J. Phys. II France
Volume 3, Numéro 5, May 1993
Page(s) 749 - 757
References of J. Phys. II France 3 749-757
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  15. Note that here we use the Laplace and not the Fourier transform of the time dependent stress tensor ; the quantity $\omega$ which determines stability is thus real, and the time dependence is e $^{\omega t}$.
  16. Since $\Pi_{x}(\omega)$ and $\Pi_{y}(\omega)$ already have terms linear in $H(x, y ; \omega) = h(x, y ; \omega) - h_{0}$, we evaluate the boundary conditions at z = h0.
  17. This is consistent with the third Navier-Stokes equation, the analogue of equation (12) for w. Using the incompressibility condition, equation (8), to solve for w indicates that w is proportional to q2. Thus to first order terms in q in the Navier-Stokes equations, we can set wzz = 0 implying that $\Pi_{z} = 0$.
  18. This can also be seen from the full dispersion relationship of equation (19). Similar restabilization has been considered in the context of jet breakup in references [ 13, 14]. While the present, dynamic argument yields the growth rate of unstable modes, an estimate of the modulus needed to prevent instability can also be obtained from energetic considerations.
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  21. It is of interest that at this level, the modulus of a polymer brush in a good solvent is significantly larger than that of a melt brush, $(E_\textrm{sol}/E_\textrm{melt}) = 1/(b \sqrt{\sigma}) = s/b \gg$ 1, where $s = \sigma^{-1/2}$ is the mean interanchor spacing $(s \gg b)$ and the segmental volume is taken as $\nu = b^{3}$.
  22. Experimentally, it is found that the longest relaxation times in entangled polymer systems scale as N$^{3\:3}$, but this does not change the present discussion (see Ferry J. D., Viscoelastic Properties of Polymers, 3rd Edition, (Wiley, Ny, 1980)).
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