Average stretching rate

Equation (35) indicates that initially, the stretching grows exponentially with time, which can be seen in Fig. 5. Since (A (0)> = (2iVa /3ir) for a random-walk conformation, the average stretching rate is given by... 

Since we have already assumed that is independent of r, consistency requires that we take the stretch rate dv /dz to be independent of r and equal to dv ldz, in which case obtaining the average of the right side of Equation 7.31 requires averaging only the viscosity hence, we have... 

However, when we have mobile solutions, the methods described above are impossible, and we have to try to create an extensional flow within a flowing system. This has been done in a number of ways, as shown in figure 26. The method used depends on the liquid of interest being spinnable or not spinnable. If the solution is spinnable, then it is possible to wind up the liquid thread on a rotating drum. In all these cases the average extension rate is measured using a camera system to record the profile of the stretched liquid. 

The Lyapunov exponent is the geometric average of the local stretching rates, which assigns equal weight to all local stretching values in the domain ... 

Solution. Begin by computing the parameters necessary to evaluate the stretch rate in radial geometry with Eq. 5.40. The average grain diameter is estimated with Eq. 5.38. 

Roozemond et al. [150] related the longitudinal growth rate to the average stretch of the HMW chains. We define... 

Table 2 Average Values of the Modulus, Yield Stress, Yield Strain, and Strain at Break for Three Samples of PTEB Stretched at Different Temperatures and Deformation Rates...

Figure 12.4 A series of SFG spectra in the CO stretch region of chemisorbed CO on polycrystalline Pt in a CO-free 0.1 M H2SO4 electrolyte. The atop spectra were fit to (12.5) (see text) to extract the amplitude, frequency, and width [Lu et al., 2005 Lagutchev et al, 2006] (each displayed data point is the average of three or five spectra). The electrode potential was swept at a rate of 5 mV/s, and SFG spectra were obtained every 200 ms. Spectra were obtained at 1 mV intervals, but, to avoid congestion in the plot, averaged spectra are displayed at 10 mV intervals in the pre-oxidation region (V < 0.43 V) and at 3.3 mV intervals in the oxidation region (V > 0.43 V) [Lu et al., 2005].

Two alternative assumptions have been made for the manner in which the area variation occurs. The more realistic postulates that all elements of the surface remain in the surface throughout an oscillation cycle. Increasing surface area stretches the surface (A3, B5) and causes a velocity normal to the surface which increases the diffusion rate. For a surface of area Aq suddenly exposed at t = 0, the mass transfer product averaged over time is given by... 

The assumption of transfer by a purely turbulent mechanism in the Handlos-Baron model leads to the prediction that the internal resistance is independent of molecular diffusivity. However, such independence has not been found experimentally, even for transfer in well-stirred cells or submerged turbulent jets (D4). In view of this fact and the neglect of shape and area oscillations, models based upon the surface stretch or fresh surface mechanism appear more realistic. For rapid oscillations in systems with Sc 1, mass transfer rates are described by identical equations on either side of the drop surface, so that the mass transfer results embodied in Eqs. (7-54) and (7-55) are valid for the internal resistance if is replaced by p. Measurements of the internal resistance of oscillating drops show that the surface stretch model predicts the internal resistance with an average error of about 20% (B16, Yl). Agreement of the data for drops in liquids with Eq. (7-56) considerably improves if the constant is increased to 1.4, i.e.. 

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