Two-dimensional Boussinesq Equations Research Articles

Modeling of Filtration and Heat and Mass Transfer Processes in Groundwater

Purpose. The method of mathematical modeling is an important tool for solving complex problems involving the analysis of groundwater dynamics and heat and mass transfer processes in them when studying their contamination from various anthropogenic sources in the event of accidental spills of chemically hazardous substances, etc. The main purpose of the article is to develop a set of mathematical models for calculating the process of filtration of non-pressure groundwater, mass transfer of impurities and the process of heat transfer in groundwater. Methodology. The two-dimensional Boussinesq equation of filtration was used to predict the dynamics of groundwater. The two-dimensional equation of convective-diffusive transport of impurities was used to model the processes of mass transfer in groundwater. The process of freezing of individual sections of the groundwater flow is modeled using the Laplace equation for the velocity potential (calculation of the flow velocity field in a time-varying geometry) and the two-dimensional equation of heat transfer in groundwater. Finite difference schemes were used to solve the modeling equations of groundwater dynamics and heat and mass transfer. Findings. A set of mathematical models has been developed to calculate the process of filtration of non-pressure groundwater and its chemical contamination. The experiment has confirmed the adequacy of the constructed numerical model of filtration of a non-pressure groundwater flow. An effective mathematical model was developed that allows determining the temperature fields in groundwater during the operation of a well used to freeze certain sections of the flow. The results of computer modeling indicate the effectiveness of the developed mathematical models. Originality. Effective mathematical models for predicting the level of chemical contamination of groundwater, its dynamics and thermal regime are proposed. The constructed mathematical models make it possible to determine the dynamics of changes in the temperature regime of groundwater during the operation of wells through which refrigerant is supplied to freeze individual areas. A computer program has been developed that allows for a comprehensive assessment of groundwater conditions. Practical value. A set of computer programs has been developed to conduct a computational experiment to study the processes of filtration, chemical contamination of groundwater and heat transfer processes in them. This set of programs can be used for the scientific substantiation of engineering solutions aimed at protecting groundwater.

A Banach spaces-based fully mixed virtual element method for the stationary two-dimensional Boussinesq equations

In this paper we extend recent results obtained for the Navier–Stokes equations to propose and analyze a new fully mixed virtual element method (mixed-VEM) for the stationary two-dimensional Boussinesq equations appearing in non-isothermal flow phenomena. The model consists of a Navier–Stokes type system, modeling the velocity and the pressure of the fluid, coupled to an advection-diffusion equation for the temperature. The variational formulation is based on the introduction of the additional unknowns given by a modified pseudostress tensor, which depends on the pressure, and the diffusive and convective terms of the fluid, and the pseudoheat vector, which involves the temperature, its gradient, and the velocity. As a consequence of the former, the pressure is eliminated from the system, but computed afterwards via a post-processing formula. In turn, for the Galerkin approximation we follow the approach employed in a previous work introducing for the first time an Lp spaces-based mixed-VEM for the Navier–Stokes equations, and couple it with a similar mixed-VEM for the convection–diffusion equation modeling the temperature. The solvability analysis of the resulting coupled discrete scheme is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach-Nečas-Babuška theorem, both in subspaces of Banach spaces. The first Strang lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of the pseudostress tensor, the pseudoheat vector, and the post-processed pressure. Finally, several numerical results, illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory, are reported.

SOLITON SOLUTIONS FOR THE TWO-DIMENSIONAL LOCAL FRACTIONAL BOUSSINESQ EQUATION

In this work we study the two-dimensional local fractional Boussinesq equation. Based on the basic definitions and properties of the local fractional derivatives and bilinear form, we studied the soliton solutions of non-differentiable type with the generalized functions defined on Cantor sets by using bilinear method. Meanwhile, we discuss the result when fractal dimension is 1, and compare it with the result when fractal dimension is [Formula: see text].

Existence and Uniqueness of Generalized and Mixed Finite Element Solutions for Steady Boussinesq Equation

Herein, we mainly employ the fixed point theorem and Lax-Milgram theorem in functional analysis to prove the existence and uniqueness of generalized and mixed finite element (MFE) solutions for two-dimensional steady Boussinesq equation. Thus, we can fill in the gap of research for the steady Boussinesq equation since the existing studies for the equation are assumed the existence and uniqueness of generalized solution without providing proof.

Stability of the 2D anisotropic Boussinesq equations with mixed partial dissipation

Stability is an important property of solutions to many fluid flow equations. In this paper, we focus on the stability problem for two-dimensional Boussinesq equations with only vertical viscosity and horizontal thermal diffusion near a stratified, hydrostatic equilibrium in the domain Ω=T×R, where T=[−12,12] is a 1D periodic box. We are mainly concerned with the H2-stability of global solutions when the initial data are closed to an equilibrium state and sufficiently small. Due to the lack of horizontal viscosity in velocity equations and vertical thermal diffusion in temperature equation, this problem seems to be not trivial and becomes particularly difficult. In order to achieve our main goal, we divide the velocity field and temperature into two parts: the horizontal average and the remaining oscillation part. Besides, we fully exploit the Lt2Lx2-norm of ∂xu2 to help us dominate the nonlinear parts. Our result shows that with a small initial perturbation, there exists a unique global solution to the stated Boussinesq equations. In addition, the perturbation remains small for all time. In particular, our result also demonstrates the stabilizing effect of internal gravity waves, which are induced by the stratified background hydrostatic state.

Low regularity global well-posedness for 2D Boussinesq equations with variable viscosity

As a continuation of our previous work, we consider lower regularity global well-posedness for a model of the two-dimensional zero diffusivity Boussinesq equations with variable viscosity. More precisely, based on De Giorgi–Nash–Moser estimates and the refined logarithmic Gronwall-type inequality, we prove that it is globally well-posed, provided that the initial data belong to Hs with s > 1. Finally, we show that it is also valid for the two-dimensional zero diffusivity Boussinesq equations with variable viscosity in the non-divergence form.

The optimal start control problem for 2D Boussinesq equations

Abstract We consider the problem of the optimal start control for two-dimensional Boussinesq equations describing non-isothermal flows of a viscous fluid in a bounded domain. Using the study of the properties of admissible tuples and of the evolution operator, we prove the solubility of the optimization problem under natural assumptions about the model data. We derive a variational inequality which is satisfied by the optimal control provided that the objective functional is determined by the final observation. We also obtain sufficient conditions for the uniqueness of an optimal control.

Задача оптимального стартового управления для двумерных уравнений Буссинеска

Рассматривается задача оптимального стартового управления для двумерных уравнений Буссинеска, описывающих неизотермические течения вязкой жидкости в ограниченной области. На основе изучения свойств допустимых кортежей и оператора сдвига доказана разрешимость задачи оптимизации при естественных допущениях относительно данных модели. Выведено вариационное неравенство, которому удовлетворяет оптимальное управление при условии, что целевой функционал определяется финальным наблюдением. Кроме того, получены достаточные условия единственности оптимального управления. Библиография: 28 наименований.

On the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces

We study the existence, uniqueness as well as regularity issues for the two-dimensional incompressible Boussinesq equations with temperature-dependent thermal and viscosity diffusion coefficients in general Sobolev spaces. The optimal regularity exponent ranges are considered.

On the Boussinesq Equations with Non-monotone Temperature Profiles

In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if T' is of size at most nu ^{1/3} in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh–Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.

Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation

This article deals with the asymptotic behavior of the two-dimensional inviscid Boussinesq equations with a damping term in the velocity equation. Precisely, we provide the time-decay rates of the smooth solutions to that system. The key ingredient is a careful analysis of the Green kernel of the linearized problem in Fourier space, combined with bilinear estimates and interpolation inequalities for handling the nonlinearity.

Global existence of classical solutions to the two-dimensional compressible Boussinesq equations in a square domain

In this paper, we consider the initial boundary value problem of two-dimensional isentropic compressible Boussinesq equations with constant viscosity and thermal diffusivity in a square domain. Based on the time-independent lower-order and time-dependent higher-order a priori estimates, we prove that the classical solution exists globally in time provided the initial mass |rho _{0}|_{L^{1}} of the fluid is small. Here, we have no small requirements for the initial velocity and temperature.

A reduced-order variational multiscale interpolating element free Galerkin technique based on proper orthogonal decomposition for solving Navier–Stokes equations coupled with a heat transfer equation: Nonstationary incompressible Boussinesq equations

In the recent decade, meshless methods have been handled for solving some PDEs due to their easiness. One of the most efficient meshless methods is the element free Galerkin (EFG) method. The test and trial functions of the EFG are based upon the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG (VMEFG) procedure. In the current article, the shape functions of interpolating moving least squares (IMLS) approximation are applied to the variational multiscale EFG technique to numerical study the Navier–Stokes equations coupled with a heat transfer equation such that this model is well-known as two-dimensional nonstationary Boussinesq equations. In order to reduce the computational time of simulation, we employ a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique. In the current paper, we developed a new reduced order model based on the meshless numerical procedure for solving an important model in fluid mechanics. To illustrate the reduction in CPU time as well as the efficiency of the proposed method, we investigate two-dimensional cases.

Global well-posedness for a model of 2D temperature-dependent Boussinesq equations without diffusivity

In this paper we consider the Cauchy problem of a model of the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity. We show that there is a unique global smooth solution to this system for arbitrarily large initial data in Sobolev spaces. Our key argument is the De Giorgi-Nash-Moser estimates for the vorticity equation.

Simulation of Two Dimensional Subsurface Seepage Flow In Isolated Anisotropic Aquifer

Mathematical models have emerged as important tools for prediction of transient and steady state behavior of water table in unconfined aquifer under seepage and recharge condition. This paper presents analytical solution to predict the behavior of water table in anisotropic unconfined aquifer overlaying a leaky base in response to multiple recharge and withdrawal. The hydrological setting consists of rectangular aquifer with all four sides with no flow conditions. The aquifer is subjected to recharge and withdrawal activities through multiple recharge basin and extraction/injection wells. The subsurface seepage flow is approximated by two-dimensional Boussinesq equation and solved using Fourier cosine transform. Application of closed form solution is demonstrated using an illustrative example. Effect of aquifer parameter on formation of ground water mound and cone of depression due to recharge and withdrawal are discussed. The influence of permeability on the mound height and cone depth is also analyzed.

Existence and Uniqueness Results for Two-Dimensional Stochastic Linearised Boussinesq Equation

The water flow in saturated zones of the soil is described by two-dimensional Boussinesq equation. This paper is devoted to investigating the linearised stochastic Boussinesq problem in the presence of randomness in hydraulic conductivity, drainable porosity, recharge, evapotranspiration, initial condition and boundary condition. We use the Sabolev spaces and Galerkin method. Under some suitable assumptions, we prove the existence and uniqueness results, as well as, the continuous dependence on the data for the solution of linearised stochastic Boussinesq problem. Keywords: linearised stochastic Boussinesq equation, Galerkin method, existence and uniqueness results, and continuous dependence on the data.

On the Yudovich's type solutions for the 2D Boussinesq system with thermal diffusivity

The goal of this paper is to study the two-dimensional inviscid Boussinesq equations with temperature-dependent thermal diffusivity. Firstly we establish the global existence theory and regularity estimates for this system with Yudovich's type initial data. Then we investigate the vortex patch problem, and proving that the patch remains in Holder class \begin{document}$ C^{1+s}\; (0 for all the time.

Global Regularity for the 2D Boussinesq Equations with Temperature-Dependent Viscosity

This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.

Global regularity results for the 2D Boussinesq equations and micropolar equations with partial dissipation

This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional Λα dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small α>0 always possesses a unique global classical solution.

A deep learning enabler for nonintrusive reduced order modeling of fluid flows

In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various DNN architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equation, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a nonintrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly nonintrusive since any prior information about the underlying governing equations is not required for generating the reduced order model. Our a posteriori analysis shows that the proposed data-driven approach is remarkably accurate and can be used as a robust predictive tool for nonintrusive model order reduction of complex fluid flows.