Unconditional nonlinear stability for double‐diffusive convection in a porous medium with temperature‐dependent viscosity and density

In this study, fluid flow in a porous medium is analyzed using a Forchheimer model. The problem of double‐diffusive convection is addressed in such a porous medium. We utilize a higher‐order approximation for viscosity‐temperature and density‐temperature, such that the perturbation equations contain more nonlinear terms. For unconditional stability, nonlinear stability has been achieved for all initial data by utilizing the or norms. It also shows that the theory of is not sufficient for such unconditional stability. Both linear instability and nonlinear energy stability thresholds are tested using three‐dimensional (3D) simlations. If the layer is salted above and salted below then stationary convection is dominant. Thus the critical value of the linear instability thresholds occurs at a real eigenvalue , and our results show that the linear theory produces the actual threshold. Moreover, it is known that with the increase of the salt Rayleigh number, R_c, the onset of convection is more likely to be via oscillatory convection as opposed to steady convection. The 3D simulation results show that as the value of R_c increases, the actual threshold moves towards the nonlinear stability threshold, and the behavior of the perturbation of the solutions becomes more oscillatory.     

Effects of quadratic drag and throughflow on double diffusive convection in a porous layer

The linear stability theory is used to investigate analytically the effects of quadratic drag and vertical throughflow on double diffusive convection in a horizontal porous layer using the Forchheimer-extended Darcy equation. The boundaries of the porous layer are considered to be either impermeable or porous, but perfect conductors of heat and solute concentration. Conditions for the occurrence of stationary and oscillatory convection are obtained using the Rayleigh-Ritz method. Stability boundaries are drawn in the Rayleigh numbers plane and the throughflow is found to influence the mode of instability. It is found that, irrespective of the nature of boundaries, a small amount of throughflow in either of its direction destabilizes the system; a result which is in contrast to the single component system.     

Unconditional nonlinear stability for double-diffusive convection with temperature- and pressure-dependent viscosity

A linear and nonlinear stability analyses are carried out for a double-diffusive chemically reactive fluid layer with viscosity being a function of temperature and pressure. The linear stability analysis is studied when the stabilizing salt gradient acts against the destabilizing thermal gradient. The effect of reaction parameters and variable viscosity on the stability of the system is studied for heated below, salted above, and the heated and salted below models with Rigid–Rigid boundary conditions. Chebyshev pseudospectral method is applied to determine the numerical solutions.     

Numerical investigation of dual solutions for double diffusive convection from a permeable horizontal flat plate

Numerical simulations are carried out to study the simultaneous effects of thermal and concentration diffusions on a mixed convection boundary layer flow over a permeable horizontal flat plate with suction/injection in a viscous incompressible fluid. The non-linear coupled partial differential equations governing the flow, thermal and concentration fields are first transformed into a set of non-linear coupled ordinary differential equations by a set of suitable similarity transformations. The resulting system of coupled non-linear differential equations is solved using shooting method by converting into initial value problem. In this method, system of equations is converted into a set of first order system which is solved by fourth order Runge–Kutta method. Flows with both assisting and opposing buoyancy forces are considered in the present investigation. The study reveals that dual solutions of velocity, temperature and concentration profiles exist for certain values of suction/injection and buoyancy parameters. Suction/injection parameter, Prandtl and Schmidt numbers strongly affect thermal and concentration boundary layers, respectively.     

Visualization and numerical prediction of flow characteristics of double diffusive convection in a static crystallizer

A quantitative flow visualization of static crystallization is performed for a binary Na₂SO₄ - H₂O solution in a rectangular static crystallizer utilizing Laser Swept Velocimetry (LSV). Comparison of experimental and numerical results for this process is also included in this paper. Both experimental and numerical results reveal key features of double diffusive convection occurred during static crystallization. A good agreement quantitatively in the experimental and numerical velocity profiles has been obtained for the experimental condition applied for this research, but the experimental flow field in the buck liquid is likely to be more complex than the somewhat idealized numerical flow field.     

Three-dimensional analyses of double diffusive convection in a two-layer system at high Rayleigh number

Three-dimensional numerical computations of double diffusive natural convection were carried out at the Prandtl number Pr=6, the Lewis number Le=100, the aspect ratio A=2, the Rayleigh number Ra=10⁷ and various buoyancy ratios N for a two-layer system which consists of a pure water (upper layer) and an aqueous solution (lower layer) with lateral heating and cooling. Salt-fingers with the termination of bulbous shapes were formed owing to the penetration of convective flow in a layer into the other layer at N=0.3 and plumes were formed by the collision of solutal fragments against the interface at N=0.6 or 1.     

An optical technique for the in-situ measurement of species concentration in double diffusive convection

This paper proposes a non-intrusive experimental method based on absorptiometry, suitable for the in-situ measurement of species concentration in flowing aqueous solutions. The method is applicable to solutions containing a pre-selected species. The technique appears to be promising for use in double diffusive convection. One of the major advantages of the method lies in the fact that it is an optical method and, as such, non-intrusive, yet accurate.     

Numerical simulation of three-dimensional double diffusive free convection flow and irreversibility studies in a solar distiller

Numerical results of double-diffusive natural convection are presented in a three-dimensional solar distiller. The flow is considered laminar and caused by the interaction of the thermal energy and the chemical species diffusions. Equations of concentration, energy and momentum are formulated using vector potential-vorticity formulations in its three-dimensional form, then solved by the finite volume method. The Rayleigh number is fixed at Ra = 10⁵ and the effects of the buoyancy ratio are studied for opposed temperature and concentration gradients, with a particular interest to the three-dimensional aspects and entropy generation.     

Exact solutions of double diffusive convection in cylindrical coordinates with Le = 1

The unsteady geometrical 2D governing equation set for the double diffusive convection—a very complicated nonlinear partial differential equation set with 4 variables—is solved analytically in the cylindrical coordinates. Two special exact solutions describing the convection in a cylindrical tube and a circular tube respectively are derived with an extraordinary method of separating variables and some other skills. The solutions are valuable for the development of heat and mass transfer theory. Moreover, as benchmark solutions, they are very useful for the computational heat and mass transfer to check the accuracy, convergence and effectiveness of various numerical computation methods.     

Numerical simulation of double diffusive mixed convection in an open enclosure with different cylinder locations

This article reports numerical simulation of the double diffusive mixed convection around a cylinder in an open enclosure with an inlet and exit ports. The temperature and mass concentration of the cylinder are higher than those of the inlet flow and the cylinder can be at three different locations (lower, middle and upper) in the enclosure. The inlet flow with low temperature and mass concentration is located at the lower-left wall of the enclosure and the exit is at the upper-right wall. Other walls are assumed to be adiabatic. Effects of Lewis number Le, buoyancy ratio Br, and cylinder locations on the double diffusive mixed convection are investigated at Richardson number Ri = 1.0 and 0.01 while Prandtl number Pr is kept at 0.7. Streamlines, isotherms, isoconcentrations, and the average and local Sherwood number at different parameters are reported to characterize the double diffusive mixed convection phenomena in the open enclosure.     

Linear and nonlinear stability of Soret‐Dufour Lapwood convection near double codimension‐2 points

In this study, the Dufour and Soret effects on natural double‐diffusion convection in a horizontal porous layer was studied numerically using FORTRAN 90 programming and analytically near various convection onset thresholds. The porous layer was subject to a uniform heat and mass fluxes on the horizontal walls while the vertical walls were impermeable and adiabatic. The Darcy model along with the Boussinesq approximation was assumed in the problem formulation. The governing parameters of the problem are the thermal Rayleigh number, R_T, the buoyancy ratio, N, the Lewis number, Le, the aspect ratio of the cavity, A, and the Dufour, D_u, and Soret, S_r, numbers. For a shallow enclosure, the analytical solution was derived assuming zero convection wave number, which is valid near and above criticality. The onset of subcritical, supercritical and oscillatory convection was investigated. Two linear and nonlinear codimension‐2 points were found to exist. Whether the system was subject to constant fluxes and heat and solute, regardless of the aspect ratio of the layer, the subcritical convection behavior remained the same with similarity in the thresholds expressions for subcritical bifurcation.     

Mixed laminar convection in a horizontal tube with natural convection around its boundaries

The laminar flow and heat transfer within a horizontal tube surrounded by a liquid medium are studied both experimentally and numerically. Emphasis is given to flow regimes where a buoyant effect on the forced flow is exhibited inside the tube. The outer surface of the tube is also subjected to natural convection resulting from the temperature difference between the wall and the surrounding fluid. Detailed analyses are performed for a number of cases with various fluids, inlet temperatures and fluid flows. It is found that the variable wall temperature has a marked effect on the secondary flow patterns within the tube as well as on the heat transfer.     

Lattice Boltzmann simulation of the double diffusive natural convection and oscillation characteristics in an enclosure filled with porous medium

This article adopts lattice Boltzmann method to investigate the double diffusive natural convection around a heated cylinder in an enclosure filled with porous medium. The heated cylinder is located at the center of the enclosure with high temperature and concentration. Four surrounding walls are assumed to be low temperature and concentration. The distributions of velocity, temperature and concentration are solved by three independent lattice Bhatnagar-Gross-Krook (LBGK) equations. The influence of Darcy number Da (10^–4 ≤ Da ≤ 10^− 2), Lewis number Le (0.2 ≤ Le ≤ 10.0) and buoyancy ratio Br (− 10.0 ≤ Br ≤ 10.0) on the double diffusive natural convection are inspected numerically. Results are presented in terms of isotherms, streamlines, isoconcentrations, average Nusselt and Sherwood numbers. At Br = − 50.0, the effect of Darcy number on unsteady flow characteristics is also investigated by the time history and phase space trajectory. It is found that the flow undergoes steady-state, unsteady doubling periodic oscillation, quasi-periodic oscillation and non-periodic oscillation when Darcy number Da is varied from 10^− 4 to 10^− 2.     

Double diffusive convection in a shallow porous cavity filled with a non-newtonian fluid

The Darcy model with the Boussinesq approximations is used to study double-diffusive convection in a shallow porous cavity saturated with a non-Newtonian fluid. A power-law model is used to characterize the non-Newtonian fluid behaviour. Motions are driven by constant heat and concentration fluxes imposed across the walls of the enclosure. The problem is solved analytically, in the limit of a thin layer, using a parallel flow approximation. Solutions for the flow fields, Nusselt and Sherwood numbers are obtained explicitly in terms of the governing parameters of the problem. A good agreement is obtained between the analytical prediction and a numerical solution of the full governing equations.     

Double diffusive unsteady free convection on two-dimensional and axisymmetric bodies in a porous medium

The unsteady laminar free convection flow of an incompressible electrically conducting fluid over two-dimensional and axisymmetric bodies embedded in a highly porous medium with an applied magnetic field has been studied. The unsteadiness in the flow field is caused by the variation of the wall temperature and concentration with time. The coupled nonlinear partial differential equations with three independent variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. It is observed that the skin friction, heat transfer and mass transfer increase with the permeability parameter but decrease with the magnetic parameter. The results are strongly dependent on the variation of wall temperature and concentration with time. The skin friction and heat transfer increase or decrease as the buoyancy forces from species diffusion assist or oppose the thermal buoyancy force. However, the mass transfer is found to be higher for small values of the ratio of the buoyancy parameters than for large values.     

Linear stability analysis of mixed convection flow in a horizontal channel filled with nanofluids

The convective instability driven by buoyancy in the Poiseuille–Rayleigh–Bénard flow through two infinite parallel horizontal plates filled with nanofluids is investigated using linear stability analysis. We considered water‐based nanofluids with different volume fractions of aluminum ( $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$) and silver ( $\mathrm{A}\mathrm{g}$) nanoparticles. A spectral collocation method founded on Chebyshev polynomials is implemented and the obtained algebraic eigenvalue problem is solved. In this study, we have numerically determined the critical Rayleigh number of the onset of longitudinal and transversal rolls and the results are represented in the form of marginal stability curves. Critical wave numbers that describe the size of convective cells in the flow are also presented, analyzed, and compared with those of the Poiseuille–Rayleigh–Bénard flow without nanoparticles. The effects of the type and nanoparticle volume fractions on the onset of both longitudinal and transversal rolls are investigated.     

具有滞后非线性悬挂的机车车辆转向架运行稳定性的研究

针对一类比较典型的具有滞后非线性悬挂的机车车辆转向架非线性系统,采用以变步长龙格-库塔数值积分算法为核心的数值算法,对转向架的横向运动行为进行了数值仿真.提出的模型以1位轮对的横向振动为例对转向架的横向振动分岔进行了数值仿真,得知转向架的横向振动分岔为亚临界Hopf分岔.同时,根据极限环理论分析了该分岔的平衡点吸引域和极限环吸引域,得出了3点稳定性结论,并对其进行了数值仿真,得到了与结论相吻合的仿真结果.     

Influence of Lewis number on double‐diffusive convection in a square cavity filled with a binary gas

This paper considers double‐diffusive convection in a square cavity filled with a binary gas, due to horizontal opposing temperature and concentration gradients. The effect of Lewis number was considered under the conditions of Prandtl number Pr = 1, buoyancy ratio N = 1, and thermal Rayleigh numbers R_aT = 10⁴ and 10⁵. Numerical solutions are obtained by a Chebyshev collocation technique with high resolution. Depending on the Lewis number, three kinds of flow structures are identified: symmetric steady flow, asymmetric oscillatory flow, and symmetric oscillatory flow. Oscillatory flow occurs in the regime of thermal dominant flow, and leads to a periodic change between stable and unstable states in species stratification due to the thermo‐solutal interaction. © 2000 Scripta Technica, Heat Trans Asian Res, 30(1): 63–75, 2001