects the results of quantitative calculations because, although the main axes of deformation and stress tensor coincide, the ratio of the principal values is different in general. In the case of linear elongation in Z-direction, the deformation in the X- and Y-direction is decreased whereas the X- and Y-components of the stress tensor (defined positive in traction) increase as a result of the decreasing elongation of the chains in these directions. 

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