Layered Marangoni convection with the Navier slip condition

Abstract

A new exact solution to the problem of Marangoni layered convection is obtained. This solution describes a layered steady-state flow of a viscous incompressible fluid at varying gradients of temperature and pressure. The velocity components depend only on the transverse coordinate; the temperature and pressure fields are three-dimensional. The Marangoni effect is observed on the upper free surface of the fluid layer. On the lower solid surface of the fluid layer, three different cases of defining boundary conditions are considered: the no-slip condition, the perfect slip condition and the Navier slip condition. The obtained exact solution is determined by the interaction of three flows: a flow caused by pressure drop (the Poiseuille flow), a flow caused by heating/cooling and the effect of the gravity force (the thermogravitational flow), and a flow caused by heating/cooling and the fluid surface tension effect (the thermocapillary flow). The obtained exact solutions in the case of each of the three types of boundary conditions specified on the lower surface are analyzed in detail. It has been proved that, when certain ratios of the boundary value problem parameters are fulfilled, the velocity components may acquire stagnation points, this being indicative of the presence of counterflow areas in the fluid layer under consideration. In particular, the presence of up to two stagnation points in each of the two longitudinal velocity components may cause a stratification of the velocity field in more than two regions. The obtained exact solution of the Marangoni layered convection problem can describe flows in thin films through the variation of the geometric anisotropy factor.

Solving the Hydrodynamical System of Equations of Inhomogeneous Fluid Flows with Thermal Diffusion: A Review

“…The papers [10][11][12][13][14][15][16][17][18][19]78] lay the foundation for unidirectional convective flows. To date, several classes of exact solutions of the three-dimensional system of Oberbeck-Boussinesq equations describing the flow of a viscous incompressible fluid have been obtained [78][79][80][81][82][83][84]. The main idea in constructing classes of exact solutions of the Navier-Stokes equations is related to the modification of the velocity field, which depends linearly on the spatial acceleration.…”

Exact solutions to the Navier-Stokes equations describing stratified fluid flows

“…Introduction. When exact solutions to the Navier-Stokes equations are sought, the main attention is given to homogeneous incompressible fluids with a constant density [1][2][3][4][5][6][7][8][9][10][11][12]. These theoretical results describe a very wide class of hydrodynamic phenomena for various time and spatial scales and enable viscometers and chemical engineering devices to be designed [13][14][15][16][17][18].…”

“…Density stratification along the vertical coordinate affects the dynamics of large structures and the energy exchange between vortices of different magnitudes, and it generates the appearance of internal waves [32,[36][37][38][39]. Note that the presence of density gradients with respect to the horizontal (longitudinal) coordinates induces various convections [1,[4][5][6][7]. Undoubtedly, the density stratification is determined by a continuous time and coordinate function.…”

Exact solutions of the Navier–Stokes equations for describing an isobaric one-directional vertical vortex flow of a fluid

“…Течение Куэтта нашло свое применение в гидродинамике, протекающей на масштабах наномира и микрофлюидике [34][35][36][37][38][39][40][41]. Использование профиля Куэтта позволяет теоретически визуализировать длину скольжения при движении жидкости по гидрофобной поверхности в протяженной слое [34][35][36][37][38][39][40][41][42].…”