The Influence of the Annular Gap Thickness on the Critical Reynolds Number During the Flow of Thermoviscous Liquids

Issues related to transient regimes of fluid flow in channels with different cross sections are a priority when solving problems of hydrodynamics. Currently, research related to the influence of heat exchange on the stability of fluid flow in processes in which the change in viscosity with temperature cannot be neglected has become particularly relevant. This article examines some features of the loss of stability of a laminar fluid flow with an exponential dependence of viscosity on temperature in an annular channel with a given temperature regime on its walls. For this purpose, the generalized Orr–Sommerfeld equation was derived, which was eventually written in relation to the stream function. A numerical study of the corresponding boundary value problem was carried out using the spectral method based on Chebyshev polynomials. It was shown that taking into account the effect of temperature on the viscosity of the liquid, which implies its non-uniform distribution over the cross section of the channel, leads to a decrease in the critical Reynolds number, which is consistent with the results of previous studies. In particular, as previously noted, for a narrow channel and a small thermoviscosity parameter, the spectrum of eigenvalues is identical to the spectrum for an isothermal flow in a flat channel. A change in the relative channel width and an increase in the thermoviscosity parameter leads to a significant restructuring of the structure of the eigenvalue spectra of the generalized Orr–Sommerfeld equation. As a result of the studies carried out in the presented work, the dependencies of the critical Reynolds number on the exponential factor or, in other words, the thermoviscosity parameter, which characterizes the intensity of the change in viscosity with increasing temperature, and on the parameter determining the ratio of the width of the annular channel to the radius of the inner cylindrical surface were constructed. It has been established that with increasing parameter of the relative channel width, the value of the critical Reynolds number changes non-monotonically, and its minimum value depends on the specific liquid. The latter circumstance can serve as a theoretical justification for carrying out optimization calculations when modeling technological processes. The dependence of the critical Reynolds number on the thermoviscosity parameter has a form close to a decreasing exponential function for all sizes of annular channels.