Investigation of eigenfunctions of perturbation of the radial component of flow velocity

Abstract

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of a generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The influence of taking into account the exponential dependence of fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of incompressible fluid flow in an annular channel is investigated. The eigenvalue spectra were obtained numerically. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity disturbances, their possible growth or decay over time. Graphs of the eigenfunctions of the generalized flow stability equation in an annular channel have been constructed. It is shown that the structure of the spectra largely depends on both the properties of the liquid, determined by the functional dependence of viscosity, and on the geometry of the channel. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It has been established that at small values of the thermoviscosity parameter the spectrum is comparable to the spectrum for isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are a greater number of points at which the problem has non-zero amplitudes of transverse velocity disturbances. It is worth noting that the diversity of eigenvalue spectra corresponds to the diversity of eigenfunctions, which have a nontrivial distribution of the oscillation amplitude over the cross section in each case. Smooth curves are obtained for a narrow channel, as for the case of a flat channel. However, as the ratio of the channel radii increases, “jumps” appear. Also, note that the eigenfunctions do not have the property of symmetry; this follows from the fact that the velocity profile in the unperturbed state also does not have symmetry. The maximum values of the eigenfunctions are shifted to the right from the center of the channel, which corresponds to the fact that disturbances arise and grow intensively near the hot wall.