Rayleigh — Benard problem for Polymer Solution
УДК 519.6:532.135
Abstract
There are three mathematical models describing the motion of aqueous solutions of polymers: the second grade fluid model (Rivlin — Eriksen), the hereditary model (Voitkunsky — Amfilokhiev — Pavlovsky), and its asymptotic simplification (Pavlovsky). This work considers the problem of fluid equilibrium stability in a horizontal fluid layer heated from below or from above. Also, equations of thermal gravitational convection for all three models are derived. Three types of boundary conditions are considered: two solid boundaries; the lower solid boundary and the upper free boundary; two free boundaries (the Rayleigh problem). For the case of heating from below, the principle of perturbation monotonicity is established that ensures the spectral problem eigenvalues to be of real type. This greatly simplifies the determination of the critical Rayleigh numbers. It turned out that these numbers coincide with the critical Rayleigh numbers in the classical Rayleigh — Benard problem. In the case of heating from above at large temperature gradients, the perturbation decrements become complex, but their real parts are negative. The conclusion that the relaxation properties of a second grade fluid and an aqueous solution of polymers do not lead to a change in the critical Rayleigh number may seem strange at first glance. According to our assumption, it is explained by the base state of the liquid being a state of rest.
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Гершуни Г.З., Жуховицкий Е.М. Конвективная устойчивость несжимаемой жидкости. М., 1972.
Straughan B. Energy stability in the Benard problem for a fluid of second grade // Journal of Applied Mathematics and Physics. 1983.Vol. 34.
Bouteraa M., Vare T., Nouar C., Becker S., Ouhajjou J. Rayleigh-Benard convection in non-Newtonian fluids: Experimental and theoretical investigations // Physics of Fluids. 2021. Vol. 33. Doi:10.1063/5.0070983
Rivlin R.S., Ericksen J.L. Stress-deformation relations for isotropic materials // Arch. Rational Mech. Anal. 1955. Vol. 4.
Войткунский Я.И., Амфилохиев В.Б., Павловский В.А. Уравнения движения жидкости с учетом ее релаксационных свойств // Труды ЛКИ. 1970. Вып. LXIX.
Пухначев В.В., Фроловская О.А. Модель Войткунско-го — Амфилохиева — Павловского движения водных растворов полимеров // Труды МИАН. 2018. Т. 300. Doi:10.1134/ S0371968518010144
Павловский В.А. К вопросу о теоретическом описании слабых водных растворов полимеров // Доклады АН СССР. 1971. Т. 200.
Pellew A., Southwell R.V. On maintained convective motion in a fluid heated from below // Proc. Roy. Soc. 1940. Vol. A176. № 966.
Сорокин В.С. Вариационный метод в теории конвекции // ПММ. 1953. Т. 17. № 1.
Зеньковская С.М., Симоненко И.Б. О влиянии вибрации высокой частоты на возникновение конвекции // Изв. АН СССР МЖГ. 1966. № 5.
Зеньковская С.М. Исследование конвекции в слое жидкости при наличии вибрационных сил // Изв. АН СССР МЖГ. 1968. № 1.
Зеньковская С.М. О влиянии вибрации на возникновение конвекции. Деп. рукопись. ВИНИТИ. 1978.
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