Fluid strain rate

A fluid packet, like a solid, can experience motion in the form of translation and rotation, and strain in the form of dilatation and shear. Unlike a solid, which achieves a certain finite strain for a given stress, a fluid continues to deform. Therefore we will work in terms of a strain rate rather than a strain. We will soon derive the relationships between how forces act to move and strain a fluid. First, however, we must establish some definitions and kinematic relationships. 

Consider first the behavior in the z-r plane as illustrated in Fig. 2.6. In a short interval of time dt, the differential element has experienced translation, rotation, dilatation, and shear. 

Consider first the normal strain rates in the z and r coordinates, ezz and err. By definition, the strain rate is given as the rate at which the relative dimension of a fluid packet changes per unit time. Stated differently, the product of the strain rate and time represents a relative elongation. Consider first the relative elongation in the radial direction owing to the r-direction normal strain, 

The radial velocity itself varies over the length of the differential element that is to say, the velocity at one edge of the element is different than that at the other end. For this reason the element will stretch or shrink, meaning dilate. The extent of the dilatation, over a differential unit of time, is (dvfdr)drdt. Thus it follows easily that 

The rotation rate of the element about the axis perpendicular to the r-z plane (i.e., about the 6 axis) is measured by the angular motion d le of the diagonal line, which is shown dashed in Fig. 2.6. The angular rotation is influenced by both the dilatation and the shearing of the element. The following equations are developed geometrically from Fig. 2.6  

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In flow-induced degradation, K is strongly dependent on the chain length and on the fluid strain-rate (e). According to the rate theory of molecular fracture (Eqs. 70 and 73), the scission rate constant K can be described by the following equation [155]... 

The Stokes viscous drag equation predicts a proportionality between the molecular stress ( / ) with the product of solvent viscosity (qs) and fluid strain-rate... 

A plausible assumption would be to suppose that the molecular coil starts to deform only if the fluid strain rate (s) is higher than the critical strain rate for the coil-to-stretch transition (ecs). From the strain rate distribution function (Fig. 59), it is possible to calculate the maximum strain (kmax) accumulated by the polymer coil in case of an affine deformation with the fluid element (efl = vsc/vcs v0/vcs). The values obtained at the onset of degradation at v0 35 m - s-1, actually go in a direction opposite to expectation. With the abrupt contraction configuration, kmax decreases from 19 with r0 = 0.0175 cm to 8.7 with r0 = 0.050 cm. Values of kmax are even lower with the conical nozzles (r0 = 0.025 cm), varying from 3.3 with the 14° inlet to a mere 1.6 with the 5° inlet. In any case, the values obtained are lower than the maximum stretch ratio for the 106 PS which is 40. It is then physically impossible for the chains to become fully stretched in this type of flow. 

In earlier sections the fluid strain rate was described in terms of the velocity field. Up to this point, however, the stress has not been related to the underlying flow field. It is the quantitative relationship between fluid strain rate and stress that permits the momentum-conservation equations (Navier-Stokes equations) to be written with the velocity field as the dependent variable. 

The nondimensional Weissenberg number (Wi = jX) is defined as the product of fluid strain rate (y) and the longest relaxation time of the polymer chain (X). This number provides an approximation to the ratio of ability of the flow to stretch the polymer chain divided by the tendency of the chain to resist stretching. For large Wi, the flow generally results in stretching of the molecule for small Wi, the molecule s conformation is dominated by the Brownian motion. 

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