Nonlinear Convective Stability Research Articles

Nonlinear Convective Stability of a Critical Pulled Front Undergoing a Turing Bifurcation at Its Back: A Case Study

Nonlinear Convective Stability of a Critical Pulled Front Undergoing a Turing Bifurcation at Its Back: A Case Study

Comparison of the effects of three types of time-periodic body force on linear and non-linear stability of convection in nanoliquids

Abstract The effect of three different types of time-periodic vertical oscillations of the Rayleigh–Benard system involving Newtonian nanoliquids is studied in the paper. Twenty nanoliquids are considered for investigation and it is found that enhanced heat transport is seen in all of them compared to what is observed in the corresponding carrier liquids without the nanoparticles. The increase in the amplitude of the vertical vibration is shown to augment convection in the nanoliquids whereas increase in frequency is to inhibit convection. On comparing the extent of heat transport in the presence of modulation with that in its absence, it was found that heat transport diminished in the former case. It was also found that the trigonometric sine type of gravity modulation transports heat intermediate to that of triangular and square types. The triangular type inhibits heat transport while square type enhances heat transport leading one to conclude that heat transport in a Rayleigh–Benard system can be enhanced by using a combination of dilute concentration of nanoparticles and gravity modulation using a square wave.

Couple stresses effect on linear instability and nonlinear stability of convection in a reacting fluid

We study the problem of convective movement of a reacting solute in a viscous incompressible fluid occupying a plane layer and subjected to a couple stresses effects. The thresholds for linear instability are found and compared to those derived by a global nonlinear energy stability analysis. In particular, we analyse the effect of no-slip boundary conditions on the stability and instability of convection. The conditions of no-slip at the boundary with couple stresses effect and non constant coefficients which are analysed for the first time in this article.

Nonlinear stability analysis of Darcy's flow with viscous heating.

The nonlinear stability of a rectangular porous channel saturated by a fluid is here investigated. The aspect ratio of the channel is assumed to be variable. The channel walls are considered impermeable and adiabatic except for the horizontal top which is assumed to be isothermal. The viscous dissipation is acting inside the channel as internal heat generator. A basic throughflow is imposed, and the nonlinear convective stability is investigated by means of the generalized integral transform technique. The neutral stability curve is compared with the one obtained by the linear stability analysis already present in the literature. The growth rate analysis of different unstable modes is performed. The Nusselt number is investigated for several supercritical configurations in order to better understand how the system behaves when conditions far away from neutral stability are considered. The patterns of the neutrally stable convective cells are also reported. Nonlinear simulations support the results obtained by means of the linear stability analysis, confirming that viscous dissipation alone is indeed capable of inducing mixed convection. Low Gebhart or high Péclet numbers lead to a transient overheating of the originally motionless fluid before it settles in its convective steady state.

Nonlinear Stability of Convection in a Porous Layer with Solid Partitions

We show that for many classes of convection problem involving a porous layer, or layers, interleaved with finite but non-deformable solid layers, the global nonlinear stability threshold is exactly the same as the linear instability one. The layer(s) of porous material may be of Darcy type, Brinkman type, possess an anisotropic permeability, or even be such that they are of local thermal non-equilibrium type where the fluid and solid matrix constituting the porous material may have different temperatures. The key to the global stability result lies in proving the linear operator attached to the convection problem is a symmetric operator while the nonlinear terms must satisfy appropriate conditions.

Global nonlinear stability of convection in a heat generating fluid filled channel with a moving boundary

The energy method is employed to investigate the stability of a steady convective flow in a heat generating fluid arising due to the combined effect of buoyancy, shear and pressure gradient. By introducing a suitable generalized energy functional and using energy inequalities sufficient conditions for the existence of such a flow are found. An analysis through the variational principles is then made to find sharper limits for nonlinear stability. Comparisons are made with linear results in the literature and it is shown that the linear theory fails to capture the physics of the onset of secondary flow.

Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities

Reaction–diffusion equations on the real line that contain a control parameter are investigated. Of interest are travelling front solutions for which the rest state behind the front undergoes a supercritical Turing or Hopf bifurcation as the parameter is increased. This causes the essential spectrum to cross into the right half plane, leading to a linear convective instability in which the emerging pattern is pushed away from the front as it propagates. It is shown, however, that the wave remains nonlinearly stable in an appropriate sense. More precisely, using the fact that the instability is supercritical, it is shown that the amplitude of any pattern that emerges behind the wave saturates at some small parameter-dependent level and that the pattern is pushed away from the front interface. As a result, when considered in an appropriate exponentially weighted space, the travelling front remains stable, with an exponential in time rate of convergence.

Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities

Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities

Nonlinear convective stability problems of viscoelastic fluids in finite domains

A Chebyshev pseudospectral method is generalized to solve the nonlinear hydrodynamic stability problems of Rayleigh-Benard convection of viscoelastic fluids in finite domains, which are compatible with the experimental situations, for the range of viscoelastic parameters where the exchange of stabilities is valid. The effects of box aspect ratio, the Deborah number λ and the dimensionless retardation time e on the critical Rayleigh number and convection intensity are investigated. The comparison of these results with the experimental data might be used to guide the selection of constitutive equations and to estimate viscoelastic parameter values. The present technique of hydrodynamic stability analysis is quite versatile and can be employed to solve other hydrodynamic stability problems in finite domains.

A Solution Method of Nonlinear Convective Stability Problems in Finite Domains

A Chebyshev pseudospectral method is generalized to solve the linear and nonlinear hydrodynamic stability problems of thermal convection in a two-dimensional rectangular box with rigid sidewalls, where there may exist a heat source or a magnetic field to enhance or suppress the convection. The incompressibility condition is imposed rigorously on all boundaries. The effects of box aspect ratio, heat source, and magnetic field on the critical Rayleigh number and convection cell size are examined and compared with the results of other investigators. We have extended the present technique to nonlinear stability analysis and derived the Landau equation that describes the temporal evolution of the strength of convection in the rectangular box with rigid sidewalls. The results of nonlinear stability analysis are compared with the exact results obtained by the numerical solution of the Boussinesq equation. The present technique solves linear and nonlinear convective stability problems accurately and can be employed to solve other hydrodynamic stability problems in finite domains.

Non-linear convection in a porous medium with inclined temperature gradient and variable gravity effects

The energy method is developed to discuss the non-linear stability of convection in a horizontal porous layer subjected to an inclined temperature gradient and a variable gravity field. Both linear and non-linear stability analyses are carried out for a large number of parameter values. The eigenvalue problems in both cases are solved by the Chebyshev tau-QZ method with optimization routine. It is found that the preferred mode at the onset of convection is a longitudinal mode and that a decrease in gravity variation has a stabilizing effect on the system. Comparisons between the linear and energy stability results show that as the horizontal Rayleigh number increases the difference between the two results increases and thus indicates the possibility of subcritical instability.

Freckle formation in a solidifying binary alloy

The effect of solutal solidification on the unidirectional solidification of a binary alloy cooled from below is considered. Soon after the onset of convection a mushy region often forms, and is accompanied by vigorous convection in the molten alloy above. In contrast, the fluid in the mush appears unaffected by the neighbouring flow and remains essentially quiescent. This work considers the nonlinear convective stability of the mush and determines a criterion for channelling in the mush to occur. The basic state is a similarity solution, so that a quasi-static approximation must be applied in order to apply conventional stability theory. Moreover, although the model for solidification is relatively simple, an analytical expression for linear stability is not available. Thus the series of equations arising from the nonlinear stability analysis lead to a complicated set of symbolic and numerical calculations. Stable finite-amplitude solutions are found for Rayleigh numbers larger than critical for all values of the chosen superheat. The nonlinear solutions demonstrate the possibility of channelling within the mush dependent upon the strength of convection. These finite-amplitude solutions are extended further by the calculation of a numerical solution of the model equations. The evolution of the stream function and the mass fraction is followed, the onset of convection and freckling can then be deduced from the numerical simulation. The onset of convection in the mush is found in terms of the mush Rayleigh number, and compares favourably with linear stability theory and experimental data. The onset of freckling is also given in terms of a Rayleigh number, but is sensitive to the initial conditions. This appears to explain the large disagreement found in experiments aimed at finding a criterion for freckling.

Non-linear stability of convection in a porous medium with inclined temperature gradient

The energy method is used to discuss the non-linear stability of convection in a horizontal porous layer subjected to an inclined temperature gradient. The compound matrix method is used to solve the associated eigenvalue problem. It is noted that linear instability is superceded by subcritical finite amplitude instability.

Nonlinear Convective Stability in a Porous Medium with Temperature‐Dependent Viscosity and Inertial Drag

A nonlinear stability analysis of convection in a fluid‐saturated porous medium with temperature‐dependent viscosity and inertia drag is presented. It is shown that the quadratic drag term is mathematically important and physically significant in ensuring conditional stability.

Nonlinear stability of convection induced by inclined thermal and solutal gradients

The energy method, giving the sufficient condition for stability, is developed for the convection problem induced by inclined thermal and solutal gradients in a horizontal layer of a saturated porous medium. The boundaries are taken to be perfectly conducting and Darcy's law is employed to represent the porous medium. A nonlinear stability analysis is performed and compound matrix method is employed for numerical calculations. The optimal stability bound is computed and numerical results are compared with the linear theory for different parameter values.

Finite amplitude convection in a two-component fluid saturated porous layer

The linear and non-linear stability of convection of a two-component fluid known as thermohaline convection is considered in a horizontal porous layer heated from below. The analysis is based on the Boussinesq-Darcy equations for 2-dim. convection under the assumption that the amplitudes of convection are small. The linear theory is based on the Fourier analysis and the critical Rayleigh numbers for both marginal and overstable motions are determined. It is found that a vertical solute gradient sets up overstable motions and a physical reason for this is given. The finite amplitude study is based on a truncated representation of Fourier series and the critical Rayleigh number is determined. The effects of Prandtl number, ratio of diffusivities and the permeability parameter on convection are studied. Nusselt number, Nu, and its analog Nu s for solute are calculated and it is found that the effect of Prandtl number is very weak in contrast to the existing viscous flow results.