Linearization of tensor nonlinear constitutive relations in the problems on stability of flows

The apparatus of tensor nonlinear functions occupies an important place in the nonlinear mechanics of a continuous medium, both in hydrodynamic applications and in problems of mechanics of a deformed solid, strength and fracture [1]. Tensor nonlinear defining correlations simulate the so-called orthogonal effects of the stress-strain state (see in [2] a review on the issue), characterized by noncollinearity of voltage deviators and the corresponding kinematic tensor. Such a noncollinearity can explain the Poynting effect and ratchet [3–9]. The scientific works pays much attention both to the definition of the main flow parameters and to the stability of such a flow with respect to small perturbations belonging to a particular class. The statement of the boundary value problem in perturbations assumes the linearization of all the system equations near the main process, including the defining correlations. Along with the general form of the tensor-nonlinear determining relations, the paper considers tensor-linear isotropic media, tensor linear potential media, the Bingham body (a two-constant viscoplastic model), the Saint-Venant flow (ideally rigid-plastic model), and the Newtonian fluid.