Weak Tangency, Weak Invariance, and Carathéodory Mappings

New results on weak invariance in Hilbert spaces, both in autonomous and nonautonomous case, are given. Applications to weakly decreasing systems and to strong invariance are presented.     

Effect of temperature dependent viscosity on the onset of Bénard–Marangoni ferroconvection

The onset of coupled Bénard–Marangoni convection in a horizontal layer of ferrofluid with viscosity depending exponentially on temperature is investigated. The lower rigid and the upper free boundaries are assumed to be insulated to temperature perturbations and the free boundary at which the surface tension effects are accounted for is assumed to be non-deformable. The resulting eigenvalue problem is solved numerically using the Galerkin technique and also analytically by a regular perturbation technique with a wave number as a perturbation parameter. The analytical and numerically computed results are found to be in concurrence. The combined effect of magnetic number M₁ and the viscosity parameter B is to reinforce together and to hasten the onset of Bénard–Marangoni ferroconvection compared to their presence in isolation. Nonetheless, the effect of increasing B also shows initially some stabilizing effect on the system depending on the strength of magnetic and buoyancy forces. In addition, the nonlinearity of fluid magnetization M₃ is found to have no influence on the criterion for the onset of Bénard–Marangoni ferroconvection.     

Onset of Rayleigh–Bénard MHD convection in a micropolar fluid

The present work is concerned with the effect of a uniform magnetic field on the onset of convection in an electrically conducting micropolar fluid. A flat fluid layer bounded by horizontal rigid boundaries, subjected to thermal boundary conditions of the Neumann type, is considered. The parallel flow approximation is used to predict analytically the critical Rayleigh number for the onset of convection. The onset of motion is found to depend on the Hartmann number Ha, materials parameters K, B, $\lambda$, and the micro-rotation boundary condition n. A linear stability analysis is carried out to study numerically the onset of convection. The predictions of the analytical model are found to be in good agreement with the numerical solution. The above results are also compared with those obtained numerically for the case of a system subject to Dirichlet thermal boundary conditions.     

Linear stability analysis of Poiseuille–Bénard–Marangoni flow in a horizontal infinite liquid film

We present the first linear stability analysis of a Poiseuille–Bénard–Marangoni flow, which refers to a horizontal infinite liquid film flowing in one direction with uniform heating from below. This study concerns the two limiting cases of pure buoyancy effect (Ma = 0) and pure thermocapillary effect (Ra = 0). The stability thresholds of the flow and their variation with the control parameters (Biot, Reynolds and Prandtl numbers) are given and compared with those for a Poiseuille–Rayleigh–Bénard flow. The spatial structures of the flow are presented, and it is shown that the centers of the rolls are shifted upwards compared to the PRB case and that there is a loss of symmetry with respect to the vertical axis for the transverse rolls. These effects are directly linked to thermocapillary convection.     

Effect of pressure work and viscous dissipation in the analysis of the Rayleigh–Bénard problem

The classical Rayleigh–Bénard problem in an infinitely wide horizontal fluid layer with isothermal boundaries heated from below is revisited. The effects of pressure work and viscous dissipation are taken into account in the energy balance. A linear analysis is performed in order to obtain the conditions of marginal stability and the critical values of the wave number a and of the Rayleigh number Ra for the onset of convective rolls. Mechanical boundary conditions are considered such that the boundaries are both rigid, or both stress-free, or the upper stress-free and the lower rigid. It is shown that the critical value of Ra may be significantly affected by the contribution of pressure work, mainly through the functional dependence on the Gebhart number and on a thermodynamic Rayleigh number. While the pressure work term affects the critical conditions determined through the linear analysis, the viscous dissipation term plays no role in this analysis being a higher order effect. A nonlinear analysis is performed showing that the superadiabatic Rayleigh number replaces Ra in the functional dependence of the excess Nusselt number. Finally, a reasoning is proposed to show how the results obtained may be used as a test on the most appropriate formulation of the Oberbeck–Boussinesq model.     

Effect of heat source on Rayleigh–Bénard convection in rotating viscoelastic liquids

The influence of heat sources on instability in rotating viscoelastic liquids is studied. Linear stability analysis is done using normal modes. Computations are done for 10 boundary combinations and the results reveal that convection manifests via the oscillatory mode in this case. The critical values of the oscillatory and stationary instability have been studied. The results indicate individual stabilizing influences of rotation and strain retardation along with heat source in the case of free isothermal boundary conditions. It has quite unpredictable influences on the system stability in all the other boundary conditions for the chosen parameters. By suitable limiting processes, results pertaining to Oldroyd liquid B will lead to those of Maxwell, Newtonian, and Rivlin–Ericksen liquids. The problem finds applications in a working media consisting of viscoelastic liquids with nonisothermal systems.     

Thermal effects on Rayleigh–Marangoni–Bénard instability in a system of superposed fluid and porous layers

Thermal effects of the heat transfer at free surface (represented by Biot number) on the Rayleigh–Marangoni–Bénard instability in a system of liquid–porous layers with top free surface are investigated numerically. The results indicate that this thermal effect can evidently lead to the mode transition of convection, which is overlooked in previous works.     

The onset of Rayleigh–Bénard convection and heat transfer under two-frequency rotation modulation

The impact of 16 combinations of sinusoidal (sine) and nonsinusoidal (square, triangular, and sawtooth) time-periodic Coriolis force (rotation modulation) on Rayleigh–Bénard convection in a Newtonian liquid is studied in this paper. This consideration is made to capture the possible effects of two-frequency rotation modulation on stability, that is, the onset of convection and the simultaneous amount of heat transfer in the system. The Venezian approach has been asserted on the linearized Lorenz model to derive the correction Rayleigh number as a function of the two frequencies. The Lorenz model with nonlinearities is evaluated numerically to assess the quantity of heat transfer in the system. The present study states that in comparison with existing studies of no-modulation and single-frequency rotation modulation, two-frequency rotation modulation yields higher stability bounds and thus diminishes the heat transfer. Heat transfer is found to be enhanced by a pair of coprime integers associated with the harmonics in the system.     

The Rayleigh–Bénard convection in near-critical fluids: Influences of the specific heat ratio

This article studies the natural convection in near-critical fluids under the Rayleigh–Bénard configuration. The governing equations are non-dimensionalized by using modified scaling factors and five dimensionless parameters are determined, out of which the specific heat ratio γ is investigated thoroughly in terms of its influences on the convection. The governing equations are solved by the finite volume method along with the low Mach number approximation technique. It is revealed that the strength of the piston effect increases with γ. Since the piston effect is more efficient than the convection in terms of heat transfer, as γ is reduced, the relaxation of temperature field is weakened, while the development of convection is enhanced. By comparing the temperature and velocity fields in different cases and analyzing the heat transfer characteristic, we conclude that the final quasi-steady state is almost unaffected.     

On effect of non-uniform basic temperature gradient on Bénard–Marangoni convection in micropolar fluid

Linear stability analysis is performed to study the effect of non-uniform basic temperature gradients on the onset of Bénard–Marangoni convection in a micropolar fluid. The influence of various parameters on the onset of convection has been analysed. The possibility of delaying the onset of convection by the application of a cubic basic state temperature profile is demonstrated.     

Individual effects of four types of rotation modulation on Rayleigh–Bénard convection in a ferromagnetic fluid with couple stress

The effect of trigonometric sine, square, triangular, and sawtooth wave types of rotation modulation on Rayleigh–Bénard convection in a ferromagnetic fluid with couple stress is investigated in this paper using linear and nonlinear analyses. The expression for the critical Rayleigh number and the correction Rayleigh number is deduced from the three-mode linearized Lorenz model using the Venezian approach. The effect of rotation modulation on heat transport is studied using the generalized fifth-order Lorenz model. The study reveals that the Taylor number stabilizes the no-modulation system and decreases the heat transport, and this situation remains so in the presence of rotation modulation. It is found that the effect of all four types of modulation is to stabilize the system and diminish heat transport. It is also observed that the sawtooth wave type of modulation has the least diminishing effect on heat transport and the square wave type of modulation diminishes the most.     

The effect of a horizontal pressure gradient on the onset of a Darcy–Bénard convection in thermal non-equilibrium conditions

In this paper there is studied the effect of a horizontal pressure gradient on the onset of Darcy–Bénard convection in a fluid-saturated porous layer heated from below, when the fluid and solid phases are not in local equilibrium. In the context of a linearized stability analysis, the problem is transformed into an eigenvalue equation. The problem, when cast in dimensionless form, contains three parameters (the pressure gradient, the porosity-scaled conductivity ratio and the scaled inter-phase heat transfer coefficient). This problem is solved numerically by using two methods: Galerkin approach and the numerical solver dsolve from Maple and comparisons between these methods are performed. Critical values of Rayleigh number, wave number and frequency are obtained for various values of the problem parameters.     

Analysis of exergy loss vs heat transfer rate for Rayleigh–Bénard convection of various fluids in enclosures with curved walls

The investigation of entropy generation is highly desirable for the optimization of the thermal systems to avoid larger energy wastage and ensure higher heat transfer rate. The numerical investigation of natural convection within enclosures with the concave and convex horizontal walls involving the Rayleigh–Bénard heating is performed via entropy generation approach. The spatial distributions of the temperature (θ), fluid flow (ψ), entropy generation due to heat transfer and fluid friction (S_θ and S_ψ) are discussed extensively for various Rayleigh numbers and Prandtl numbers involving various wall curvatures. A number of complex patterns of spatial distributions of fluid flow and temperature for cavities with concave or convex isothermal walls (top and bottom) have been obtained. The zones of high entropy generation for temperature and fluid flow are detected within cavities with concave and convex horizontal walls. The optimal situation involves the high heat transfer rate with moderate or low entropy generation. Overall, case 3 (highly concave) is found to be optimal over cases 1 and 2 (concave) and cases 1–3 (convex) for all Pr and Ra.     

Onset of Bénard–Marangoni ferroconvection with a convective surface boundary condition: The effects of cubic temperature profile and MFD viscosity

The combined effects of basic cubic temperature profiles and magnetic field dependent (MFD) viscosity on the onset of Bénard-Marangoni convection in a ferrofluid layer are studied. The lower boundary is rigid-isothermal, while the upper free boundary open to the atmosphere is flat and subject to a general thermal boundary condition. The Galerkin technique is employed to extract the critical stability parameters numerically. The results indicate that the basic cubic temperature profiles have a profound influence on the stability characteristics of the system and can be effectively used to either suppress or augment the onset of Bénard–Marangoni ferroconvection. Besides, increasing the magnetic Rayleigh number and the nonlinearity of magnetization hastens, while an increase in the Biot number and MFD viscosity parameter delays the onset of Bénard–Marangoni ferroconvection. The existing results in the literature are obtained as particular cases from the present study.     

Effect of horizontal pressure gradient on Rayleigh–Bénard convection of a Newtonian nanoliquid in a high porosity medium using a local thermal non-equilibrium model

A study of linear and weakly nonlinear stability analyses of Darcy–Brinkman convection in a water–alumina, nanoliquid-saturated porous layer for stress-free isothermal boundaries, when the solid and nanoliquid phases are in local thermal nonequilibrium, is conducted. The critical eigenvalue is found using the Galerkin approach. The effect of the pressure gradient, thermal conductivity ratio, interphase heat transfer coefficient, inverse Darcy number, and Brinkman number on the heat transport and onset of convection is examined and represented graphically. The critical values of wavenumber and nanoliquid Rayleigh number are found for different problem parameter values. The effect of increasing the porosity-modified ratio of thermal conductivity advances the onset of convection and increases the amount of heat transport, whereas the remaining parameters have the opposite impact on the onset of convection and amount of heat transport. The classical results of the local thermal equilibrium case and Darcy–Bénard convection in the presence of pressure gradient are obtained as a limiting case of the present problem.     

Red Eléctrica de España S.A.: Instrument of regulation and liberalization of the Spanish electricity market (1944–2004)

To understand the regulation system of the Spanish electricity market it is first necessary to understand on the one hand the system of tariffs and prices, and on the other the organization of the market for high voltage distribution. This article is concerned with this second aspect and traces its history from 1944, this is because before that date it was not possible to speak of a truly national market, but rather only of regional monopolies. In the 1940s, with Franco's new political regime, and the development of the Spanish electricity sector, it became necessary to completely rethink business strategies in relation to competition and cooperation, as well as the regulatory function of the state. In the 1950s, the main feature of the sector was the system of business self-regulation permitted by the state. Throughout the remaining years of Franco's government state intervention was particularly focussed on the subject of tariffs, but with the onset of democracy the state was to involve itself in the transmission network as well. A debate began as to whether it should be run by a private or public operator. In this dispute were ranged, on the one hand, the economic policy concepts of the major parties (PSOE and PP), and against them the strategic interests of the companies. Although the high voltage transmission network was nationalized by the state in the mid-1980s, establishing a ‘traditional’ model of regulation, the 1990s saw the triumph of a market-based regulation, strongly influenced by the dominant ideas in the European Union, which has converted Red Eléctrica into a private company. Currently the TSO (Transmission System Operators) model has been extended to Portugal and has entered into competition–cooperation with the other models of the European electricity market.