Oberbeck-Boussinesq System Research Articles

On the existence of solutions to generalized Navier–Stokes–Fourier system with dissipative heating

AbstractWe consider a flow of non-Newtonian incompressible heat conducting fluids with dissipative heating. Such system can be obtained by scaling the classical Navier–Stokes–Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck–Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck–Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.

Rigorous Derivation of the Oberbeck–Boussinesq Approximation Revealing Unexpected Term

We consider a general compressible viscous and heat conducting fluid confined between two parallel plates and heated from the bottom. The time evolution of the fluid is described by the Navier–Stokes–Fourier system considered in the regime of low Mach and Froude numbers suitably interrelated. Surprisingly and differently to the case of Neumann boundary conditions for the temperature, the asymptotic limit is identified as the Oberbeck–Boussinesq system supplemented with non-local boundary conditions for the temperature deviation.

The Oberbeck–Boussinesq system with non-local boundary conditions

We consider the Oberbeck–Boussinesq system with non-local boundary conditions arising as a singular limit of the full Navier–Stokes–Fourier system in the regime of low Mach and low Froude numbers. The existence of strong solutions is shown on a maximal time interval [ 0 , T m a x ) [0, T_{\mathrm {max}}) . Moreover, T m a x = ∞ T_{\mathrm {max}} = \infty in the two-dimensional setting.

Distributed Optimal Control of the 2D Cahn–Hilliard–Oberbeck–Boussinesq System for Nonisothermal Viscous Two-Phase Flows

We analyze a distributed optimal control problem where the state equation is governed by the coupling of the two-dimensional Cahn–Hilliard and Oberbeck–Boussinesq systems modelling incompressible viscous two-phase flows with convective heat transfer. Pointwise constraints are imposed on the controls that act as external sources in the fluid and convection–diffusion equations. The objective functional is of tracking-type that consists of a weighted energy of the difference between the state and a desired target. We establish the existence of optimal controls, the differentiability of the control-to-state operator, and the necessary and sufficient optimality conditions. For initial and target data with finite energy norms, limited space–time regularity of the adjoint states arises due to convection and surface tension.

A Multiscale Problem for Viscous Heat-Conducting Fluids in Fast Rotation

In the present paper, we study the combined incompressible and fast rotation limits for the full Navier-Stokes-Fourier system with Coriolis, centrifugal and gravitational forces, in the regime of small Mach, Froude and Rossby numbers and for general ill-prepared initial data. We consider both the isotropic scaling (where all the numbers have the same order of magnitude) and the multi-scale case (where some effect is predominant with respect to the others). In the case when the Mach number is of higher order than the Rossby number, we prove that the limit dynamics is described by an incompressible Oberbeck-Boussinesq system, where the velocity field is horizontal (according to the Taylor-Proudman theorem), but vertical effects on the temperature equation are not negligible. Instead, when the Mach and Rossby numbers have the same order of magnitude, and in absence of the centrifugal force, we show convergence to a quasi-geostrophic equation for a stream function of the limit velocity field, coupled with a transport-diffusion equation for a new unknown, which links the target density and temperature profiles. The proof of the convergence is based on a compensated compactness argument. The key point is to identify some compactness properties hidden in the system of acoustic-Poincar\'e waves. Compared to previous results, our method enables first of all to treat the whole range of parameters in the multi-scale problem, and also to consider a low Froude number regime with the somehow critical choice $Fr=\sqrt{Ma}$, where $Ma$ is the Mach number. This allows us to capture some (low) stratification effects in the limit.

On Marangoni shear convective flows of inhomogeneous viscous incompressible fluids in view of the Soret effect

A new exact solution is obtained for the Oberbeck-Boussinesq equations describing the steady-state layered (shear) Marangoni convection of a binary viscous incompressible fluid with the Soret effect. When layered (shear) flows are considered, the Oberbeck-Boussinesq system is overdetermined. For it to be solvable, a class of exact solutions is constructed, which allows one to satisfy identically the “superfluous” equation (the incompressibility equation). The found exact solution allows the Oberbeck-Boussinesq system of equations to be reduced to a system of ordinary differential equations by the generalized method of separation of variables. The resulting system of ordinary differential equations has an analytical solution, which is polynomial. The polynomial velocity field describes counterflows in the case of a convective fluid flow. It is demonstrated that the components of the velocity vector can have one stagnant (zero) point inside the region under study. In this case, the corresponding component of the velocity vector can be stratified into two zones, in which the fluid flows in opposite directions. The exact solution describing the velocity field for the Marangoni convection of a binary fluid has non-zero helicity, the flow itself being almost everywhere vortex.

A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system

A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system

A Lorenz Model for Almost Compressible Fluids

Starting from the almost compressible 2-D Oberbeck–Boussi nesq system, characterized by an extra buoyancy term which is pressure-dependent, we write the corresponding Lorenz system and observe a larger instability with respect to the classical model in which the buoyancy is proportional to the temperature deviation only.

Existence and nonlinear stability of convective solutions for almost compressible fluids in Bénard problem

We study the nonlinear almost compressible 2D Oberbeck-Boussinesq system, characterized by an extra buoyancy term where the density depends on the pressure, and a corresponding dimensionless parameter β, proportional to the (positive) compressibility factor β0. The local in time existence of the perturbation to the conductive solution is proved for any “size” of the initial data. However, unlike the classical problem where β0 = 0, a smallness condition on the initial data is needed for global in time existence, along with smallness of the Rayleigh number. Removing this condition appears quite challenging, and we leave it as an open question.We study the nonlinear almost compressible 2D Oberbeck-Boussinesq system, characterized by an extra buoyancy term where the density depends on the pressure, and a corresponding dimensionless parameter β, proportional to the (positive) compressibility factor β0. The local in time existence of the perturbation to the conductive solution is proved for any “size” of the initial data. However, unlike the classical problem where β0 = 0, a smallness condition on the initial data is needed for global in time existence, along with smallness of the Rayleigh number. Removing this condition appears quite challenging, and we leave it as an open question.

The Bénard Problem for Slightly Compressible Materials: Existence and Linear Instability

We study a generalization of the Oberbeck–Boussinesq system, which consists in a buoyancy term where the density depends also on the pressure. A new pressure equation is introduced, which is deduced from the divergence-free condition on the velocity; such an equation cannot be decoupled from the system and is studied under Robin’s boundary conditions. Then, the existence of regular periodic solutions is proved for the full system. In Benard’s problem, the two-dimensional linear instability of the solution depends on a dimensionless parameter that is proportional to the compressibility factor: the related critical Rayleigh number decreases as it increases.

Convective layered flows of a vertically whirling viscous incompressible fluid. Velocity field investigation

Обсуждается разрешимость переопределенной системы уравнений тепловой конвекции в приближении Буссинеска. Система уравнений Обербека-Буссинеска, дополненная уравнением несжимаемости, является переопределенной. Количество уравнений превосходит количество неизвестных функций, поскольку изучаются неоднородные слоистые потоки вязкой несжимаемой жидкости (одна из компонент вектора скорости тождественно равна нулю). Проведено исследование разрешимости нелинейной системы уравнений Обербека-Буссинеска. Исследование разрешимости переопределенной системы нелинейных уравнений в частных производных Обербека-Буссинеска осуществлялось при помощи построения нескольких частных точных решений. Приведен новый класс точных решений для описания трехмерных нелинейных слоистых течений вертикальной завихренной вязкой несжимаемой жидкости. Вертикальная компонента завихренности в невращающейся жидкости генерируется неоднородным полем скоростей на нижней границе бесконечного горизонтального слоя жидкости. Конвекция в вязкой несжимаемой жидкости индуцируется линейными источниками тепла. Основное внимание уделено исследованию свойств поля скоростей течения. Исследована зависимость структуры этого поля от величины вертикальной закрутки. Показано, что одна из компонент вектора скорости при ненулевой вертикальной закрутке допускает расслоение на пять зон по толщине рассматриваемого слоя (четыре застойные точки). Анализ поля скоростей показал, что кинетическая энергия жидкости может дважды принимать нулевой значение по толщине слоя.

Rayleigh–Bènard convection in the generalized Oberbeck–Boussinesq system

The original Oberbeck–Boussinesq (OB) equations which are the coupled two dimensional Navier–Stokes(NS) and heat conduction equations have been investigated by E.N. Lorenz half a century ago with Fourier series and opened the way to the paradigm of chaos. In our former study—Chaos, Solitons and Fractals 78, 249 (2015)—we presented fully analytic solutions for the velocity, pressure and temperature fields with the aim of the self-similar Ansatz and gave a possible explanation of the Rayleigh–Bènard convection cells. Now we generalize the Oberbeck–Boussinesq hydrodynamical system, going beyond the first order Boussinesq approximation and consider a non-linear temperature coupling. We investigate more general, power law dependent fluid viscosity or heat conduction material equations as well. Our analytic results obtained via the self-similar Ansatz may attract the interest of various fields like meteorology, oceanography or climate studies.

Exact Solutions for Stationary and Unsteady Layered Convection of a Viscous Incompressible Fluid with the Specified Velocities at the Bottom

The layered convective flow of a viscous incompressible fluid is considered with the specified velocities at the bottom of an infinite layer. A new exact stationary and nonstationary solution of the Oberbeck-Boussinesq system is presented. The account of fluid velocity at the bottom is characterized by the presence of two stagnant points, this being indicative of the nonmonotonic kinetic energy profile with two local extrema.

Asymptotic Laws of Thermovibrational Convecton in a Horizontal Fluid Layer

Theoretical study of convective instability is applied to a horizontal layer of incompressible single-component fluid subjected to the uniform steady gravity, longitudinal vibrations of arbitrary frequency and initial temperature difference. The mathematical model of thermovibrational convection has the form of initial boundary value problem for the Oberbeck-Boussinesq system of equations. The problems are solved using different simulation strategies, like the method of averaging, method of multiple scales, Galerkin approach, Wentzel-Kramers-Brillouin method and Floquet technique. The numerical analysis has shown that the effect of vibrations on the stability threshold is complex: vibrations can either stabilize or destabilize the basic state depending on values of the parameters. The influence of the Prandtl number on the instability thresholds is investigated. The asymptotic behaviour of critical values of the parameters is studied in two limiting cases: (i) small amplitude and (ii) low frequency of vibration. In case (i), the instability is due to the influence of thermovibrational mechanism on the classical Rayleigh-Benard convective instability. In case (ii), the nature of the instability is related to the instability of oscillating counter-streams with a cubic profile.

Initial–boundary layer associated with the nonlinear Darcy–Brinkman–Oberbeck–Boussinesq system

In this paper, we study the vanishing Darcy number limit of the nonlinear Darcy–Brinkman–Oberbeck–Boussinesq system (DBOB). This singular perturbation problem involves singular structures both in time and in space giving rise to initial layers, boundary layers and initial–boundary layers. We construct an approximate solution to the DBOB system by the method of multiple scale expansions. The convergence with optimal convergence rates in certain Sobolev norms is established rigorously via the energy method.

On Chorin’s Method for Stationary Solutions of the Oberbeck–Boussinesq Equation

Stability of stationary solutions of the Oberbeck–Boussinesq system (OB) and the corresponding artificial compressible system is considered. The latter system is obtained by adding the time derivative of the pressure with small parameter \({\varepsilon > 0}\) to the continuity equation of (OB), which was proposed by A. Chorin to find stationary solutions of (OB) numerically. Both systems have the same sets of stationary solutions and the system (OB) is obtained from the artificial compressible one as the limit \({\varepsilon \to 0}\) which is a singular limit. It is proved that if a stationary solution of the artificial compressible system is stable for sufficiently small \({\varepsilon > 0}\), then it is also stable as a solution of (OB). The converse is proved provided that the velocity field of the stationary solution satisfies some smallness condition.

A theoretical study of the first transition for the non-linear Stokes problem in a horizontal annulus

For any aspect ratio Ro/Ri of the cylinder radii, the non-linear stability of the steady 2D-solutions of the non-linear Stokes system, which is an approximation of the Oberbeck–Boussinesq system, is theoretically studied. The sufficient condition for the stability shows a critical Ra which is a function of the aspect ratio. It is the same of the associated homogeneous linear problem and it can be found by looking for the largest eigenvalue of a suitable symmetric operator. The critical Ra so defined proves to be uniformly bounded from below in the space of dimensionless parameters, while it is non-uniformly bounded from above for the aspect ratio going to infinity. A scheme to evaluate it as a function of the aspect ratio is given. The results do not depend on the Prandtl number Pr.

Theoretical Results on Steady Convective Flows between Horizontal Coaxial Cylinders

For arbitrary Rayleigh number, $\mathrm{Ra}$, Prandtl number, and any ratio of the cylindrical radii, existence of steady solutions of the two-dimensional Oberbeck–Boussinesq system is proved in Theorem 3.5. We show nonlinear stability for large values of the aspect ratio and $\mathrm{Ra}<\mathrm{Ra}_L$, for some number $\mathrm{Ra}_L$ which is bounded from below (cf. Theorem 4.4). We prove that for large aspect ratio, $\mathrm{Ra}_L$ approaches the critical Rayleigh number for the stability of the rest state solution of a suitable linear problem.

SINGULAR LIMIT OF THE EQUATIONS OF MAGNETOHYDRODYNAMICS IN THE PRESENCE OF STRONG STRATIFICATION

We consider a low Mach, Péclet, Froude and Alfvén number limit in the complete Navier–Stokes–Fourier system coupled with Maxwell's equations for gases with large specific heat at constant volume. The target system is shown to be the anelastic Oberbeck–Boussinesq system coupled with Maxwell's equations. The proof allows an intrinsic view into the process of separation of fast oscillating acoustic waves, governed by a Lighthill-type equation, from the equations describing the slow fluid flows.

Rigorous derivation of the anelastic approximation to the Oberbeck–Boussinesq equations

It is shown that the anelastic Oberbeck-Boussinesq system is a small Mach, small Peclet and small Froude number limit of the complete Navier-Stokes-Fourier system for gases with large specific heat at constant volume. This result is obtained on an arbitrary large time interval. The proof allows an intrinsic view into the process of separation of fast oscillating acoustic waves, governed by a Lighthill-type equation, from the equations describing the slow fluid flows. This is a very useful information for numerical analysts. keywords: anelastic approximation