ISSN 2658–5782
DOI 10.21662
DOI 10.21662
Electronic Scientific Journal
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.
thermoviscous liquid,
hydrodynamics instability,
eigenfunctions,
annular channel
The purpose of this work is to study the hydrodynamic stability of the flow of thermoviscous liquids with a monotonic dependence of viscosity on temperature and to search for critical parameters that characterize the flow regime.
Solution methods. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of the previously obtained generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind.
As a result of the study, it was established:
At present, sufficient groundwork has been accumulated in the study of the stability of liquid flows in flat channels, however, when studying this problem, the influence of the temperature factor on the change in flow regime is often neglected. Flows of viscous liquids arise in a number of industries during the operation of various technical installations and devices, during the implementation of a number of technological processes. In such cases, the problem of identifying the features of such a flow under different regimes is important. From the point of view of energy efficiency, the laminar regime is important; on the other hand, when taking into account the efficiency of heat and mass transfer, the turbulent regime is important. Fluid viscosity is an important parameter that determines flow patterns.
The flow of a viscous model fluid in an annular channel with a given temperature field is considered. 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 a flat channel at different values of wall temperature is investigated. At small values of the viscosity change parameter, the velocity profile of the undisturbed flow is similar to the Poiseuille profile, and when it increases, the maximum value of the velocity profile tends towards the heated channel wall. This phenomenon is explained by the fact that the viscosity of the liquid near the hot wall is less than the viscosity of the liquid near the cold wall, and therefore the speed in the region of the heated wall will be greater. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of the previously obtained generalized Orr-Sommerfeld equation by the spectral method of expansion in Chebyshev polynomials of the first kind. Spectral pictures of the eigenvalues of the generalized equation are constructed. 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. 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. It has been established that at small values of the thermoviscosity parameter the spectrum is comparable to the spectrum for isothermal fluid flow, 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 a nontrivial solution. 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 is shown that at fixed values of the Reynolds number and wave number, the flow can become unstable as the thermoviscosity parameter increases.
Conclusions. It has been established that taking into account the dependence of viscosity on temperature quite strongly influences conclusions regarding hydrodynamic stability, which is certainly important when analyzing flow regimes in heat exchanger channels. At the same values of Reynolds numbers and wave numbers describing stable flow regimes, an increase in the thermoviscosity parameter can lead to the emergence of both stable and unstable regimes. Spectral characteristics of flow are an important part in the analysis of fluid flow regimes. It should be noted that in this case there is also a qualitative change in the structure of the eigenfunctions.
