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

IF 0.8 Q2 MATHEMATICS
A. D. Nizamova, V. N. Kireev, S. F. Urmancheev
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Abstract

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.

Abstract Image

热粘性液体流动过程中环形间隙厚度对临界雷诺数的影响
摘要 在解决流体力学问题时,与不同截面通道中流体流动的瞬态有关的问题是一个重点。目前,在不能忽略粘度随温度变化的过程中,与热交换对流体流动稳定性的影响有关的研究变得尤为重要。本文研究了层流流体在环形通道中失去稳定性的一些特征,该层流流体的粘度与温度呈指数关系,通道壁上有给定的温度机制。为此,推导了广义的奥尔-索默费尔德方程,并最终将其写入流函数。利用基于切比雪夫多项式的频谱法对相应的边界值问题进行了数值研究。结果表明,考虑到温度对液体粘度的影响(这意味着液体粘度在通道横截面上的分布不均匀),临界雷诺数会降低,这与之前的研究结果是一致的。特别是,如前所述,对于狭窄的通道和较小的热粘滞参数,特征值谱与平坦通道中等温流动的谱相同。相对通道宽度的变化和热粘度参数的增加会导致广义 Orr-Sommerfeld 方程特征值谱结构的显著调整。本文研究的结果是,临界雷诺数与指数因子或热粘度参数(表征粘度随温度升高而变化的强度)的关系,以及与决定环形通道宽度与内圆柱面半径之比的参数的关系。结果表明,随着相对通道宽度参数的增加,临界雷诺数的值会发生非单调变化,其最小值取决于特定的液体。后一种情况可以作为技术过程建模时进行优化计算的理论依据。临界雷诺数与热粘滞度参数的关系在所有尺寸的环形水道中都接近于指数递减函数。
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来源期刊
CiteScore
1.50
自引率
42.90%
发文量
127
期刊介绍: Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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