Influence of the Marangoni Effect on the Emergence of Fluid Rotation in a Thermogravitational Boundary Layer
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INFLUENCE OF THE MARANGONI EFFECT ON THE EMERGENCE OF FLUID ROTATION IN A THERMOGRAVITATIONAL BOUNDARY LAYER V. A. Batishchev∗
UDC 532.526
Abstract: Steady axisymmetric regimes of the fluid flow in a thermogravitational boundary layer near the free surface with a nonuniform temperature distribution on this boundary are calculated for equations of fluid motion in the Oberbeck–Boussinesq approximation, where the viscosity and thermal diffusivity are small. With due allowance for the thermocapillary effect, it is demonstrated that nonhomogeneous fluid flow regimes with rotation can arise in the boundary layer owing to a bifurcation in the case of local cooling of the free surface, whereas rotation outside this layer is absent. Keywords: free boundary, nonhomogeneous fluid, boundary layer, bifurcation, thermocapillary effect, rotation. DOI: 10.1134/S002189442003013X
In a homogeneous fluid with small diffusion coefficients, there may emerge thin boundary layers in the vicinity of the free surface under the action of shear stresses on this surface. These stresses arise, e.g., if there is a temperature gradient along the free boundary. As a result, a thermocapillary fluid flow is formed (Marangoni effect). Napolitano [1] was the first researcher who obtained self-similar solutions that describe thermocapillary flows of a homogeneous fluid. An important cycle of investigations of the Marangoni effect was performed by Pukhnachev and his team. For example, the properties of unsteady and steady Marangoni boundary layers were considered in [2, 3]. Nonlinear boundary layers may be formed in the vicinity of the free boundary if there are no shear stresses on this boundary. This phenomenon occurs in the absence of the Marangoni effect in a nonhomogeneous fluid whose flow is described by the equations of motion in the Oberbeck–Boussinesq approximation. Owing to the influence of the temperature field on the velocity field, a thermogravitational boundary layer is formed in the fluid near the free boundary. Depending on the values of the problem parameters, the Marangoni effect can produce a small or finite action on the nonhomogeneous fluid dynamics. The influence of the Marangoni effect on the thermogravitational fluid flow in a thin layer was investigated in [4]. The present study is aimed at considering the Marangoni effect on the thermogravitational flow in the boundary layer near the free surface of a nonhomogeneous fluid, which occupies a finite-thickness layer. Conditions at which a fluid flow with rotation can arise near the free boundary are determined.
South Federal University, Rostov-on-Don, 344090 Russia; ∗ [email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 61, No. 3, pp. 120–128, May–June, 2020. Original article submitted September 17, 2019; revision submitted November 29, 2019; accepted for publication December 23, 2019. ∗ Corresponding author. c 2020 by Pleiades Publishing, Ltd. 0021-8944/20/6103-0417
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1. MATHEMATICAL MODEL A steady flow of a nonhomogeneous fluid under axial symmetry conditions is considered. The
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