# zbMATH — the first resource for mathematics

Hydrodynamic stability of unidirectional shear flow of linear and branched polymeric melts. (English) Zbl 1115.76323
Summary: The linear stability of homogeneous shear flow of polymer melts has been investigated by using a prototypical reptation-based model, namely, the Pompom model. The hydrodynamic stability characteristics of the flow have been determined by using an eigenvalue analysis. Particular attention has been paid to accurate determination of the discrete and continuous eigenvalues as well as identification and isolation of the spurious eigenvalues. Specifically, it has been shown that the eigenspectrum for the Pompom model has four continuous spectra, three of which are associated with the orientation tensor, S, i.e., one regular and two branch-cuts. One spectrum arises as a result of the stretch, $$\lambda$$ evolution equation, which always occurs as the leftmost spectrum. The discrete modes are classified as centered and non-centered eigenvalues, depending on their imaginary parts. It is shown that there is only one pair of non-centered eigenvalues (a non-centered eigenvalue possesses $$\sigma_{i} \neq -\alpha/2$$, where $$\sigma_{i}$$ denotes the imaginary part of the eigenvalue and $$\alpha$$ denotes the wavenumber in the streamwise direction). For large deformations rates, the real parts of these eigenvalues decay in a similar fashion as the Gorodstov-Leonov eigenvalue pair, observed in planar flows of the Upper Convected Maxwell model (UCM). The number of centered eigenvalues however, depends strongly on the choice of parameters. We observe that the leading eigenvalues always belong to the right-most continuous mode (branch-cut), which possess highly singular eigenfunctions. Although, the flow is linearly stable when a maximum in the shear stress versus shear rate curve is not observed, the ballooning of the right most eigenspectrum that arises due to the singular nature of its eigenfunctions could lead to erroneous determination of onset conditions for the instability.
##### MSC:
 76E05 Parallel shear flows in hydrodynamic stability 76A10 Viscoelastic fluids
Full Text: