Stratified Liquid Research Articles

The densities of Bushveld Melts: textural and hydrodynamic criteria

A caveat applies to the use of densities of evolving magmatic melts in the modelling of fluid dynamic processes in magma chambers. Models that are dependent on the concepts of stratified liquids within a magma chamber, or the rising of liquid plumes through such columns until a level of neutral buoyancy is reached, often assume that the melts are entirely crystal-free. Both field and physical evidence suggest, however, that this assumption may not be true and that melts at or below the liquidus commonly retain some crystals in suspension. The effective density of the melt will then include the contribution of the suspended crystals. Because of the large-scale density difference between the liquid and some early-crystallized solid phases, as little as ca. 1.5% of suspended load will overwhelm any density changes that may be attributed to the evolution of a cooling, fractionating melt that is assumed to have shed its crystal load. Although there are many factors involved in the dynamics of magma chambers (e.g., evolving viscosity), this density effect alone raises profound doubts about the validity of some current models of magma chambers.

Multifractality of large turbulent fluctuations and the topology of strange attractors.

The connection between the local self-generation of turbulence by the Zaslavsky-Rachko scenario (Zh. Eksp. Teor. Fiz. 76, 2052 (1979) [Sov. Phys. JETP 49, 1039 (1979)]) and the multifractality of turbulent energy is investigated. The effect of the topological singularities of the Zaslavsky attractor on the multifractal asymptotic is shown. The calculated value of the asymptotic generalized dimension is in good agreement with the experimental data. The model has also turned out to be applicable to passive scalar dissipation fields and to the quantitative description of turbulent diffusion in a stable stratified liquid.

Numerical Simulation of Stratified Coating Flow by a Variational Method

We present a method for calculating the two-dimensional steady state flow of stratified Newtonian liquids. Lagrange multipliers are used to impose the free surface conditions. When surface tension is not vanishing, contact angles are easily introduced. Inlet and outlet free sections are treated by means of a technique which allows us to calculate layer thicknesses as part of the result. Parametric studies are thereby facilitated. The solution procedure is derived from a variational approach. At each iteration, a linear system is obtained by linearizing the weak form of the problem. A finite element discretization is carried out on this continuous system. The mesh is adapted to the successive configurations of the free boundaries. III-shaped elements are corrected by exerting artificial forces on the mesh vertices. The global iterative scheme exhibits a quasi-quadratic convergence. The outflow from a slot and the curtain coating flow are analysed and illustrate the power of the method.

Propagation of a boundary disturbance in a stratified gas for arbitrary knudsen number

of waves on a running stream," Quart. J. Mech. Appl. Math., 22, No. 2 (1969). 5. L. V. Cherkesov, Surface and Internal Waves [in Russian], Naukova Dumka, Kiev (1973). 6. A. K. Pramanik, "Waves due to a moving oscillatory surface pressure in a stratified fluid," Trans. ASME, Ser. E. J. Appl. Mech., 41, No. 3 (1974). 7. A. K. Pramanik, "Generation of internal gravity waves in a stratified ocean," Bull. Cal. Math. Soc., 75, No. 1 (1983). 8. F. Tricomi, Differential Equations [Russian translation], IL, Moscow (1962). 9. V. V. Goncharov, "Some features of internal waves in the ocean," in: Tsunami and Internal Waves [in Russian], MGI, Sevastopol' (1976). i0. N.A. Zavol'skii, "Features of the propagation of linear internal waves in a continuously stratified liquid," Izv. Akad. Nauk SSSR, MZhG, No. 1 (1987). ii. R. G. Rehm and H. S. Radt, "Internal waves generated by a translation oscillating body," J. Fluid Mech., 68, No. 2 (1975).

Modeling of the dynamics of the homogeneous turbulence of steadily stratified media

Numerical modeling of the dynamics of the homogeneous turbulence of a steadily stratified liquid has been accomplished in the presence of vertical constant density gradient on the basis of a second-order model, right up to the final stage of degeneration. The structure of relict turbulence has been investigated for values of the molecular Prandtl numbers corresponding to air and a sea wave.

Mixing of stratified liquids by the motion of gas bubbles: application to magma mixing

Mafic magma underlying more silicic magma in a reservoir can exsolve volatiles as it crystallizes, forming bubbles that segregate upwards to the interface with the overlying silicic magma. We describe laboratory experiments designed to investigate how gas bubbles migrate into the more viscous upper layer. In one regime, bubbles move individually across the interface, entraining a small quantity of lower layer fluid and the two layers become progressively stirred into a homogeneous mixture. In a second regime, bubbles form a thin foam layer at the interface which becomes gravitationally unstable, giving rise to two-phase plumes which rise and produce a coarse mixture of the liquids. For a given viscosity ratio, the transition between these regimes occurs when the gas flux is greater than a critical value, which we found to decrease with increasing viscosity ratio. We estimate the critical gas flux for magmatic conditions, and how vesiculation may vary with pressure, degree of crystallization and the composition of volatile components. Mafic inclusions, which are often found in silicic to intermediate lavas, can form in the plume regime; analysis of the instability of the foam predicts wavelengths of between 1 cm and 1 m, which is consistent with the observed sizes of inclusions. Volatiles from the mafic magma may be efficiently transported by this mechanism into the viscous silicic magma as bubble plumes despite low Stokes' velocities for individual vesicles. At small viscosity ratios or low gas fluxes, the bubbling regime can produce hybrid magmas.

Estimation of the dynamic characteristics of an acoustic velocimeter in a continuously stratified liquid

Estimation of the dynamic characteristics of an acoustic velocimeter in a continuously stratified liquid

A new subseismic governing system of equations and its expansions

By appealing to the subseismic approximation (SSA), we establish a new system of two first-order partial differential equations involving displacement field only, which govern the normal modes of a rotating self-gravitating stratified compressible inviscid liquid core. To treat a slightly ellipsoidal Earth model we use a coordinate transformation method which introduces modified spheroidal and toroidal fields so as to avoid trouble with the validity of Taylor expansions near a surface of discontinuity in material properties. The governing equations are expanded in these modified spheroidal and toroidal fields, so that the governing partial differential equations in three-dimensional space become systems of ordinary differential equations in one-dimensional space. As one application of the new subseismic governing system of equations (SGSE) and its expansions, we make an analytical study of the effects of the liquid core on the Chandler wobble (CW) and free core nutation (FCN) using an Earth model which consists of a much simplified mantle and inner core and a more realistic liquid outer core. The results show that the periods of the CW and FCN are independent of the structure of the Earth's liquid core when the radial variation of ellipticity of isopycnal surfaces in the liquid outer core is neglected. As a second application of the SGSE we calculate eigenperiods of some gravity/inertia modes for both non-rotating and rotating stably spherically stratified liquid core with a rigid mantle and inner core. This computation shows the advantage of the new SGSE over the subseismic wave equation (SSWE).

Thermally induced flow of two immiscible stratified liquids in a porous body at a low Rayleigh number

If nuclear waste is buried deep in the rock where the ground water is saline, the generated heat will force the saline water to move upward. If the generated heat is great enough, the saline water will rise through the fresh ground water, which is floating on top of the saline ground water. A theoretical estimate and a simple laboratory experiment with a combined Hele-Shaw and heat conduction cell show that the critical power, required for the interface between saline and fresh water to reach the free surface, is proportional to the relative density difference and to the thickness of the fresh water layer, whereas it is inversely proportional to the volume expansion coefficient of water. The result also shows that the critical power is independent of the distance between the depository and the interface.

The cauchy-poisson waves in an inviscid rotating stratified liquid

Based upon the Boussinesq approximation, an initial-value investigation is made of the axisymmetric Cauchy-Poisson waves generated in an inviscid, rotating, stratified liquid of infinite depth by the initially prescribed free-surface elevation. The asymptotic analysis of the integral solution is carried out by the stationary-phase method to describe the solution for large time and large distance from the source of the disturbance. The asymptotic solution is found to consist of the classical free-surface gravity waves and the internal-inertial waves.

A new method in nonlinear problems of waves in a heavy stratified liquid excited by a vertically moving body

A new method in nonlinear problems of waves in a heavy stratified liquid excited by a vertically moving body

Fluid motion investigation by the photochromic flow visualization technique

The development and application of the photochromic flow visualization method (PFV) in experimental hydrodynamics are described. Tracers in the form of colored lines with a lifetime of 20–300 s are created in a photochromic solution by pulse radiation for 10 −9 s. The displacement of the tracers in the flow is followed by a high-speed movie camera. The information given by the PFV method permits the flow pattern velocity distribution, localization of circulation, and stagnation zones as well as other characteristics of the flow to be found. Several applications are considered such as flows in channels with complicated shapes, flows behind prosthetic heart valves, and thermocapillary and thermogravitational convective flows in homogeneous and stratified liquids.

Real-time coloured visualization of phase flows by the schlieren method

A coloured real-time visualizer of optical inhomogeneities comprising a bichromatic schlieren system, video camera and colour monitor has been developed. The schlieren system represents a function Foucault-Hilbert transformation provided with an amplitude spatial frequency filter, or a quadrant Foucault knife edge. Two colour-coded complementary Toepler-grams are obtained in the exit plane of this schlieren system. Their summed image is then recorded by the video camera and displayed on the screen of the colour monitor. The schlieren photograph of internal gravity waves, generated by the cylindrical body motion in the reservoir filled with the stratified liquid, is presented.

Near field of internal gravity waves excited by a source in a moving stratified liquid

Near field of internal gravity waves excited by a source in a moving stratified liquid

High-speed asymptotic form of the wave resistance of bodies in a uniformly stratified liquid

High-speed asymptotic form of the wave resistance of bodies in a uniformly stratified liquid

Propagation of plane surface waves over an underwater obstacle and a submerged plate

15. V. S. Belyaev and Yu. D. Chashechkin, "Free thermoconcentration convection regimes over a localized heat source," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2 (1989). B. R. Morton, "Buoyant plumes in a moist atmosphere," J. Fluid Mech., ~, Part 2 (1957). A. B. Tsinober, Y. Yahalom, and D. J. Shlien, "A point source of heat in a stable salinity gradient," J. Fluid Mech., 135, 199 (1983). D. E. Mowbray, "The use of schlieren and shadowgraph techniques in the study of flow patterns in density stratified liquids," J. Fluid Mech., 27, Part 2 (1967). S. N. Netreba, "Reaction of stratified rotating media to local thermal effects," Prikl. Mat. Mekh., 50, No. 5 (1986). A. S. Kabanov and S. N. Netreba; "Free convection from a point heat source in a stably stratified medium," Prikl. Mat. Mekh., 46, No. i (1982). Oceanology, Physics of the Ocean, Vol. 2: Ocean Hydrodynamics [in Russian], Nauka, Moscow (1978). : G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Univ. Press (1973). F. Olver, Introduction to Asymptotics and Special Functions, Academic Press, New York (1974). L. N. Sretenskii, Theory of Fluid Wave Motions [in Russian], Nauka, Moscow (1977).

Core dynamics: the two-potential description and a new variational principle

SUMMARY The mathematical description of the normal modes of the rotating self-gravitating stratified compressible liquid core is shown (in the absence of dissipation) to rest exactly on only two scalar fields. These two potentials are governed by a coupled pair of second-order linear partial differential equations which follow from a new variational principle (VP). The latter is simpler for computational purposes than the conventional VP involving four field variables. One of the two new equations reduces to the subseismic wave equation (SSWE) when the effect of flow pressure on compression is neglected. A VP for the SSWE exists only when the Earth model is further constrained, but the VP for the two-potential formulation of core dynamics is valid, without approximation, for any core stratified in accord with hydrostatic equilibrium, with elastically deformable oblate boundaries.

Influence of the marangoni effect on the stability of flows of stratified liquids

A study is made of the effect of a change in interfacial tension on the stability of shear flows of two-component systems. A linear analysis of the stability of motion is performed using the concept of waves of negative energy.

An investigation of the motion of homogeneous and stratified liquids in rectangular cavities by the method of photochromic visualization

A method of photochromic visualization has been developed for investigating thermogravitational flows in homogeneous and stratified fluids in a closed volume.

Vibrations of a flat disk at the interface between two stratified liquids

The problem of the vibrations of a flat two-sided disk at the interface between two stratified liquids is considered. The explicit solution of the problem is constructed, and both the behaviour of the solution for long times and the states of stationary vibrations are studied using the methods of the theory of dynamic potentials.