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Transient and asymptotic stability of granular shear flow. (English) Zbl 0821.76026

The paper addresses the linear stability of pure shear flows of dry particles that interact by colliding against one another nearly elastically. Analysis enables to capture transient as well as long-time phenomena. It is based on the norm of the matrix exponential or the fundamental matrix rather than on eigenvalues and their relation to eigenvalue-based stability analysis. At first the governing equations are represented in the limits of a simple model describing the interaction of the disturbances with the mean flow. The above equations constitute a system of differential equations with variable coefficients.
Two cases are investigated: (i) wavelike disturbances with time-constant wavenumber vector, and (ii) disturbances that change their wave structure in time owing to a shear-induced tilting of the wavenumber vector. Both cases lead to similar stability criteria, although their mathematical structures are markedly different. For the first case, regions of asymptotic instability are found in two-dimensional wavenumber plane, and second case is found to be asymptotically stable for all physically meaningful parameter combinations.
In the stability analysis the authors focus on the linear stability in the asymptotic limit of large time which is accurately described by the spectrum of the linear evolution operator, and then the norm of the matrix exponential is considered that gives insight into the short times stability character of the flow. The features of the pronounced transient growth are established and studied. It is shown that the transient effects might trigger high-amplitude phenomena (i.e. nonlinearities), although this analysis predicts asymptotically decaying perturbations for all physical parameter combinations. Owing to the asymptotic stability of the turning wave vector scenario, large growth and development of finite- amplitude effects can be achieved with the help of the transient instability mechanism shown in the paper.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
76T99 Multiphase and multicomponent flows
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
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