Double-diffusive Convection System Research Articles

Blow-up criteria for double-diffusive convection systems in sum spaces

Applying some energy estimates, we show blow-up criteria of strong solutions to the 3D incompressible double-diffusive convection system.

On continuation criteria for the double-diffusive convection system in Vishik spaces

This paper proves the continuation criteria of the strong solution for the 3D double-diffusive convection system involving the deformation tensor in Vishik spaces. As a bi-product, our theorem also improve some well-known results on conditional regularity for the particular case of classical Navier–Stokes equations.

A two-relaxation-time lattice Boltzmann study on the Soret and Dufour effects of double-diffusive convection over a rough surface

This study examines the Soret and Dufour effects of double-diffusive convection (DDC) over a rough surface by using a lattice Boltzmann (LB) model with two-relaxation-time (TRT) collision operators, where the cross-diffusion terms are modeled by introducing additional collision operators that can be used to avoid the need for special treatment of the gradient terms. The TRT model is applied to improve the stability of the LB method for flows at high Rayleigh numbers. We investigate how rough surfaces with different distributions and compactness of rough elements affect heat and mass transfer as well as the flow structure of the DDC system with the Soret and Dufour effects. We examined five rough models with different spatial distributions of the triangular rough elements as well as different values of the Rayleigh number (Ra=106 and 109), Prandtl number (Pr=0.7), Lewis number (Le=1 and 2), buoyancy ratio (−2≤Nc≤6.5), Soret factor (0≤STC≤1), Dufour factor (0≤DCT≤1), and the height of the roughness (1/96≤h≤1/2). Specifically, the influence of the buoyancy ratio on the DDC system was investigated, and the relationship of heat and mass transfer with the buoyancy ratio was divided into four stages: conduction-dominated, transition, steady convection-dominated, and oscillatory convection-dominated stages. The rough surfaces reduced heat and mass transfer at low Rayleigh numbers and low height of the roughness, and the third rough model (Model III) yielded the highest reduction. The flow showed periodic behavior after long-term reversals in some cases, which appear to have not been observed in previous research. Moreover, the effects of the Soret and Dufour factors were examined, and it was found that mass and heat transfer could be enhanced in most cases by increasing their values. To enhance heat and mass transfer, the height of the roughness should be increased as well. Model III delivered the best performance, with a scenario completely opposite to that in case of a low roughness height. Finally, approximate values of the critical height of the roughness with different rough models were obtained such that heat and mass transfer were enhanced only when the height of the roughness was greater than a critical value.

Global regularity and stability of solutions to the 3D double-diffusive convection system with Navier boundary conditions

This paper is concerned with the initial-boundary value problem to 3D double-diffusive convection system. First, we establish the global weak solutions for the double-diffusive convection system in a domain $\Omega$ with Navier boundary condition by Galerkin approximation method. Then by making higher-order estimates of the approximation function with higher regularity assumption on the initial data, combined with compactness argument we obtain the local strong solution with uniqueness. Secondly, we obtain the classical Serrin-type blow-up criterion for the local strong solution in the Lorentz space. Lastly, we focus on global stability of strong large solutions for the double- diffusive convection system with Navier boundary conditions, it is shown that the large solution is stable with the additional assumption on some suitable integrable property of solution.

A lattice Boltzmann model for the coupled cross-diffusion-fluid system

In this paper, we propose a lattice Boltzmann (LB) model for the generalized coupled cross-diffusion-fluid system. Through the direct Taylor expansion method, the proposed LB model can correctly recover the macroscopic equations. The cross diffusion terms in the coupled system are modeled by introducing additional collision operators, which can be used to avoid special treatments for the gradient terms. In addition, the auxiliary source terms are constructed properly such that the numerical diffusion caused by the convection can be eliminated. We adopt the developed LB model to study two important systems, i.e., the coupled chemotaxis-fluid system and the double-diffusive convection system with Soret and Dufour effects. We first test the present LB model through considering a steady-state case of coupled chemotaxis-fluid system, then we analyze the influences of some physical parameters on the formation of sinking plumes. Finally, the double-diffusive natural convection system with Soret and Dufour effects is also studied, and the numerical results agree well with some previous works.

A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations

We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in $$\mathrm {L}^4$$ . As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable $$\mathrm {H}(\mathrm {div})$$ -type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Necas–Babuska theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order $$k\ge 0$$ for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree $$\le k + 1$$ are employed for the velocity, and piecewise polynomials of degree $$\le k$$ for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

Café latte: spontaneous layer formation in laterally cooled double diffusive convection

Abstract

Blowup Criterion of Strong Solutions to the Three-Dimensional Double-Diffusive Convection System

In this paper, we investigate some sufficient conditions for the breakdown of local strong solutions to the three-dimensional double-diffusive convection system. We obtain the classical blowup criterion for strong solutions $$(u, \theta , s)$$, i.e., $$u \in L^{q}\left( 0, T ; L^{p}\left( {\mathbb {R}}^{3}\right) \right) $$, for $$\frac{2}{q}+\frac{3}{p}=1$$, $$3<p \le \infty $$ or $$\nabla u \in L^{q}\left( 0, T ; L^{p}\left( {\mathbb {R}}^{3}\right) \right) $$, for $$\frac{2}{q}+\frac{3}{p}=2$$, $$\frac{3}{2}<p \le \infty $$. Moreover, we get a further blowup criterion, which involves partial derivatives of the partial components of the velocity field (i.e., $$\nabla _{h}{\tilde{u}}$$ ) belonging to the Besov spaces, where $$\nabla _{h}=(\partial _{1}, \partial _{2})$$ and $$ {\tilde{u}}=(u_{1}, u_{2})$$.

The well-posedness for the Cauchy problem of the double-diffusive convection system

In this paper, we consider the well-posedness of the solution for the Cauchy problem of the double-diffusive convection system in R3. We establish the local existence and uniqueness of the solution for the double-diffusive convection system in H1(R3) with large initial data and the global well-posedness under the assumption that the L2 norm of the initial data is small. Moreover, we also prove that there exists a global unique solution in HN(R3) for any N ≥ 2, without any other smallness condition of the initial data.

The suitable weak solution for the Cauchy problem of the double-diffusive convection system

ABSTRACTIn this paper, we consider the existence of the suitable weak solutions for the Cauchy problem of the double-diffusive convection system in . We establish the global existence of the suitable weak solutions of the double-diffusive convection system, especially, we obtain some generalized energy estimates which will play a very important role in studying the partial regularity of the suitable weak solutions of the double-diffusive convection system.

DUAL WAVELENGTH INTERFEROMETRY-BASED THREE-DIMENSIONAL SIMULTANEOUS RECONSTRUCTION OF TEMPERATURE AND CONCENTRATION FIELDS IN DOUBLE-DIFFUSIVE SYSTEMS: PART I

The present study is concerned with the development and application of dual wavelength inter-ferometry technique for investigating the transients associated with the three-dimensional temperature and solutal fields in a heat-salt based double-diffusive convection system. The double-diffusive convection phenomenon has been arranged in an octagonal chamber. The dual color interferometer employs two distinct monochromatic lasers, i.e., He-Ne (λ1 = 632.8 nm) and Ar-ion (λ2 = 457.9 nm) as light sources, which individually map the refractive index fields in real time. Experiments have been reported with an initial stability ratio of Rρ ≈ 2.94. The operating experimental conditions constitute an initial concentration step of ΔCi = 0.2 (wt/wt)% and temperature differential of ΔT = 1°C at each horizontal wall. The projection data of the coupled thermal and solutal changes have been recorded in the form of dual-color interferograms from four view angles (0, 45, 90, and 135°). Results have been presented in the form of contour plots of retrieved thermal and solutal fields and reconstructed two-dimensional thermal and solutal fields at select horizontal planes. It has been observed that the diffusive layer convection system undergoes transition from quasi-steady mode convection to other modes of convection, which are functions of both temperature and salinity gradients. With decreasing stability ratio of the system, the thermal transport from the isothermal wall to the fluid layer increases, which leads to development of instabilities in the fluid layer and subsequently at the interface. The development of instabilities in the interface results in enhanced transport of heat and salt between the two layers. Despite the presence of lower levels of thermal gradients, the transition in the flow field and interface is affected by the salinity step at the interface.

Existence of Time Periodic Solution to Some Double-Diffusive Convection System in the Whole Space Domain

This paper is concerned with the existence of time periodic solutions to some system which describes double-diffusive convection phenomena in the whole space $${\mathbb {R}} ^N $$ with $$N = 3$$ and 4. In previous results for periodic problems of parabolic type equations with non-monotone perturbation terms, it seems that either of the smallness of given data or the boundedness of space domain is essential. In spite of the presence of non-monotone terms, the solvability of our problem in the whole space is shown for large external forces via the convergence of solutions to approximate equations in bounded domains.

Global solvability for double-diffusive convection system based on Brinkman--Forchheimer equation in general domains

In this paper, we are concerned with the solvability of the initial boundary value problem of a system which describes double-diffusive convection phenomena in some porous medium under general domains, especially unbounded domains. In previous works where the boundedness of the space domain is imposed, some global solvability results have been already derived. However, when we consider our problem in general domains, some compactness theorems are not available. Hence it becomes difficult to follow the same strategies as before. Nevertheless, we can assure the global existence of a unique solution via the contraction method. Moreover, it is revealed that the global solvability holds for higher space dimension and larger class of the initial data than those assumed in previous works.

Global solvability of some double-diffusive convection system coupled with Brinkman-Forchheimer equations

In this paper, the global solvability of the initial boundary value problem and the periodic problem are discussed for double-diffusive convection systems under the homogeneous Neumann boundary condition in a bounded domain. This system is coupled with the so-called Brinkman-Forchheimer equation, which is similar to the Stokes equation and contains some convection terms similar to that in Navier-Stokes equations. However, in contrast to Navier-Stokes equations, it is shown that the global solvability in L^2 -spaces holds true for the 3-dimensional problems.

Heat and mass fluxes across density interfaces in a grid-generated turbulence

We present heat and salt fluxes measurements across a single density-stratified interface in a grids-generated turbulence system. The turbulent field is measured in a homogeneous medium (water) by Particle Image Velocimetry (PIV). Stratifications have been followed in time by recording the vertical temperature and density profiles. Measured buoyancy heat and mass fluxes show two different behaviors similar to that in double diffusive convection systems. This was also highlighted by measurement of entrainment rate near the interface, which showed that entrainment depends on the diffusivity for high Richardson number Ri. For low Ri, molecular diffusion has less effect on the transport process than turbulent mixing.

Systematics of Periodic Windows in the Double-Diffusive Convection System11The project supported partially by National Natural Science Foundation of China.

The dynamics of truncated double-diffusive convection system on a parameter axis is studied with symbolic dynamics. It is found that the systematics of periodic windows interspersed in chaotic region for its one-dimensional (1D) return maps (to be defined in the text) can be fitted into the universal form for the 1D mappings with two critical points and the antisymetric property.

Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations

In Rayleigh-Bénard convection, the spatially uniform motionless state of a fluid loses stability as the Rayleigh number is increased beyond a critical value. In the simplest case of convection in a pure Boussinesq fluid, the instability is a symmetry-breaking steady-state bifurcation that leads to the formation of spatially periodic patterns. However, in many double-diffusive convection systems the heat-conduction solution actually loses stability via Hopf bifurcation. These hydrodynamic systems provide motivation for the present study of spatiotemporally periodic pattern formation in Euclidean equivariant systems. We call such patterns planforms . We classify, according to spatio-temporal symmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a group-invariant equilibrium. Instead of focusing on plan- forms periodic with respect to a specified planar lattice, as has been done in previous investigations, we consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular dif­ferential equation.

Predicting time-dependent motion in the double-diffusive convection system.

Predicting time-dependent motion in the double-diffusive convection system.

Fiber optic sensor for liquid mixture composition

Abstract Information on the concentration and temperature distributions is important in the study of the heat and mass transfer in liquid mixture systems. This paper reports on a miniature optical fiber sensor that continuously measures the concentration of the liquid mixture. It is found that the reproducibility of sensor is very good and that the concentration of the liquid mixture can be measured with a long time reproducibility of 0.1%. The relationship between measuring sensitivity and concentration is investigated. The influence of temperature on the measurement is also examined. An application of the sensor to the continuous measurement of liquid mixture concentration in a double diffusive convection system is described.

Diffusion of trace elements during bottom crystallization of double-diffusive convection systems: The magnetitite layers of the Bushveld Complex

Magmas that crystallize to produce layered intrusions may behave as double-diffusive convective systems. In such an intrusion, the magma would be stratified as a stacked series of thick convecting layers separated by thin, static “diffusive” boundary layers. Crystallization is often inferred to occur at the base of the chamber, and chemical changes due to fractional crystallization of the lowermost convecting layer are transmitted by diffusion across the thin “diffusive” boundary layers separating the thick stacked layers of convecting magma. In this work the Rayleigh distillation law is extended to describe changes in the concentration of a compatible trace element due to fractional crystallization of the convecting magma layer at the base of the stack as well as diffusion of the trace element into the layer across its top diffusive boundary. General equations are presented defining predicted geochemical profiles in terms of partition coefficient, rate of accumulation of the crystal pile, diffusion rates and thickness of discrete magma layers. Extraction of a single dense phase from the lowermost convecting layer progressively decreases the density of the residual liquid in the layer. Ultimately, the densities of the basal and immediately overlying layers of magma become equal, the diffusive boundary layer breaks down, and the two basal layers mix. This results in a reversal in the trace-element concentration of the crystals being precipitated. A “sawtooth” trace-element concentration profile results. The Bushveld Complex provides a good test for the double-diffusive convection model envisaged here. Available data on chromium concentration with respect to height in the main magnetitite layer of the Bushveld Complex are well described by these equations.