Oscillatory Mode Of Convection Research Articles

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.

On Mixed Oscillatory and Steady Modes of Nonlinear Convection in Mushy Layers

We consider the problem of mixed oscillatory and steady modes 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 two- and three-dimensional weakly nonlinear mixed solutions, and the stability of these solutions with respect to arbitrary three-dimensional disturbances is then investigated. The present investigation is an extension of the problem of mixed oscillatory and steady modes of convection, which was investigated by Riahi (J Fluid Mech 517: 71–101, 2004), where some calculated results were inaccurate due to the presence of a singular point in the equation for the linear frequency. Here we resolve the problem and find some significant new results. In particular, over a wide range of the parameter values, we find that the properties of the preferred and stable solution in the form of particular subcritical mixed standing and steady hexagons appeared to be now in much better agreement with the available experimental results (Tai et al., Nature 359:406–408, 1992) than the one reported in Riahi (J Fluid Mech 517:71–101, 2004). We also determined a number of new types of preferred supercritical solutions, which can be preferred over particular values of the parameters and at relatively higher values of the amplitude of convection.

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.

On nonlinear convection in mushy layers. Part 2. Mixed oscillatory and stationary modes of convection

This paper presents part 2 of a study of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. Part 1 dealt with only the oscillatory modes of convection (Riahi, J. Fluid Mech. vol. 467, 2002, pp. 331–359). In the present paper we consider the particular range of parameters where the critical values of the scaled Rayleigh number just above its lowest subcritical value where convection is possible. Such a preferred solution has properties mostly in agreement with the experimental results due to Tait et al. (Nature, vol. 359, 1992, pp. 406–408) in the sense that the flow is downward at the cell centres, upward at the cell boundaries and there is some tendency for channel formation at the cell nodes.

The onset of penetrative double-diffusive convection

The onset of double-diffusive convection in a horizontal fluid layer is studied. The density is assumed to depend quadratically on the temperature and linearly on the solute concentration. Under the Boussinesq approximation, the linear stability of the conduction state is investigated with respect to the oscillatory and steady convection modes. For steady onset, the critical thermal Rayleigh number is found to be a double-valued function of the solutal Rayleigh number as long as the relative maximum of the density profile exists within the fluid layer. Driving mechanisms of the steady convections are discussed.

On the Linear Stability of Large Stefan Number Convection in Rotating Mushy Layers for a New Darcy Equation Formulation

The Coriolis effect on a solidifying mushy layer is considered. A near-eutectic approximation and large far-field temperature is employed in the current study for large Stefan numbers. The linear stability theory is used to investigate analytically the Coriolis effect on convection in a rotating mushy layer for a new formulation of the Darcy equation. It was found that a large Stefan number scaling allows for the presence of both the stationary and oscillatory modes of convection. In contrast to the problem of a stationary mushy layer, rotating the mushy layer has a stabilising effect on convection. It was observed that increasing the Taylor number or the Stefan number encouraged the oscillatory mode of convection.

On nonlinear convection in mushy layers Part 1. Oscillatory modes of convection

We consider the problem of nonlinear convection in horizontal mushy layers during the solidification of binary alloys. We analyse the oscillatory modes of convection in the form of two- and three-dimensional travelling and standing waves. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the solutions to the nonlinear problem by using a perturbation technique, and the stability of two- and three-dimensional solutions in the form of simple travelling waves, general travelling waves and standing waves is investigated. The results of the stability and the nonlinear analyses indicate that supercritical simple travelling rolls are stable over most of the studied range of parameter values, while supercritical standing rolls can be stable only over some small range of parameter values, where the simple travelling rolls are unstable. The results of the investigation of the onset of plume convection and chimney formation leading to the occurrence of freckles in the alloy crystal indicate that the chimney of the plume can be generated internally or near the lower boundary of the mushy layer. The roles of a Stefan number, a permeability parameter and a concentration ratio on the flow instability in both linear and nonlinear regimes are also determined.

Low‐Frequency Nonradial Oscillations in Rotating Stars. I. Angular Dependence

We obtain the θ-dependence of the displacement vector of rotationally modulated low-frequency nonradial oscillations by numerically integrating Laplace's tidal equation as an eigenvalue problem with a relaxation method. This method of calculation is more tractable than our previous method in which the θ-dependence was represented by a truncated series of associated Legendre functions. Laplace's tidal equation has two families of eigenvalues. In one of these families, an eigenvalue λ coincides with l(l + 1) when rotation is absent, where l is the latitudinal degree of the associated Legendre function, Pml(cos θ). The value of λ changes as a function of ν ≡ 2Ω/ω, where Ω and ω are the angular frequencies of rotation and of oscillation (seen in the corotating frame), respectively. These eigenvalues correspond to rotationally modulated g-mode oscillations. In the domain of | ν | > 1, another family of eigenvalues exists. Eigenvalues belonging to this family have negative values for prograde oscillations, while they change signs from negative to positive for retrograde oscillations as | ν | increases. Negative λ's correspond to oscillatory convective modes. The solution associated with a λ that has a small positive value after changing its sign is identified as an r-mode (global Rossby wave) oscillation. Amplitudes of g-mode oscillations tend to be confined to the equatorial region as | ν | increases. This tendency is stronger for larger λ. On the other hand, amplitudes of oscillatory convective modes are small near the equator.

Oscillatory instabilities of the liquid and mushy layers during solidification of alloys under rotational constraint

Linear flow instabilities due to oscillatory disturbances of the liquid and mushy regions during solidification of binary alloys are investigated under a rotational constraint where the rotation axis is inclined to gravity vector. Results of stability analyses and numerical computations for a preferred centrifugal mode of general oscillatory disturbances at zero and non-zero rotation rates are determined which provide information about the preference of oscillatory flow and its role on the solidification system as modified by the rotational effects. The main results are due to a preferred oscillatory mode of convection which is more significant for non-zero rotation case and is restricted mostly to the mushy region. The preferred oscillatory mode of convection is a traveling wave in the presence of rotation, but it is a standing wave in the absence of rotation. The results for different Prandtl numbers indicate that the freckles formation tendency for metallic alloys is less than that for aqueous solutions. Freckles are imperfections that reduce the quality of the solidified materials.