Nonlinear Oscillatory Convection Research Articles

The Effect of Throughflow on Weakly Nonlinear Convection in a Viscoelastic Saturated Porous Medium

In this paper we have investigated the effect of throughflow on thermal convection in a viscoelastic fluid saturated porous media. The governing equations are modelled in the presence of throughflow. These equations are made dimensionless and the obtained nonlinear problem solved numerically. There are two types of throughflow effects on thermal instability inflow and outflow investigated by finite amplitude analysis. This finite amplitude equation is obtained using the complex Ginzburg-Landau amplitude equation (CGLE) for a weak nonlinear oscillatory convection. The heat transport analysis is given by complex Ginzburg-Landau amplitude equation (CGLE). The numerical results indicate that due to the non-uniform throughflow there is instability at the bottom plate and influence the heat transfer in the system. The vertical throughflow is having both stable and unstable modes depending on flow direction. The nature of viscoelastic fluid is having both effects either stabilize or destabilize. Further, it is found that the nonlinear throughflow effects have dual role on heat transport. The solutions of the present problem are obtained numerically by using Runge-Kutta fourth order method.

Weakly Nonlinear Oscillatory Convection in a Viscoelastic Fluid Saturated Porous Medium with Throughflow and Temperature Modulation

The effect of vertical throughfow and temperature modulation on a viscoelastic fluid saturated porous medium has been investigated. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small and the disturbances are expanded in terms of power series of amplitude of convection. A weak nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the non autonomous complex Ginzburg- Landau equation, is calculated. The effect of vertical through flow is found to stabilize the system irrespective of the direction of through flow in the case of permeable boundary conditions. The time relaxation has a destabilizing effect, while the time retardation parameter has a stabilizing effect on the system. The effects of amplitude and frequency of modulation on heat transport have been analyzed and depicted graphically. The study shows that the heat transport can be controlled effectively by a mechanism that is external to the system. Further, it is also found that heat transfer is more in oscillatory mode of convection rather than in stationary mode of convection.

Weakly Nonlinear Oscillatory Convection in an Electrically Conduction Fluid Layer Under Gravity Modulation

The time dependant and constant gravity field is investigated on electrically conducting fluid layer using weakly nonlinear stability analysis. The small amplitude perturbation is imposed on physical quantities like fluid velocity, energy and magnetic flux for simplifying nonlinear model. An amplitude equation is derived from solvability condition for the oscillatory mode of convection. Using this amplitude heat transfer is measured in terms of the Nusselt number. Also the effect of various system parameters are discussed on heat transfer. The effect of Prandtl number, magnetic Prandtl number and amplitude of modulation is to enhance the heat transfer. While the Chandrasekhar number, modulation frequency is to reduce the heat transfer. Further, it is found that, heat transfer is more for oscillatory mode than the stationary mode. Finally it is also found that, gravity modulation can be effectively used in either enhancing or diminishing the heat transfer.

Weakly Nonlinear Oscillatory Convection in a Rotating Fluid Layer Under Temperature Modulation

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal rotating fluid layer. The temperature at the boundaries has been taken to be time-periodic, governed by the sinusoidal function. A weakly nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex form of Ginzburg–Landau equation (CGLE), is calculated. The influence of external controlling parameters such as amplitude and frequency of modulation on heat transfer has been investigated. The dual effect of rotation on the system for the oscillatory mode of convection is found either to stabilize or destabilize the system. The study establishes that heat transport can be controlled effectively by a mechanism that is external to the system. Further, the bifurcation analysis also presented and established that CGLE possesses the supercritical bifurcation.

Weak Nonlinear Oscillatory Convection in a Viscoelastic Fluid-Saturated Porous Medium Under Gravity Modulation

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal porous medium saturated with viscoelastic fluid. Modified Darcy’s law is used to explain characteristics of fluid motion. The time periodic gravity field has been considered, and its effect on the system has been investigated. A weak nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex non-autonomous Ginzburg–Landau equation, is calculated. The influence of relaxation and retardation times of viscoelastic fluid on heat transfer has been discussed. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Finally, it is found that overstability advances the onset of convection, and hence increases heat transfer.

Weakly nonlinear oscillatory convection in a viscoelastic fluid saturating porous medium under temperature modulation

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal porous medium saturated with viscoelastic fluid. Modified Darcy’s law is used to explain characteristics of fluid motion. The time-periodic temperature on the boundaries has been considered and its effect on the system has been investigated. A weak nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex non-autonomous Ginzburg–Landau equation, is calculated. The influence of relaxation and retardation times of viscoelastic fluid on heat transfer has been discussed. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Finally, it is found that supercritical flow advances the onset of convection hence increases heat transfer.

Weak non-linear oscillatory convection in a viscoelastic fluid layer under gravity modulation

Using a complex non-autonomous Ginzburg–Landau equation, we have investigated oscillatory Rayleigh–Bénard convection under Gravity modulation, considering a viscoelastic fluid layer. The influence of (stress) relaxation and (strain) retardation times of viscoelastic fluid on heat transfer has been discussed. The study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Modulation has a destabilization effect at low frequencies and a stabilization effect at high frequencies, which increases with increasing the amplitude of modulation. Finally, it is found that overstability advances the onset of convection, hence increases heat transfer.

On oscillatory modes of nonlinear compositional convection in mushy layers

We consider the problem of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of weakly nonlinear oscillatory solutions, and the stability of these solutions with respect to arbitrary disturbances is then investigated. The present investigation is an extension of the problem of the oscillatory modes of convection, which was investigated by [D.N. Riahi, On nonlinear convection in mushy layers. Part 1. Oscillatory modes of convection, J. Fluid Mech. 467 (2002) 331–359] in the absence of the main permeability parameter K 1 and very recently by [P. Guba, M.G. Worster, Nonlinear oscillatory convection in mushy layers, J. Fluid Mech. 553 (2006) 419–443] for two-dimensional cases in the presence of K 1 , to include the effects of K 1 , over a range of values of the other parameters, and for both two- and three-dimensional motion. It was found, in particular, that the results reported in [D.N. Riahi, On nonlinear convection in mushy layers. Part 1. Oscillatory modes of convection, J. Fluid Mech. 467 (2002) 331–359; P. Guba, M.G. Worster, Nonlinear oscillatory convection in mushy layers, J. Fluid Mech. 553 (2006) 419–443] are recovered if K 1 is zero or sufficiently small. In such cases two-dimensional solutions in the form of simple-travelling rolls are mostly the only stable and preferred solutions. However, as in the more realistic cases, if K 1 is not sufficiently small, then such solutions are replaced by preferred and stable three-dimensional solutions, which are mostly simple-travelling waves in the form of rectangles, squares or hexagons.

Nonlinear oscillatory convection in mushy layers

We study the problem of nonlinear development of oscillatory convective instability in a two-dimensional mushy layer during solidification of a binary mixture. We adopt the near-eutectic limit, making the problem analytically tractable using standard perturbation techniques. We consider also a distinguished limit of large Stefan number, which allows a destabilization of the system to an oscillatory mode of convection. We find that either travelling waves or standing waves can be supercritically stable, depending strongly on the sensitivity of permeability of the mushy layer to variations in the local solid fraction: mushy-layer systems with relatively weak sensitivity are more likely to select travelling waves rather than standing waves in the nonlinear regime. Furthermore, the decrease in permeability is found to promote the subcritical, and hence more unstable, primary oscillatory states. In addition to mapping out the location of different stable oscillatory patterns in the available parameter space, we give the detailed spatio-temporal structure of the corresponding thermal, flow and solid-fraction fields within the mushy layer, as well as the local bulk composition in the resulting eutectic solid.

Nonlinear oscillatory convection in rotating mushy layers

We consider the problem of nonlinear oscillatory convection in a horizontal mushy layer rotating about a vertical axis. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the stable and unstable oscillatory solutions of the weakly nonlinear problem by using perturbation and stability analyses. It was found that depending on the values of the parameters, supercritical simple travelling modes of convection in the form of hexagons, squares, rectangles or rolls can become stable and preferred, provided the value of the rotation parameter increases beyond some value, and each subcritical form of the oscillatory convection is unstable. In contrast to the non-rotating case, qualitative properties of the left-travelling modes of convection are different from those of the right-travelling modes, and such qualitative difference is found to be due to the interactions between the local solid fraction and the Coriolis term in the momentum-Darcy equation.

Fully nonlinear oscillatory convection in a rotating layer

Two-dimensional overstable convection in a rotating layer is studied for large Taylor numbers. In this regime, the leading order nonlinearity arises from the distortion of the horizontally averaged temperature profile. Fully nonlinear solutions in the form of traveling and standing waves are obtained via an asymptotic expansion in the Taylor number. The formulation leads to a nonlinear eigenvalue problem for the vertical structure and frequency of each solution type and allows the computation of the Nusselt number and frequency as functions of the applied Rayleigh number.

Pattern selection in rotating convection with experimental boundary conditions.

present the results of weakly nonlinear calculations that enable us to compute accurately both the critical rotation rate beyond which rolls are Kuppers-Lortz unstable as well as the angle with which the instability sets in. These quantities are a sensitive function of the Prandtl number, and here the results differ considerably from those with more idealized boundary conditions. The remainder of the paper is concerned with weakly nonlinear oscillatory convection. The calculations are restricted to two spatial dimensions so that the nonlinear problem concerns only traveling and standing waves. We determine the relative stability between these two patterns and show that either may be stable, depending on the system parameters. In this discussion we are careful to consider the impact of distant sidewalls in preventing the appearance of the mean flows that are associated with traveling patterns in unbounded systems. As in the Kiippers-Lortz calculation we consider stability only with respect to perturbations of the same wave number as the basic pattern. Although this procedure excludes a large class of potentially important perturbations, prior experience suggests that the selected perturbations are the most dangerous. In addition this restriction enables us to rely on results obtained from equivariant bifurcation theory. This technique [6] allows us to classify the possible weakly nonlinear and spatially periodic states and establish their relative stability. In addition it identifies precisely the computations that have to be carried out on the partial diiferential equations in order to discriminate among the possible scenarios. The paper is organized as follows. In the following section we summarize the relevant bifurcation theory results. In Sec. III we formulate the hydrodynamical equations and describe the technique we use to solve them. In Sec. IV we solve the linear stability problem for the conduction solution and identify regions in parameter space where oscillatory and steady-state convection takes place. In Sec. V we restrict the region of stable smallamplitude rolls by computing the location of the onset of

Weakly nonlinear oscillatory convection in a rotating fluid

The problem of weakly nonlinear two- and three-dimensional oscillatory convection in the form of standing waves is studied for a horizontal layer of fluid heated from below and rotating about a vertical axis. The solutions to the nonlinear problem are determined by a perturbation technique and the stability of all the base flow solutions is investigated with respect to both standing wave and travelling wave disturbances. The results of the stability and the nonlinear analyses for various values of the rotation parameter τ and the Prandtl number P (0 ≼ P < 0.677) indicate that there is no subcritical instability and that all the base flow solutions are unstable. Disturbances with highest growth rates are found to be some particular disturbances superimposed on two-dimensional base flow. Particular standing wave disturbances parallel to two-dimensional base flow are the most unstable ones either for sufficiently small P or for intermediate values of P with τ below some critical value τ *. Travelling wave disturbances inclined at an angle of about 45° to the wave vector of two-dimensional base flow are the most unstable disturbances either for P sufficiently close to its upper limit or for intermediate values of P with τ ≽ τ *. The dependence on P and τ of the nonlinear effect on the frequency and of the heat flux are also discussed.

Nonlinear oscillatory convection in the presence of a vertical magnetic field

Convection rolls in a fluid layer heated from below become unstable to disturbances in the form of waves travelling along the axis of the rolls when the Rayleigh number exceeds a critical value R II . This transition to a time-dependent form of convection also occurs in the presence of a vertical magnetic field when the fluid is electrically conducting. In this paper the finite-amplitude properties of these waves are investigated for the values 0.1 and 0.025 of the Prandtl number. It is shown that the onset of oscillations reduces the heat transport by convection and that a mean flow in the direction of propagation is associated with the waves. Although the magnetic field has an inhibiting influence on steady convection, the inhibiting influence on the onset of oscillations is even stronger such that in some cases a higher heat transport is obtained in the presence of a magnetic field than in its absence. For similar reasons subcritical finite-amplitude onset of travelling-wave convection occurs for sufficiently large magnetic field strengths. Finally the stability of travelling-wave convection is investigated and the Rayleigh number R III for the transition to asymmetric wave convection is determined.

Nonlinear oscillatory convection: A quantitative phase dynamics approach

The oscillatory instability of convective rolls, for both free-slip and rigid boundary conditions, is shown to be related to the translational and Galilean invariances of the Oberbeck-Boussinesq equations, which implies that the phase dynamics of the basic roll pattern is second order in time. In the free-slip case the invariance is exact and the instability comes in at zero wavenumber. It is argued that, for low Prandtl number fluids, the effect of the rigid boundaries is to weakly break the Galilean invariance, thereby shifting the instability's critical wavenumber to a finite value. We derive the equations governing the nonlinear phase dynamics and show that, when supercritical, the instability always saturates into travelling waves (as observed in experiments and numerical simulations). In the rigid case, the description is made quantitative by numerically computing both the linear and nonlinear coefficients of the phase equation. It is found that, depending of the values of Prandtl number P and basic roll pattern wavenumber α, the oscillatory instability can be supercritical or subcritical. In the case of mercury ( P = 0.025), our model predicts transition from supercritical onset to subcritical onset, for α slightly below the critical wavenumber for convection onset.

Nonlinear oscillatory convection

A numerical analysis has been performed of three-dimensional time-dependent solutions which bifurcate supercritically from two-dimensional convection-roll solutions at the onset of the oscillatory instability. The bifurcating solutions describe a periodic shifting forward and backward of the convection rolls and lead to a strong deformation of the rolls as the Rayleigh number increases. Since the bifurcating solution is stable in the form of a travelling wave, the computational expense can be reduced by assuming a moving coordinate. Travelling-wave solutions have been computed in the case of rigid boundaries as a function of the Prandtl number and of the two basic wavenumbers αx, αy of the problem. The onset of oscillations reduces the heat transport in comparison with that of two-dimensional rolls because the occupation of a new degree of freedom of motion by the oscillation reduces the energy of the heat-transporting component of convection. A limited stability analysis of finite-amplitude travelling waves has been performed and the onset of an asymmetric mode of oscillations is determined as a function of the parameters of the problem. This mode appears to be identical with a mode that was observed in the numerical simulations of Lipps (1976) and McLaughlin & Orszag (1982).

Oscillatory and competing instabilities in a nonlinear model for Langmuir circulations

A five-mode truncation of the full partial differential equations which describe Langmuir circulations is studied. Instabilities of different kinds are possible depending on the relative strengths of a stabilizing vertical density stratification and a destabilizing vertical Stokes drift gradient. For sufficiently small buoyancy, steady convection obtains, whereas for larger values of the buoyancy oscillatory convection obtains. In addition the competition can create multiple instabilities. Techniques from dynamical systems theory are used to derive equations which describe nonlinear oscillatory convection and the multiple bifurcation, whose character has been explored in the literature. The center manifold and normal form transformations are explained in a direct and elementary manner in order to be clear to readers encountering these methods for the first time. The problem treated may also be interpreted as one of double diffusion, but with boundary conditions not usually considered. The results are discussed in terms of the physics of Langmuir circulations.