Exact solutions to the oberbeck–boussinesq equations for convective stokes flows

The article proposes several classes of exact solutions to the Oberbeck–Boussinesq equations to describe convective flows of micropolar fluids. The possibility of using families of exact solutions for convective flows of classical incompressible fluids to micropolar incompressible fluids is discussed. It is shown that the three-dimensional Oberbeck–Boussinesq equation for describing steady and unsteady flows of micropolar fluids satisfies the class of Lin–Sidorov–Aristov exact solutions. The Lin–Sidorov–Aristov ansatz is characterized by a velocity field with a linear dependence on two spatial coordinates. These coordinates are called horizontal or longitudinal. The coefficients of the linear forms of the velocity field depend on the third coordinate (vertical or transverse) and time. The pressure field and the temperature field are described using quadratic forms. Generalizations of the Ostroumov–Birikh class are considered a special case of the Lin–Sidorov–Aristov family for describing unidirectional flows and homogeneous shear flows. An overdetermined system of Oberbeck–Boussinesq equations is investigated for describing non-homogeneous shear flows of non-trivial complex topology in 3D metric space. A compatibility condition is obtained in the Lin–Sidorov–Aristov class. Finally, a class of exact solutions with a vector velocity field that is nonlinear in part of the coordinates is presented in our analysis; such partially invariant solutions correspond to theoretical findings regarding symmetric/asymmetric properties of flow fields in solutions topology in a part of the existence appropriate for symmetry for the obtained invariant solutions.

The article presents a class of exact solutions of the Oberbeck-Boussinesq equations for media flows taking into account energy dissipation (transformation of mechanical energy into thermal energy). The construction of a family of exact solutions is based on the use of the Lin-Sidorov-Aristov class. The velocity field is described by linear forms relative to two spatial coordinates. These coordinates are called horizontal or longitudinal. The coefficients of linear forms depend on the third coordinate (vertical or transverse) and time. The pressure field and temperature field are described by quadratic forms with a functional dependence similar to the velocity vector. The article presents a system of nonlinear partial differential equations for determining unknown functions of hydrodynamic fields. Exact solutions are constructed to describe unidirectional flows of the Birikh-Ostroumov type with the dissipation of mechanical energy. The problems for convective flows of the Ekman type of a solid-state rotating fluid with the first Coriolis parameter are formulated. To describe inhomogeneous shear motions of a viscous incompressible fluid, an overdetermined system of nonlinear partial differential equations was investigated for the existence of solutions. It is shown that there is a nontrivial exact solution for flows with spatial acceleration.

This paper presents two exact explicit solutions for the three dimensional dual-phase lag (DLP) heat conduction equation, during the derivation of which the method of trial and error and the authors’ previous experiences are utilized. To the authors’ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.

The Fokker–Planck (FP) equation represents the drift and diffusive processes in kinetic models. It can also be regarded as a model for the collision integral of the Boltzmann-type equation to represent thermo-hydrodynamic processes in fluids. The lattice Boltzmann method (LBM) is a drastically simplified discretization of the Boltzmann equation for simulating complex fluid motions and beyond. We construct new two FP-based LBMs, one for recovering the Navier–Stokes equations for fluid dynamics and the other for simulating the energy equation, where, in each case, the effect of collisions is represented as relaxations of different central moments to their respective attractors. Such attractors are obtained by matching the changes in various discrete central moments due to collision with the continuous central moments prescribed by the FP model. As such, the resulting central moment attractors depend on the lower-order moments and the diffusion tensor parameters, and significantly differ from those based on the Maxwell distribution. The diffusion tensor parameters for evolving higher moments in simulating fluid motions at relatively low viscosities are chosen based on a renormalization principle. Moreover, since the number of collision invariants of the FP-based LBMs for fluid motions and energy transport are different, the forms of the respective attractors are quite distinct. The use of such central moment formulations in modeling the collision step offers significant improvements in numerical stability, especially for simulations of thermal convective flows under a wide range of variations in the transport coefficients of the fluid. We develop new FP central moment LBMs for thermo-hydrodynamics in both two and three dimensions, and demonstrate the ability of our approach to simulate various cases involving thermal convective buoyancy-driven flows especially at high Rayleigh numbers with good quantitative accuracy. Moreover, we show significant improvements in the numerical stability of our FP central moment LBMs when compared to other existing central moment LBMs using the Maxwell distribution in achieving high Peclet numbers for mixed convection flows involving shear effects.

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.

view Abstract Citations (7) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Large-Scale Flows and Solar Luminosity Variations Arendt, S. Abstract The influence of a variation of the large-scale flows present in the solar convection zone on solar luminosity is examined. The convection zone is modeled as a plane-parallel fluid layer through which heat is transported by small-scale convection as described by the mixing-length theory, and the large-scale flow is treated as a small perturbation superposed on the layer. The excess energy transported by the flow is calculated and is found to depend critically on the mixing length. The excess luminosity is found to be carried as small-scale convective heat flow rather than bulk enthalpy transport throughout all but the top few percent of the convection zone. The results are applied to the specific examples of supergranules and giant cells. Publication: The Astrophysical Journal Pub Date: April 1992 DOI: 10.1086/171217 Bibcode: 1992ApJ...389..421A Keywords: Convective Flow; Convective Heat Transfer; Solar Activity Effects; Solar Constant; Solar Interior; Stellar Luminosity; Boundary Conditions; Enthalpy; Mixing Length Flow Theory; Nimbus 7 Satellite; Solar Cycles; Solar Maximum Mission; Solar Temperature; Solar Physics; CONVECTION; SUN: ACTIVITY; SUN: GENERAL; SUN: INTERIOR full text sources ADS |

Multiparametric families of exact solutions are obtained for the steady-state two-dimensional Navier–Stokes equations in Cartesian, polar, and cylindrical coordinates. General exact solutions depending on arbitrary functions are derived for the equations of a nonstationary axisymmetric boundary layer. New noninvariant solutions of the heat conduction equation are presented. Various modifications of the method of generalized (and functional) separation of variables are used in seeking exact solutions.

Modulation of mantle plumes and heat flow at the core mantle boundary by plate-scale flow: results from laboratory experiments

In this paper, we study the zero dissipation limit of the initial boundary-value problem of the multi-dimensional Boussinesq equations with viscosity and heat conductivity. Such equations are used as models for the motion of multi-dimensional incompressible fluids in atmospheric and oceanographic turbulence. In particular, they describe the thermal convection of an incompressible flow, and constitute the relations between the velocity field, the pressure and the local temperature. Under the Navier slip boundary condition in the velocity field and the thermal isolation boundary condition for the temperature, we prove the existence of weak amplitude characteristic boundary layers. Then, by a standard energy method, we prove theL2convergence of the solutions when both the viscosity and the heat conductivity coefficients tend to 0.

The exact solution of the plane problem of the impact of a finite liquid strip on a rigid barrier is obtained in the linearized formulation. The velocity components, the pressure and other elements of the flow are determined by means of a velocity potential that satisfies a two-dimensional wave equation. The final expressions for them are given in terms of elementary functions that clearly reflect the wave nature of the motion. The exact solution has been thoroughly analyzed in numerous particular cases. It is shown directly that in the limit the solution of the wave problem tends to the solution of the analogous problem of the impact of an incompressible strip obtained in [1]. A logarithmic singularity of the velocity parallel to the barrier in the corner of the strip is identified. A one-dimensional model of the motion, which describes the behavior of the compressible liquid in a thin layer on impact and makes it possible to obtain a simple solution averaging the exact wave solution, is proposed. Inefficient series solutions are refined and certain numerical data on the impact characteristics for a semi-infinite compressible liquid strip, previously considered in [2–4] in connection with the study of the earthquake resistance of a dam retaining water in a semi-infinite basin, are improved. The solution obtained can be used to estimate the forces involved in the collision of solids and liquids. It would appear to be useful for developing correct and reliable numerical methods of solving the nonlinear problems of fluid impact on solids often examined in the literature [5].

A method was developed (and implemented as a software package) for the numerical simulation of the dynamics of a large number of particles in a viscous flow near a plane wall. The fluid motion is considered in the Stokes approximation with allowance for the hydrodynamic interaction of all the particles. The velocities and trajectories of the particles are computed, and fluid streamlines are constructed. In the case of even two particles falling onto the wall, it is found that a complex vortex flow is formed between them. This gives rise to the particles having a velocity component along the wall. The dynamics of a group of several dozen particles moving parallel and perpendicular to the wall are simulated.

This paper presents an exact analytical solution for unsteady conductive heat transfer in a cylindrical multilayer composite laminate. Here, it is supposed that fibers have been wound around the cylinder in each lamina. In order to find the exact solution, the Laplace transformation is applied on anisotropic heat conduction equation to convert the time scale of problem to frequency scale and the separation of variable method is used to solve the resulted partial differential equations. The effect of fibers arrangements of multilayer cylindrical laminates and thermal boundary conditions on unsteady conductive heat transfer of these orthotropic materials is studied based on the exact solution that is presented in the current investigation. The analytical results illustrated that the unsteady temperature distribution in any multilayer composite laminates is in a state between the temperature distribution in single layer laminates with fibers’ angle equal to 0 deg and 90 deg.

The flow of thin liquid layer on an inclined substrate is investigated numerically. The mathematical modelling is based on the Oberbeck-Boussinesq equations and the generalized conditions on the thermocapillary boundary simplified during the parametrical analysis. In the framework of the long-wave approximation the evolution equation which determines the thickness of the liquid layer in the case of moderate Reynolds numbers is derived. The results of numerical modelling of the liquid flow with evaporation at the interface are obtained.

The current paper presents a numerical technique in solving the 3D heat conduction equation. The Finite Volume method is used in the discretisation scheme. Gauss's theorem has also been employed for solving the integral parts of the general heat conduction equation in solving problems of steady and unsteady states. The proposed technique is applicable to unstructured (tetrahedral) elements for dealing with domains of complex geometries. The validation cases of the developed, FORTRAN based, heat conduction code in 1D, 2D and 3D representations have been reviewed with a grid independence check. Comparisons to the available exact solution and a commercial software solver are attached to the manuscript.

Relevance. Many branches of engineering require, to varying degrees, the calculation of pipeline systems, the analysis of fluid flow, and the investigation of heat transfer phenomena occurring during fluid motion near solid boundaries. Depending on the fluid viscosity and flow velocity, the flow regime may be laminar or turbulent, which significantly influences the temperature distribution along the pipeline. In oil and gas fields operating under harsh climatic conditions, electric heating systems are widely used for oil, gas, and water pipelines installed on the surface or for individual sections exposed at the surface in well clusters, process facilities, booster pumping stations, preliminary water and gas separation units, compressor stations, and oil collection points. The selection of the electric heating configuration and power is determined by thermal engineering calculations. However, existing engineering approaches are largely based on simplified empirical models and do not adequately account for the coupled interaction of hydrodynamic, thermal, and electrical processes, which leads to increased energy consumption and reduced reliability of fluid and gas transportation. This article presents a calculation of the velocity distribution in the cross-sectional and longitudinal directions of a pipe in order to determine the corresponding temperature distribution. Aim. To determine the fluid velocity distribution in the pipe cross-section and the temperature distribution along the pipe under laminar flow conditions based on the Navier–Stokes equations, the continuity equation, and the heat conduction equation. Methods. Analytical method is employed to solve the Navier–Stokes equations and the continuity equation. Result and conclusions. The authors have obtained the analytical expressions for the fluid velocity distribution in the pipe cross-section for both steady-state and transient laminar flow regimes. The paper demonstrates the effect of the flow regime on the heat transfer coefficient at the fluid–pipe wall interface. The authors determined the Nusselt eigenvalues and eigenfunctions, enabling the fluid temperature field to be represented as a series and the temperature distribution along the pipeline to be calculated. The results can be applied in thermal engineering calculations and in the design of electric heating systems for pipelines.