The Effect of Temperature Dependence of the Viscosity on Stationary Convective Flows in Hele–shaw Cell

Abstract

ȼɜɟɞɟɧɢɟ. ȼ ɪɚɛɨɬɟ ɩɪɨɜɟɞɟɧɨ ɱɢɫɥɟɧɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɜɥɢɹɧɢɹ ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ ɜɹɡɤɨɫɬɢ ɢ ɬɟɩɥɨɜɵɯ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɧɚ ɫɬɚɰɢɨɧɚɪɧɵɟ ɤɨɧɜɟɤɬɢɜɧɵɟ ɞɜɢɠɟɧɢɹ ɜ ɜɟɪɬɢɤɚɥɶɧɨ ɨɪɢɟɧɬɢ-ɪɨɜɚɧɧɨɣ ɹɱɟɣɤɟ ɏɟɥɟ-ɒɨɭ ɩɪɢ ɪɚɜɧɨɦɟɪɧɨɦ ɩɨɞɨɝɪɟɜɟ ɫɧɢɡɭ. ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɭɫɥɨɜɢɹɯ ɩɪɢ ɧɟɛɨɥɶɲɢɯ ɩɟɪɟɩɚɞɚɯ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɠɢɞɤɨɫɬɢ ɢɦɟɟɬ ɦɟɫɬɨ ɦɟɯɚɧɢɱɟɫɤɨɟ ɪɚɜɧɨɜɟɫɢɟ, ɚ ɩɪɢ ɦɚ-ɥɵɯ ɧɚɞɤɪɢɬɢɱɧɨɫɬɹɯ ɜɨɡɧɢɤɚɸɬ ɫɬɚɰɢɨɧɚɪɧɵɟ ɨɞɧɨ-ɢɥɢ ɞɜɭɯɜɢɯɪɟɜɨɟ ɤɨɧɜɟɤɬɢɜɧɵɟ ɞɜɢɠɟɧɢɹ ɠɢɞɤɨɫɬɢ, ɨɛɥɚɫɬɶ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɤɨɬɨɪɵɯ ɢ ɮɨɪɦɚ ɢɫɫɥɟɞɨɜɚɥɢɫɶ ɜ ɞɚɧɧɨɣ ɪɚɛɨɬɟ. ȼɩɟɪɜɵɟ ɨɛ-ɫɬɨɹɬɟɥɶɧɨɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɟ ɧɚɛɥɸɞɟɧɢɟ ɪɚɡɥɢɱɧɵɯ ɥɚɦɢɧɚɪɧɵɯ ɤɨɧɜɟɤɬɢɜɧɵɯ ɪɟɠɢɦɨɜ ɜ ɩɨɞɨɝɪɟɜɚɟɦɨɣ ɫɧɢɡɭ ɜɟɪɬɢɤɚɥɶɧɨɣ ɹɱɟɣɤɟ ɏɟɥɟ-ɒɨɭ ɩɪɨɜɨɞɢɥɨɫɶ ɜ ɪɚɛɨɬɟ [1]. Ʌɢɧɟɣɧɵɣ ɚɧɚ-ɥɢɡ ɭɫɬɨɣɱɢɜɨɫɬɢ ɬɟɱɟɧɢɣ ɜ ɫɥɭɱɚɟ ɢɞɟɚɥɶɧɨ ɬɟɩɥɨɩɪɨɜɨɞɧɵɯ ɲɢɪɨɤɢɯ ɝɪɚɧɟɣ ɛɵɥ ɜɵɩɨɥɧɟɧ ɜ [2]. Ɉɞɧɚɤɨ, ɢɡ-ɡɚ ɫɩɟɰɢɮɢɱɟɫɤɢɯ ɬɟɩɥɨɜɵɯ ɭɫɥɨɜɢɣ ɧɚ ɲɢɪɨɤɢɯ ɝɪɚɧɹɯ ɢ ɜɵɛɪɚɧɧɨɝɨ ɫɨɨɬɧɨɲɟ-ɧɢɹ ɫɬɨɪɨɧ 2:10:20, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɪɟɝɭɥɹɪɧɵɯ ɪɟɠɢɦɨɜ ɨɤɚɡɚɥɚɫɶ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɨɣ. ɉɨ-ɫɥɟ ɩɪɨɯɨɠɞɟɧɢɹ ɩɨɪɨɝɚ ɭɫɬɨɣɱɢɜɨɫɬɢ ɪɚɜɧɨɜɟɫɢɹ ɜ ɫɢɫɬɟɦɟ ɫɧɚɱɚɥɚ ɪɨɠɞɚɥɫɹ ɨɞɧɨɜɢɯɪɟɜɨɣ ɫɬɚɰɢɨɧɚɪɧɵɣ ɪɟɠɢɦ. ɉɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ ɧɚɞɤɪɢɬɢɱɧɨɫɬɢ ɨɫɶ ɫɢɦɦɟɬɪɢɢ ɨɫɧɨɜɧɨɝɨ ɜɢɯɪɹ ɧɚ-ɱɢɧɚɥɚ ɨɬɤɥɨɧɹɬɶɫɹ, ɚ ɫɤɨɪɨɫɬɶ ɜɪɚɳɟɧɢɹ ɪɚɫɬɢ. ȼ ɨɩɪɟɞɟɥɟɧɧɵɣ ɦɨɦɟɧɬ ɫɢɫɬɟɦɚ ɩɪɨɯɨɞɢɥɚ ɬɨɱ-ɤɭ ɛɢɮɭɪɤɚɰɢɢ, ɜ ɤɨɬɨɪɨɣ ɪɨɠɞɚɥɫɹ ɚɜɬɨɤɨɥɟɛɚɬɟɥɶɧɵɣ ɱɟɬɵɪɟɯɜɢɯɪɟɜɨɣ ɪɟɠɢɦ ɫ ɩɟɪɟɡɚɦɵɤɚ-ɧɢɟɦ ɭɝɥɨɜɵɯ ɜɢɯɪɟɣ. ɉɨɡɞɧɟɟ ɩɨɫɥɟɞɨɜɚɥɚ ɫɟɪɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɪɚɛɨɬ [3][4][5], ɜ ɤɨɬɨɪɵɯ ɢɫ-ɩɨɥɶɡɨɜɚɥɚɫɶ ɩɨɥɨɫɬɶ ɫ ɫɨɨɬɧɨɲɟɧɢɟɦ ɫɬɨɪɨɧ 2:20:40 ɢ ɛɵɥɨ ɨɛɧɚɪɭɠɟɧɨ ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɟɟ ɪɚɡɧɨɨɛɪɚɡɢɟ ɤɚɤ ɩɟɪɟɯɨɞɧɵɯ, ɬɚɤ ɢ ɭɫɬɨɣɱɢɜɵɯ ɤɨɧɜɟɤɬɢɜɧɵɯ ɬɟɱɟɧɢɣ. ȼ ɱɚɫɬɧɨɫɬɢ ɛɵɥ ɨɛɧɚ-ɪɭɠɟɧ ɢ ɨɩɢɫɚɧ ɧɨɜɵɣ, ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɩɭɥɶɫɚɰɢɨɧɧɵɣ ɪɟɠɢɦ, ɤɨɥɟɛɚɬɟɥɶɧɨɟ ɩɨɜɟɞɟɧɢɟ ɤɨɬɨɪɨ-ɝɨ ɩɪɨɢɫɬɟɤɚɟɬ ɢɡ ɧɟɭɫɬɨɣɱɢɜɨɫɬɢ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɞɜɭɯɜɢɯɪɟɜɨɝɨ ɬɟɱɟɧɢɹ, ɤɨɝɞɚ ɫɩɨɧɬɚɧɧɨ ɨɞɢɧ ɜɢɯɪɶ ɧɚɱɢɧɚɟɬ ɞɨɦɢɧɢɪɨɜɚɬɶ ɧɚɞ ɞɪɭɝɢɦ. ȼ ɤɚɤɨɣ-ɬɨ ɦɨɦɟɧɬ ɛɨɥɶɲɨɣ ɜɢɯɪɶ ɰɟɥɢɤɨɦ ɩɨɝɥɨɳɚɟɬ ɦɚɥɟɧɶɤɢɣ, ɢ ɬɟɱɟɧɢɟ ɧɚ ɜɪɟɦɹ ɫɬɚɧɨɜɢɬɫɹ ɨɞɧɨɜɢɯɪɟɜɵɦ. Ɂɚɬɟɦ ɧɚ ɮɨɧɟ ɨɞɧɨɜɢɯɪɟɜɨɝɨ ɬɟɱɟɧɢɹ ɜ ɨɞɧɨɦ ɢɡ ɧɢɠɧɢɯ ɭɝɥɨɜ ɩɨɥɨɫɬɢ ɪɨɠɞɚɟɬɫɹ ɧɟɛɨɥɶɲɨɣ ɜɢɯɪɶ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɡɚɤɪɭɬɤɢ. ɗɬɨɬ ɜɢɯɪɶ ɫɨ ɜɪɟɦɟɧɟɦ ɪɚɫɬɟɬ, ɞɨɫɬɢɝɚɟɬ ɜ ɪɚɡɦɟɪɚɯ ɩɪɢɦɟɪɧɨ 1/3 ɜɵɫɨɬɵ ɩɨɥɨɫɬɢ, ɢ, ɜɡɚɢɦɨɞɟɣɫɬɜɭɹ ɫ ɨɫɧɨɜɧɵɦ ɬɟɱɟɧɢɟɦ, ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɨɩɹɬɶ ɩɨɝɥɨɳɚɟɬɫɹ ɢɦ. ɑɟɪɟɡ ɧɟɤɨɬɨɪɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ ɷɬɨɬ ɩɪɨɰɟɫɫ ɩɨɜɬɨɪɹɟɬɫɹ. ȼ ɢɬɨɝɟ ɧɚɛɥɸɞɚɸɬɫɹ ɩɭɥɶɫɚɰɢɢ ɧɢɠɧɟɝɨ ɜɢɯɪɹ ɧɚ ɮɨɧɟ ɨɫɧɨɜɧɨɝɨ ɨɞɧɨɜɢɯɪɟɜɨɝɨ ɬɟɱɟɧɢɹ.ɗɤɫɩɟɪɢɦɟɧɬɵ, ɩɪɨɜɟɞɟɧɧɵɟ ɧɚ ɬɪɚɧɫɮɨɪɦɚɬɨɪɧɨɦ ɦɚɫɥɟ, ɩɪɢɜɟɥɢ ɤ ɧɟɬɪɢɜɢɚɥɶɧɨɦɭ ɪɟ-ɡɭɥɶɬɚɬɭ. Ɉɤɚɡɚɥɨɫɶ, ɱɬɨ ɚɜɬɨɤɨɥɟɛɚɬɟɥɶɧɨɟ ɱɟɬɵɪɟɯɜɢɯɪɟɜɨɟ ɬɟɱɟɧɢɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɪɚɡɧɵɦɢ ɩɨ ɜɟɥɢɱɢɧɟ ɜɟɪɯɧɢɦɢ ɢ ɧɢɠɧɢɦɢ ɜɢɯɪɹɦɢ, ɱɬɨ ɛɵɥɨ ɨɛɴɹɫɧɟɧɨ ɜ [5] ɧɚɥɢɱɢɟɦ ɭ ɞɚɧɧɨɣ ɠɢɞɤɨ-ɫɬɢ ɫɢɥɶɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɜɹɡɤɨɫɬɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ. ȼ ɬɨɠɟ ɜɪɟɦɹ ɜ ɨɬɦɟɱɟɧɧɵɯ ɪɚɛɨɬɚɯ ɧɟ ɨɛɪɚ-Ɇɚɬɟɦɚɬɢɤɚ

Thermal convection in a HeleShaw cell with the dependence of thermal diffusivity on temperature

“…In [2,3] the temperature dependence of viscosity was taken into account in order to explain the up-down symmetry violation of steady and oscillatory convective flows in a vertical Hele-Shaw cell heated from the bottom. Thus in the presence of strong environmental conditions those dependencies can not be trivially ignored.…”

“…Setting the exponential decay rate λ to zero, we shall be interested in neutral disturbances. If dependence of thermal diffusivity on temperature is absent, the temperature distribution along vertical is linear and the equations system for amplitudes ξ and θ of small perturbations can be represented in the following form [3]:…”

“…In a limiting case of ε = 0 the analysis of (3.8), (3.9) leads to important result that the three most dangerous mode for the cavity with the aspect ratio 2:20:40 are (2, 1), (3,1) and (1,1).…”