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.     

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.     

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.     

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.     

Global stability for thermal convection in a couple-stress fluid

We show that the global nonlinear stability threshold for convection in a couple-stress fluid is exactly the same as the linear instability boundary. This optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of convection. It is also found that the couple-stress fluid is thermally more stable than the ordinary viscous fluid and then the effect of couple stress parameter on the onset of convection is also analyzed.     

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

This paper presents double‐diffusive convection in a square cavity filled with 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 number Ra_T = 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 it leads to a periodic change between stable and unstable states in species stratification due to the thermo‐solutal interaction. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(1): 85–97, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10073     

Further results on a discriminating monopolist model

This note represents an extension of a paper which appeared in this Journal in 1981—‘A Discriminating Monopolist Model of Natural Gas Markets in the US’. One of the conclusions of the previous paper was that the supply side of the market needed to be examined. Here we make some additional inferences about whether the companies are in fact monopolists, by looking at optimal marginal revenues and average costs.     

A nonlinear stability analysis for thermoconvective magnetized ferrofluid with magnetic field dependent viscosity

The generalized energy method which gives sufficient condition for the stability is developed for convection problem in a magnetized ferrofluid with magnetic field dependent (MFD) viscosity heated from below. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. Both linear and nonlinear analyses are carried out and comparison of results shows a marked difference in the stability boundaries and thus indicates that the sub-critical instabilities are possible. The effect of various parameters on the sub-critical region has also been analyzed.     

Composite relation for laminar free convection in inclined channels with uniform heat flux boundaries

A composite correlation of the average Nusselt number and the channel Rayleigh number for buoyant air flow through inclined channels with uniform heat flux boundaries is presented. The form of the correlation is based on dimensional analysis and is a superposition of the developing and fully developed flow limits. In the limit of fully developed flow, an analytical solution for the Nusselt number is derived. The developing flow limit follows the format of the correlation for a single plate. The composite relationship based on the top wall temperature is $\overline{\text{Nu}}={\left[\frac{6.25(1+r)}{{\text{Ra}}^{″}\sin ?}+\frac{1.64}{({\text{Ra}}^{″}\sin ?{)}^{2/5}}\right]}^{-1/2}$, where r is ratio of the heat flux at the top and bottom wall. At inclination angles of $30°\le ?\le 90°$, this correlation predicts the available data base for $10\le {\text{Ra}}^{″}\le {10}^5$ and agrees with the analytical solution for $1\le {\text{Ra}}^{″}\le {10}^2$.     

Double diffusive convection in a porous enclosure submitted to cross gradients of temperature and concentration

Numerical results of two dimensional double diffusive natural convection in a square porous cavity submitted to cross gradients of heat and solute concentration are reported in this study. The parameters governing the problem are the thermal Rayleigh number (100⩽Ra⩽200), the Lewis number (0.1⩽Le⩽10) and the buoyancy ratio (−10⩽N⩽10). The effects of the governing parameters on the flow structure and heat and mass transfer are analyzed. It is demonstrated that the solutal buoyancy force induced by horizontal concentration gradients eliminates the multiplicity of solutions obtained in pure thermal convection when N exceeds some critical value, which depends on Le and Ra. For N>0/(N<0), the monocellular trigonometric/(clockwise) solution is maintained for all the explored ranges of the governing parameters considered in this study.     

Double diffusive natural convection in an enclosure filled with a step type porous layer: Non-Darcy flow

The double diffusive natural convection between a saturated porous layer and an overlying fluid layer in an enclosure has been investigated using the non-Darcy flow model. The problem has been investigated for two cases; namely case I where the interface between fluid and porous layer is horizontal, and case II where the interface contains a step that has a height a. The fluid flow and heat and mass transfer has been investigated for different values of the step height and the Rayleigh and Darcy numbers. The results show that the height of the step at the interface has a significant effect on the flow field and heat and mass transfer from the left-hand to the right-hand walls in the composite enclosure. This is very important for insulation problems and for heat and mass blockage in enclosure systems.     

Generalized double integral inequalities and their applications in studying the stability of nonlinear integro-differential systems with time delay

In this paper, we generalize some integral inequalities, then we establish bounds on the solutions and show the usefulness of our results in investigating the stability on the solutions of integral equations, differential equations and integro-differential systems.     

Modeling of double diffusive buoyant flow in a solar collector with water‐CuO nanofluid

The behavior of a prism‐shaped solar collector with a right triangular cross sectional area is investigated numerically. The water‐CuO nanofluid is taken as the functioning liquid through the solar collector. The leading differential equations with boundary conditions are solved by the penalty finite element method using Galerkin's weighted residual scheme. The performance of parameters in terms of temperature, mass, velocity distributions, radiative, convective heat and mass transfer, mean temperature and concentration of nanofluid, mid height horizontal‐vertical velocities, and sub‐domain average velocity field are investigated systematically. These parameters include the Rayleigh number Ra and the solid volume fraction φ. The outcome explains that the performance of the solar collector can be enhanced with the largest Ra and φ. The code validation shows excellent concurrence with the hypothetical outcome obtainable in the literature. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21039     

Role of rotation and hall currents on free convection and mass transfer flow through a porous medium

Effects of Hall current and rotation on free convection and mass transfer flow through a porous medium bounded by a vertical surface have been analysed. The problem is solved analytically. The velocity profiles are shown on graphs. Effects of m (Hall parameter), K★ (permeability parameter), E (Ekman number) and Sc (Schmidt number) on velocity are discussed.     

The effect of rotation on the onset of convection in a horizontal anisotropic porous layer

The effect of rotation and anisotropy on the onset of convection in a horizontal porous layer is investigated using a linear theory and a weak nonlinear theory. The linear theory is based on the usual normal mode technique and the nonlinear theory on the truncated Fourier series analysis. Darcy model extended to include time derivative and Coriolis terms with anisotropic permeability is used to describe the flow through porous media. A modified energy equation including the thermal anisotropy is used. The effect of rotation, mechanical and thermal anisotropy parameters and the Prandtl number on the stationary and overstable convection is discussed. It is found that the effect of mechanical anisotropy is to allow the onset of oscillatory convection instead of stationary. It is also found that the existence of overstable motions in case of rotating porous medium is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The steady finite amplitude analysis is performed using truncated Fourier series to find the Nusselt number. The effect of various parameters on heat transfer is investigated.     

Linear and nonlinear feedback control of chaos in a thermal convection loop

In this paper, the linear and nonlinear active feedback controls of chaos in thermal convection loop are studied theoretically. The one‐dimensional partial differential equations consisting of mass, momentum, and energy balances are expanded to an infinite set of ordinary differential equations. The first mode of this set of equations is three equations which are similar to the celebrated Lorenz equations. These equations can be decoupled from the rest of the set and can be solved independently of other equations without need of truncation. The temperature difference as the controlling signal and the power input as the controlled signal are used. The linear and nonlinear active feedback control are used to adjust the wall temperature. It is found that the linear and nonlinear active feedback control can change the flow structure in a thermal convection loop. The chaos suppression or enhancement can be realized and some unstable periodic orbits can be stabilized by nonlinear feedback control successfully. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(6): 430–437, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10053