Oberbeck-Boussinesq Model Research Articles

On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating

Abstract This article surveys some results in the study of Iannelli [Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento, Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for t → ∞ t\to \infty of the time-dependent solutions is proved.

Thermal Cnvection of Two Immiscible Liquids in a 3D Channel with a Velocity Field of a Special Type

The three-dimensional stationary flow of two immiscible liquids in a layer bounded by solid parallel walls is investigated. The upper wall is thermally insulated, and the lower one has a temperature field quadratic in horizontal coordinates. Velocity fields in liquids have a special form: their horizontal components are linear in the coordinates of the same name. The resulting conjugate boundary value problem in the framework of the Oberbeck–Boussinesq model is inverse and is reduced to a system of ten integro-differential equations. For small Marangoni numbers (creeping current), the problem is solved analytically. The nonlinear problem is solved by the tau method. It is shown that the solution of the nonlinear problem with a decrease in the Marangoni number is approximated by the solution of the creeping flow problem. The analysis of the influence of physical and geometric parameters, as well as the behavior of temperature on the substrate, on the structure of convection in layers is carried out.

On the Oberbeck–Boussinesq approximation for gases

The Oberbeck–Boussinesq (OB) approximation for a compressible fluid in Bénard’s problem geometry is studied. We present a new derivation of the classical and ’extended’ OB models for a fluid obeying the ideal gas state equation by means of an asymptotic limit for vanishing temperature difference. The ‘extended’ OB approximation takes account of the compressibility effects in the energy balance, due to the work of hydrostatic pressure inside the stratified fluid, while the isochoric character of the motion is unchanged. The scaling laws on the material and experimental parameters realizing the asymptotic systems are characterized. Moreover, the ranges of validity of the approximations are described in terms of temperature difference and thickness of the fluid layer and two other characteristic lengths associated to the material parameters. The problem of non-linear stability of the rest state in the extended OB model is traced back to the known stability analysis of the classical OB model by means of a rescaling of the temperature perturbation.

Boundary value and extremum problems for generalized Oberbeck–Boussinesq model

Boundary value and extremum problems for generalized Oberbeck–Boussinesq model

On the Conditions for Existence of Unidirectional Motions of Binary Mixtures in the Oberbeck—Boussinesq Model

Compatibility conditions are obtained for the nonstationary Oberbeck—Boussinesq equations describing the unidirectional motions of a liquid binary mixture in a horizontal strip. We examine the case of polynomial dependence of temperature on the longitudinal coordinate is considered, and the influence of the dependence of the kind on the remaining unknown functions from the original system. It is shown that a nonstationary unidirectional motion between two solid walls can be described by the Oberbeck—Boussinesq model only for the quadratic or linear law of temperature distribution along the horizontal coordinate. Some initial-boundary value problems are posed.

On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck–Boussinesq model

On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck–Boussinesq model

A Priori Estimates of the Solution of the Problem of the Unidirectional Thermogravitational Motion of a Viscous Liquid in the Plane Channel

We consider an initial boundary-value problem describing the unidirectional motion of a liquid in the Oberbeck–Boussinesq model in a plane channel with rigid immovable walls on which the temperature distribution is given (or the upper wall is heat-insulated). For this problem, we obtain a priori estimates, find an exact stationary solution, and determine conditions under which the solution converges to its stationary regime.

A numerical study on the impact of thermal alterations in porous media during hot fluid injection process employing a modified Boussinesq model

The Oberbeck-Boussinesq (OB) approximation is widely employed as a simplifying assumption for density-dependent flow problems. It reduces the governing differential equations to simpler forms, which can be handled analytically or numerically. In this study, a modified OB model is formulated to account for the variation of rock permeability and porosity with temperature during the hot fluid injection process in an oil-saturated porous medium under the assumption of local thermal equilibrium (LTE). The mathematical model is solved numerically using a fully implicit control volume finite difference discretization with the successive over relaxation (SOR) method to handle the non-linearity. Subsequently, the numerical model is validated with the analytical solution of the simplified problem successfully. Through detailed sensitivity analyses, the simulation results reveal the hot fluid injection rate as the most important operational parameter to be optimized for a successful thermal flood. The numerical runs show that that for single-phase core-flood simulation, the effect of temperature on the rock absolute permeability and porosity can be neglected without introducing any significant errors in the estimated recovery and temperature profile.

Stationary Oberbeck–Boussinesq model of generalized Newtonian fluid governed by multivalued partial differential equations

A generalization of the Oberbeck–Boussinesq model consisting of a system of steady state multivalued partial differential equations for incompressible, generalized Newtonian of the p-power type, viscous flow coupled with the nonlinear heat equation is studied in a bounded domain. The existence of a weak solution is proved by combining the surjectivity method for operator inclusions and a fixed point technique.

Unidirectional flows of binary mixtures within the framework of the Oberbeck–Boussinesq model

Existence conditions are investigated for the unidirectional flows of binary mixtures describable by the equations of motion in the Oberbeck–Boussinesq approximation with account for the thermodiffusion effect. Possible solutions are classified, known and novel classes of exact solutions being distinguished. For the solutions obtained different formulations of boundary-value problems are proposed. Flows between two rigid walls with given heat distribution laws are described.

Ostroumov–Birikh solution of convection equations with nonlinear buoyancy force

The thermodiffusion convection equations with nonlinear buoyancy force are studied. Some particular solutions for describing of thermoconcentration flows are found. Two boundary value problems are solved for thermal convection case. The comparison with the linear Oberbeck–Boussinesq model is given.

Control Problems for Heat-Conducting Viscous Fluid Flow in Manufacturing Processes

This work is concerned with control and inverse problems for the Oberbeck-Boussinesq model of heat convection. Optimality system describing the first-order necessary optimality conditions is deduced. Numerical algorithm based on Newton's method is proposed.

Solutions of special type describing the three dimensional thermocapillary flows with an interface

Abstract The convective fluid flows with an interface are modeled using the classical Oberbeck–Boussinesq model of convection. The three dimensional solutions for the infinite domains with fixed heat-insulated boundaries and with the interface under action of a longitudinal temperature gradient are studied. Construction of the solutions for the flows of two immiscible fluids in a channel with a rectangular cross-section is carried out using a complete problem statement. The kinematic and dynamic conditions are prescribed at the interface. The additional condition of continuity of the tangential velocities, the conditions of continuity of temperature and of the thermal fluxes are assumed to be fulfilled on the interface. In the present paper the fluid flows are studied in the stationary case under conditions of gravity and microgravity. To investigate this problem numerically an iteration algorithm is introduced. This algorithm is based on a finite difference scheme (the alternating direction method) and it allows to find all the components of velocity for both phases and temperature distributions. The examples of flows which can be characterized as a combination of the translational and progressively rotational types of motion are presented.

Modeling of microconvection in a fluid between heat conducting solids

Unsteady convection in a fluid under weak gravity is modeled. Convection in a rectangular domain elongated in the direction of gravity and enclosed between two heat-conducting solids is investigated in the case of heat insulation of the ends of the rectangle and the periodic heat flow through the outer boundaries of the solids. In this case, the condition of zero total heat flux is satisfied. Convective fluid motions are described using two mathematical models: the classical Oberbeck-Boussinesq model and the microconvection model for an isothermally incompressible fluid. Results of the numerical studies confirm the quantitative and qualitative differences between the flow characteristics calculated using the two convection models. Fluid particle trajectories are presented. Effects due to various physical characteristics of the problem are studied.

Direct Numerical Simulation of Kinematics and Thermophoretic Deposition of Inhalable Particles in Turbulent Duct Flows

Three-dimensional, incompressible turbulent air-particle flows in a channel with a temperature gradient are simulated by direct numerical simulations (DNS). The calculations used the fractional projection method to directly solve the Navier-Stokes equations. For obtaining more accurate results, the Oberbeck-Boussinesq model was used for considering the convective heat transfer and applied two-way coupling between the particles and the air phase to accurately simulate flow field state. The particles motions including mutual collisions were calculated with the direct simulation Monte-Carlo method (DSMC). The particles agglomeration and deposition in the turbulent channel flow with a temperature gradient were simulated by the Dahneke model. The research focused on the effects of the Reynolds number, the temperature gradient and particle concentration which simultaneity affect particle kinematics, impacts, agglomerations, and deposition characteristics. The numerical results show that the thermophoresis domin...

Effect of pressure work and viscous dissipation in the analysis of the Rayleigh–Bénard problem

The classical Rayleigh–Bénard problem in an infinitely wide horizontal fluid layer with isothermal boundaries heated from below is revisited. The effects of pressure work and viscous dissipation are taken into account in the energy balance. A linear analysis is performed in order to obtain the conditions of marginal stability and the critical values of the wave number a and of the Rayleigh number Ra for the onset of convective rolls. Mechanical boundary conditions are considered such that the boundaries are both rigid, or both stress-free, or the upper stress-free and the lower rigid. It is shown that the critical value of Ra may be significantly affected by the contribution of pressure work, mainly through the functional dependence on the Gebhart number and on a thermodynamic Rayleigh number. While the pressure work term affects the critical conditions determined through the linear analysis, the viscous dissipation term plays no role in this analysis being a higher order effect. A nonlinear analysis is performed showing that the superadiabatic Rayleigh number replaces Ra in the functional dependence of the excess Nusselt number. Finally, a reasoning is proposed to show how the results obtained may be used as a test on the most appropriate formulation of the Oberbeck–Boussinesq model.

Mathematical and Numerical Modelling of Microconvection in the Long Rectangular Domains

The thermal gravitational convection of liquids under conditions of microgravity is studied in a 2D long rectangular domain elongated in the direction of the gravity force. The liquid is located between two solid regions of equal thickness. The solid parts are heat conducting. The mathematical modelling of the coupled problem is presented. Two mathematical models of convection are used to describe a motion of liquid: the classical Oberbeck-Boussinesq model and the microconvection model of isothermally incompressible liquid. The numerical experiments of convection are performed and demonstrate the qualitative and quantitative differences in the flow characteristics.

Convection under low gravity: Correctness of mathematical models

The alternative mathematical models of convective fluids flows (the microconvection model of isothermally incompressible fluid, the model of convection of weakly compressible fluid) and the classical Oberbeck-Boussinesq model with temperature dependent viscosity are applicable to investigation of many applied problems of convection: convection under low gravity, in small scales and at fast changes of the boundary thermal regimes. A characteristic property of the alternative mathematical models is the nonsolenoidality of the velocity fields.

Stability of steady nonisothermal flow in a vertical layer with permeable boundaries in the microconvection model

The problem of the stability of steady convective viscous incompressible fluid flow in a vertical layer with boundaries at different temperatures is considered in the presence of transverse blowing through the layer. The complete spectral problem is solved for a silicon melt. The neutral curve is constructed and the critical Grashof number is found. The numerical calculations show that the presence of transverse blowing significantly affects the flow stability. As compared with the Oberbeck-Boussinesq model, in the microconvection model the instability develops at lower wavenumbers.

Exact solutions of linearized equations of convection of a weakly compressible fluid

A mathematical model of fluid convection under microgravity conditions is considered. The equation of state is used in a form that allows considering the fluid as a weakly compressible medium. Based on the previously proposed mathematical model of convection of a weakly compressible fluid, unsteady convective motion in a vertical band, with a heat flux periodic in time set on the solid boundaries of this band, is considered. This model of convection allows one to study the problem with the boundary thermal model oscillating in an antiphase rather than in-phase mode, while the latter was required for the model of microconvection of an isothermally incompressible fluid. Exact solutions for velocity components and temperature are derived, and the trajectories of fluid particles are constructed. For comparison, the trajectories predicted by the classical Oberbeck–Boussinesq model of convection and by the microconvection model are presented.