Rayleigh-Taylor Problem Research Articles

Singularities in water waves and Rayleigh–Taylor instability

This paper is concerned with singularities in inviscid two-dimensional finite-amplitude water waves and inviscid Rayleigh–Taylor instability. For the deep water gravity waves of permanent form, through a combination of analytical and numerical methods, we present results describing the precise form, number and location of singularities in the unphysical domain as the wave height is increased. We then show how the information on the singularity can be used to calculate water waves numerically in a relatively efficient fashion. We also show that for two-dimensional water waves in a finite depth channel, the nearest singularity in the unphysical region has the form as for deep water waves. However, associated with such a singularity, there is a series of image singularities at increasing distances from the physical plane with possibly different behaviour. Further, for the Rayleigh–Taylor problem of motion of fluid over vacuum, and for the unsteady water wave problem, we derive integro-differential equations valid in the unphysical region and show how these equations can give information on the nature of singularities for arbitrary initial conditions. We give indications to suggest that a one-half point singularity on its approach to the physical domain corresponds to a spike observed in Rayleigh-Taylor experiment.

Onset of convection for autocatalytic reaction fronts: Laterally bounded systems.

The linear stability of exothermic autocatalytic reaction fronts that convert unreacted fluid into a lighter reacted fluid is considered using the viscous thermohydrodynamic equations. For upward front propagation and a thin front, the discontinuous jump in density at the front is reminiscent of the Rayleigh-Taylor problem of an interface between two immiscible fluids, whereas the vertical thermal gradient near the front is reminiscent of the Rayleigh-B\'enard problem of a fluid layer heated from below. The problem is also similar to flame propagation, except that here the front propagation speed is limited by catalyst diffusion rather than by activation kinetics. For a thin ascending front and small density changes in a laterally unbounded system, the curvature dependence of the front speed stabilizes perturbations with short wavelengths \ensuremath{\lambda}${\ensuremath{\lambda}}_{\mathit{c}}$, whereas long wavelengths are unstable to convection, indicating that the density discontinuity dominates over thermal gradients. Simple analytical results for the critical wavelength ${\ensuremath{\lambda}}_{\mathit{c}}$ for onset of convection, the growth rate near onset of convection, and the maximum growth rate are found. Agreement with experiments on iodate--arsenous acid solutions in vertical tubes motivates linear and nonlinear calculations in cylindrical geometries.

Onset of convection for autocatalytic reaction fronts: Laterally unbounded system.

The linear stability of exothermic autocatalytic reaction fronts that convert unreacted fluid into a lighter reacted fluid is considered using the viscous thermohydrodynamic equations. For upward front propagation and a thin front, the discontinuous jump in density at the front is reminiscent of the Rayleigh-Taylor problem of an interface between two immiscible fluids, whereas the vertical thermal gradient near the front is reminiscent of the Rayleigh-B\'enard problem of a fluid layer heated from below. The problem is also similar to flame propagation, except that here the front propagation speed is limited by catalyst diffusion rather than by activation kinetics. For a thin ascending front and small density changes in a laterally unbounded system, the curvature dependence of the front speed stabilizes perturbations with short wavelengths \ensuremath{\lambda}${\ensuremath{\lambda}}_{\mathit{c}}$, whereas long wavelengths are unstable to convection, indicating that the density discontinuity dominates over thermal gradients. Simple analytical results for the critical wavelength ${\ensuremath{\lambda}}_{\mathit{c}}$ for onset of convection, the growth rate near onset of convection, and the maximum growth rate are found. Agreement with experiments on iodate--arsenous acid solutions in vertical tubes motivates linear and nonlinear calculations in cylindrical geometries.

Estimating limits on resolution and predictability in computations of hydrodynamically unstable flows

Unstable hydrodynamic flows containing different materials such as Rayleigh-Taylor instability rapidly evolve to a chaotic state with limited predictability and a distribution of materials having a fractral dimension. Here it is shown how Lagrange points, either existing as part of the hydrodynamic solution method or added as passive tracers, can be used to test for self-similarity and to compute the dimension and spatial resolution scaling for the Rayleigh-Taylor instability mixing region. The efficiency of the integral or point counting methods of estimating dimension is discussed and it is shown how it can be increased by order of magnitude factors using random sampling. The problem of how to compute the rate of loss of memory of initial conditions in the transient case is discussed and a quantity similar to a largest Lyapunov exponent is defined and used to determine the predictability time for the Rayleigh-Taylor problem.

Ill-posedness of the rayleigh-taylor and helmholtz problems for incompressible fluids

(1988). Ill-posedness of the rayleigh-taylor and helmholtz problems for incompressible fluids. Communications in Partial Differential Equations: Vol. 13, No. 10, pp. 1265-1295.

Stability in systems with moving contact lines

An energy stability theory is formulated for systems having moving contact lines. The method derives from criteria obtained from the integral mechanical-energy balance manipulated to reflect general material and dynamical properties of moving-contact-line regions. The method yields conditions for both stability and instability and is applied to the two-dimensional Rayleigh-Taylor problem in a vertical slot.

Linear stability of a viscous–inviscid interface

In this paper the stability of the interface separating fluids of widely differing viscosities has been examined. It is shown that a viscous–inviscid (V–I) model offers a consistent zeroth-order approximation to the stability problem. The zeroth-order solution is obtained by neglecting the smallest-order effect, viz., viscosity on the less viscous side of the interface. In this sense, the V–I model significantly differs from the Kelvin–Helmholtz (K–H) approach where both the viscosities are dropped in a single step. A closed form solution for the stability criterion governing the V–I model has been obtained, and a novel instability mechanism is described. It is shown that the V–I model is also a consistent zeroth-order approximation for the Rayleigh–Taylor problem of a viscous–viscous, nonflowing interface when the viscosity ratio tends to zero. For the interface separating two viscous, nonflowing, incompressible fluids, exact solutions for the velocities, pressures, and interface displacement for a disturbance of a given wavelength have been provided for the stable (lighter fluid on top) wave motion. By discussing the roles played by the dynamic and kinematic viscosities, it is made clear why neither the V–I nor the K–H model should apply to the air–water interface. The results of the V–I model compare well with experimental observations. The V–I model serves as an excellent basis for comparison in detailed numerical studies of the viscous–viscous interface.

A minimum dissipation principle for the Rayleigh-Taylor problem in viscous magnetohydrodynamics

A variational principle for magnetohydrodynamic gravitational modes, including isotropic viscosity and the shear of magnetic field lines, is given. It is shown to imply a minimum dissipation requirement for the modes. A sufficient condition for stability given in the literature for the case of ordinary fluids is corrected here.

Thermal instability of nonhomogeneous fluids

The present paper is an attempt to investigate theoretically the stability of a continuously stratified layer of an incompressible, inviscid fluid, statically confined between two horizontal boundaries and heated uniformly from above. The basic density stratification is assumed to be unstable and of the exponential type, namely, p = p o e δz ( δ > 0 being a constant) depending only on the vertical coordinate ( z) while the fluid extends indefinitely in the horizontal directions. The principal results established in the paper are as follows: 1. (a) Neutral modes do not exist. 2. (b) The oscillatory modes of the system are damped irrespective of the unstable basic density stratification while the nonoscillatory modes are amplified irrespective of the stable nonadverse 1 1 The heating is qualified as nonadverse since, on account of thermal expansion, the fluid at the bottom will be heavier than the fluid at the top. heating. 3. (c) An upper bound for the modulus of the dimensionless growth rate of an arbitrary unstable mode is (R 2) 1 2 , where R 2 represents a dimensionless numbers corresponding to the gravitational effects of the basic nonhomogeneity of the fluid. 4. (d) The results are compared with the classical Bénard and Rayleigh-Taylor problems for inviscid fluids.

The Rayleigh—Taylor problem with a vertical magnetic field, including the effects of Hall current and resistivity

The influence of resistivity and Hall current on the Rayleigh-Taylor problem involving two superposed fluids of finite density in the presence of gravitational and magnetic fields normal to the fluid interface is examined. Unlike the related problem in which the magnetic field is parallel to the interface, it appears that the dispersion relation does not exhibit singular behaviour in the zero resistivity limit. The ‘potentially stable’ situation is considered throughout. The results are compared with earlier ideal and resistive theories, and an apparent anomaly regarding the existence of normal modes in such systems is resolved.

Rayleigh—Taylor problem for a Hall plasma

Recent calculations for hydromagnetic analogues of the classical Rayleigh– Taylor problem with Hall effect are compared. It is shown that an early calculation is the limiting situation of the work of Taiwar & Kaira, which confirms that the Hall effect introduces new instabilities.

On the Rayleigh-Taylor problem in magneto-hydrodynamics with finite resistivity

In order to elucidate the importance of the infinite conductivity assumption in MHD a simple problem has been studied. This is a Rayleigh-Taylor problem of two superposed fluids under gravity partially stabilized by a uniform, horizontal magnetic field. It is found that the inclusion of a small, but finite resistivity introduces new and unexpected solutions. For instance, moderately long, stabilized’ waves are now found to grow aperiodically and unexpectedly rapidly at a rate ∝ ($\rm {resistivity}^{\frac{1}{3}$). Other modes are found to be periodic and damped at a rate ∝ ($\rm {resistivity}^{\frac{1}{3}$).