Strain ratio, mean-square

Turbulent mass burning rate versus the turbulent root-mean-square velocity by Karpov and Severin [18]. Here, nis the air excess coefficient that is the inverse of the equivalence ratio. (Reprinted from Abdel-Gayed, R., Bradley, D., and Lung, F.K.-K., Combustion regimes and the straining of turbulent premixed flames. Combust. Flame, 76, 213, 1989. With permission. Figure 2, p. 215, copyright Elsevier editions.)... 

In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio X of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number n, of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n 5 por polymer SI). 

For a given chain, the strain ratios 5 (q) fully determine the mean-square dimensions of the chain. In the following, they will be derived from free-energy self-consistent optimization. 

In the preceding, oc iq) (without the tilde) is the mean-square strain ratio of the q mode with respect to the unperturbed real chain with screened interactions (i.e., at r = 0), not with respect to the phantom chain accordingly, cco(q) s 1. In agreement with our assumption of a small solvent strength, will be proportional to T—, and we shall write... 

Here the elongation ratio A = L/Lq is given by the ratio of lengths L and Lq after and before straining, respectively. Under applied strain, the complicated interplay of changes of angle and mean-square dimension from initial segment orientation and position leads to a second moment which assumes its minimum value near but not exactly at A = 1. 

Predictions for the Parameters k and The parameter f is not far from zero, which is to he expected since the surroundings of jimctions cause their deformation to be nearly affine with the macroscopic strain. The primary parameter ic is defined as the ratio of the mean-square junction fluctuations in the equivalent phantom network, ie, in the absence of constraints, to the mean-square jimction fluctuations about the centers of domains of entanglement constraints (in the absence of the network) in the isotropic state. Thus in a phantom network, the absence of constraints leads to /c = 0. In an affine one, the complete suppression of fluctuations is equivalent to /c = oo. It has been proposed that k should be proportional to the degree of interpenetration of chains and junctions (165). Since an increasing number of junctions in a volume pervaded by a chain leads to stronger constraints on these jimctions, k was taken to be... 

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