Layer Fluid System Research Articles

The effect of porosity and permeability variations on convective stability of a two-layer system under longitudinal vibration in zero gravity

The effect of the porosity and permeability variations on the onset of average convection in a single-component fluid within an non-uniformly heated horizontal layer partially filled with a porous zone under zero gravity conditions is studied. The system of the fluid and porous layers oscillates as a whole with high frequency and small amplitude in the longitudinal direction. Permeability is uniform within the porous zone and related to porosity by the Carman–Kozeny formula. The method of constructing a fundamental system of solutions as well as the orthogonalization of vectors for partial solutions are applied to simulate a linear stability problem with respect to the fluid quasi-equilibrium state in the layers numerically. The instability threshold relative to the short-wave and long-wave convective rolls is found. An abrupt change in the type of instability from the short-wave to long-wave ones occurs with the growth of the porosity from 0.3 to 0.8. A similar variation also presents in the case of thermal gravitational convection in layered fluid systems with porous zones under Earth conditions. Convection in weightlessness distinguishes from that in the gravitational field by a non-monotonic behavior of the instability threshold with increasing porosity at different fixed frequencies of vibration. The onset of convection delays at low frequencies and, on the contrary, speeds up at high enough frequencies, if the porosity belongs to the interval from 0.3 to 0.8. In addition, one studies the effect of two types of boundary conditions for tangential velocities near the interface between the layers on the instability threshold in the system with a low permeable porous zone.

The third-order asymptotic solutions in the Lagrangian description for interfacial internal waves in a three layer fluid system

In this paper, we discuss the interfacial internal waves with a rigid boundary in a three-layer fluid system, where the density of the upper layer fluid is smaller than that of the lower layer. With the Lagrangian matching conditions at the interfaces, the first-order solutions, the second-order solutions and the third-order asymptotic solutions for the interfacial internal waves are obtained in the Lagrangian description using the perturbation method, and the mass transport velocity, the wave frequency, the mean level and the particle trajectory are also given. The results show that the discontinuities across the interfaces appear for the mass transport velocity, wave frequency and mean level, but we find that these discontinuities may disappear if the water depth ratio and the density ratio of the three layer fluids satisfy certain conditions.

Three dimensional simulation of natural convection and entropy generation in an air and MWCNT/water nanofluid filled cuboid as two immiscible fluids with emphasis on the nanofluid height ratio's effects

Abstract This paper reports a numerical study of the natural convection and entropy generation for a layered fluid system in a cuboid enclosure which is differentially heated from sides and filled by two immiscible gas/liquid fluids (air and MWCNTs-water nanofluid). This study emphasize on the effects of the liquid interface aspect ratio, solid volume fraction of nanofluid, and Rayleigh number on the fluid flow, heat transfer and entropy generation. The fluid flow streamlines, contours of local fluid friction irreversibility, temperature, and local heat transfer irreversibility in the 3D enclosure are obtained. The local and averaged Nusselt number and volumetric entropy generation are also discussed. The results show that the nanofluid interface aspect ratio has significant influences on the fluid flow, heat transfer performance and entropy generation. Moreover, it is shown that the total entropy generation and mean Nusselt number are reduced by increasing of nanofluid interface aspect ratio and are enhanced by increasing of Rayleigh number.

A multi-layer model for nonlinear internal wave propagation in shallow water

AbstractIn this paper, a multi-layer model is developed for the purpose of studying nonlinear internal wave propagation in shallow water. The methodology employed in constructing the multi-layer model is similar to that used in deriving Boussinesq-type equations for surface gravity waves. It can also be viewed as an extension of the two-layer model developed by Choi & Camassa. The multi-layer model approximates the continuous density stratification by an $N$-layer fluid system in which a constant density is assumed in each layer. This allows the model to investigate higher-mode internal waves. Furthermore, the model is capable of simulating large-amplitude internal waves up to the breaking point. However, the model is limited by the assumption that the total water depth is shallow in comparison with the wavelength of interest. Furthermore, the vertical vorticity must vanish, while the horizontal vorticity components are weak. Numerical examples for strongly nonlinear waves are compared with laboratory data and other numerical studies in a two-layer fluid system. Good agreement is observed. The generation and propagation of mode-1 and mode-2 internal waves and their interactions with bottom topography are also investigated.

Generation of internal and surface waves by seafloor movement in a two layer fluid system

Internal and surface waves generated by the deformations of the solid bed in a two layer fluid system of infinite lateral extent and uniform depth are investigated. An integral solution is developed for an arbitrary bed displacement on the basis of a linear approximation of the complete description of wave motion using a transform method (Laplace in time and Fourier in space) analogous to that used to study the generation of tsunamis by many researchers. The theoretical solutions are presented for three interesting specific deformations of the seafloor; the spatial variation of each seafloor displacement consists of a block section of the seafloor moving vertically either up or down while the time-displacement history of the block section is varied. The generation process and the profiles of the internal and surface waves for the case of the exponential bed movement are numerically illustrated, and the effects of the deformation parameters, densities and depths of the two layers on the solutions are discussed. As expected, the solutions derived from the present work include as special cases that obtained by Kervella et al. [Theor Comput Fluid Dyn 21:245-269, 2007] for tsunamis cased by an instantaneous seabed deformation and those presented by Hammack [J Fluid Mech 60:769-799, 1973] for the exponential and the half-sine bed displacements when the density of the upper fluid is taken as zero.

Sloshing of a layered fluid with a free surface as a Hamiltonian system

The aim of this paper is the investigation of a layered sloshing fluid system using both a new Hamiltonian mathematical model and new laboratory experiments. The mathematical model is defined for a cylindrical tank with an arbitrary shape and subjected to an arbitrary rigid motion. The model consists of a pure evolution system of partial differential first order equations in the canonical four unknowns: water elevation at the upper free surface and at the separation surface and the gap in momentum potential density computed at each fluid surface. The system of equations is obtained by avoiding the construction of the Hamiltonian and its variational derivatives. An important advantage of this formulation, with respect to the Lagrangian formulation, is that the nonevolution constraint, which imposes for each fluid the same velocity component along the normal direction of the separation surface, is fulfilled by the model itself. The model implementation needs to define the so-called Neumann–Dirichlet operators, which are computed by an efficient algorithm, for any instantaneous configuration of the two fluid domains. A numerical integration of the model is performed by a suitable Galerkin projection of the evolution equations. New laboratory experiments, simulating the sloshing of a layered fluid system, inside a tank with a squared cross section, were performed. The experiments, with a two-dimensional sloshing, were carried out by varying the forcing frequency, the oscillation amplitude and the ratio of the two fluids’ depths. In some experiments, a traveling wave, with a shape similar to a moving hydraulic jump, was observed at the separation surface. Measurements of the space-time evolution of both the free and the separation surfaces were performed and compared with the model’s predictions. A good agreement between the model predictions and laboratory measurements is found, even for strong nonlinear cases such as the traveling wave.

Analysis of experimental data on internal waves with statistical method

PurposeThis study seeks to develop a systematic means of identifying regression models using a complex regression model with a statistical method.Design/methodology/approachAs a widely adopted statistical scheme for analyzing multifactor data, regression analysis provides a conceptually simple algorithm for examining functional relationships among variables. This investigation assesses the proposed relationship using a sample of data in regression analysis and then estimates the fit using statistics. Furthermore, several algorithms and added variable plots are presented to obtain an appropriate regression model and the relationship between response variables y, p and explanatory variables x0,x1,x2, … ,xp.FindingsThe proposed statistical scheme is demonstrated by the analysis of experimental data on internal waves, in which the results can well illustrate what has been investigated in laboratory experiment and may be applicable to the naturally occurring reflection of internal waves from sloping bottoms.Practical implicationsIn previous studies, field observations of internal waves were carried out. Owing to the limit of stationary measurement in situ, sufficient experiment data were not easily obtained. On the other hand, data collected by laboratory experiments express more information on wave mechanism, such as energy dissipation, mixing efficiency, and stratification thickness in a stratified layer fluid system. In present study, the authors supply valuable experiment data and analytic method, which will be a great contribution to the geophysical fluid dynamics. The results illustrate well what has been investigated in laboratory experiment and may be applicable to the naturally occurring reflection of internal waves from sloping bottoms.Originality/valueMore recently, it has been proposed that internal wave mixing may contribute significantly to internal mixing in the ocean and hence has an important influence on world climatic changes. Based on the statistical algorithm and regression model, the reduction in internal wave energy can be predicted on a sloping bottom due to frictional effect. Since, interaction between internal waves and uniform slopes has occurred in an estuary, a lake or in the ocean, the results available in this paper would benefit future study on internal wave hydrodynamics.

Hamiltonian formulation for the motion of a two fluid system with free surface

In this work, a theoretical investigation is performed on modeling interfacial and surface waves in a layered fluid system. The physical system consists of two immiscible liquid layers of different densities ρ1 > ρ2 with an interfacial surface and a free surface, inside a prismatic-section tank. On the basis of the potential formulation of the fluid motion, we derive a nonlinear system of partial differential equations using the Hamiltonian formulation for irrotational flow of the two fluids of different density subject to conservative force. As a consequence of the assumption of potential velocity, the dynamics of the system can be described in terms of variables evaluated only at the boundary of the fluid system, namely the separation surface and the free surface. This Hamiltonian formulation enables one to define the evolution equations of the system in a canonical form by using the functional derivatives.

Baroclinic solitary water waves in a two-layer fluid system with diffusive interface

Experiments are described that have been performed in a 10 m long and 0.33 m wide channel on the baroclinic wave propagation in a two-layered free surface fluid system with an upper fresh-water and a lower salt-water layer. The interface between the two layers is diffuse and has a finite thickness which grows with the life time of the layered fluid system. Soliton-type disturbances of the interface are generated at one end of the channel. They move along the channel and encounter an obstruction (sill) in the middle, where they break into reflected and transmitted signals. Two types of solitary waves are produced by the interaction: a fast, ground mode soliton and a slower (by a factor of approximately 3), mode-one soliton due to the diffusive nature of the interface. Using the shallow water theory of a Boussinesq fluid with a continuous density profile and Benney's (1996) theoretical description it is demonstrated that the observed fast and slow speed wave signals are indeed due to the given density structure. It is also demonstrated that the ground mode wave speed decreases with increasing thickness of the diffusive interface whereas that of the first higher mode increases with good agreement between theory and experimental observation.

Steady natural convection in a double layer of immiscible liquids with density inversion

Natural convection of a layered fluid system composed of two immiscible liquids, silicone oil on top of water, is studied numerically. The flow in the two layers is viscously and thermally coupled. Two counter-rotating natural convection roll cells of opposite vorticity develop when one side wall temperature in the density inversion fluid is above and the other below the density inversion temperature. In contrast, only one roll cell develops in a liquid layer with Boussinesq properties. The viscous coupling between both immiscible layers strengthens the roll cell with opposite vorticity in the density inversion layer. At large Rayleigh number the flow pattern in the density inversion layer becomes very complex. The largest heat transport occurs in the upper Boussinesq layer. The two-roll cell pattern in the density inversion layer is impeding the total horizontal heat transfer. A vertical heat transport exists across the interface from the density inversion layer into the encapsulating upper layer. The moving interface between the immiscible liquids improves the heat transfer in each layer when compared to the cavity cases.

Numerical simulation of time-dependent thermocapillary convection in layered fluid systems

We consider steady and time-dependent thermocapillary convection in encapsulated layers of moderate Prandtl number fluids. Assuming flat free surfaces and fluid/fluid interfaces, the two-dimensional, time-dependent Navier-Stokes equations and the energy equation are integrated by a time-accurate method on a stretched, staggered mesh. Particular attention is focused on the nature of time dependence in water encapsulation of Fluorinert FC-75 and in ethylene glycol encapsulated by FC-75 and hexadecane. We show that similar mechanisms for time-dependent thermocapillary convection exist for single-layer and multiple-layer fluid systems, and that shear effects at a thermocapillary interface can reduce convection in an encapsulated fluid layer. Based on our estimates for the thermal coefficients of interface tension, an apparent benefit of fluid encapsulation is to lessen the tendency toward time dependence. Nomenclature AR = aspect ratio Cp = specific heat at constant pressure k = thermal conductivity Lx, Ly = x and y dimensions of the cavity Ma = Marangoni number, crTATLx/iJLK Nu(x) = Nusselt number, Eq. (14) Pr = Prandtl number, V!K p =

Flow and temperature measurements of two immiscible liquid layers

Surface tension driven flow of two immiscible liquid layers subjected to a horizontal temperature gradient has been observed. The objective of this study is the visualization of the thermoconvective flow of a double layer fluid system and the implementation of the liquid crystal tracer technique. This flow configuration simulates the encapsulation of electronic melts, where the lower liquid layer represents the melt and the upper layer represents the confining liquid having the task to stop the fluid motion of the melt component. The influence of different encapsulation conditions on the resultant flow pattern of the melt layer has been observed using liquid crystal tracers. The liquid crystal tracer technique was successfully applied and bears the potential to become a powerful diagnostic tool, especially with respect to experiments in space.

Spherical wave propagation in layered poro-elastic and fluid systems

A numerical procedure for computing sound propagation from a point source in an environment consisting of a stratified fluid half-space above a layered poro-elastic ground is described. The wave amplitudes are computed from depth-separated wave equation by forming a system of linear equations arising from the boundary conditions at each and every interface between layers and solving the resulting global matrix. The inverse Hankel transform is then performed by fast Fourier method (FFP) to obtain the pressure and particle velocity at any elevation as a function of range. Wave propagation within the porous elastic layers is calculated according to the modified Biot–Stoll theory which predicts two compressional and one shear wave types in the medium. Six boundary conditions are therefore required at each ground intefaces and four at the fluid–ground interface, while only two are needed at the fluid–fluid interfaces. Using this code the role of the ground elasticity on above-ground propagation and in acoustically induced seismic excitation are explored. It is predicted that on certain ground types, characterized by a thin layer with small compressional and shear velocities, and at frequencies that coincide with a peak in the seimic-to-acoustic coupling ratio spectrum there is an extra sound attenuation.

The wave attenuation on the mud bed

In the paper the wave attenuation in a two layer fluid system is studied. The fluid in the top layer is ideal and that in the lower layer is the Voigt model of the viscoelastic medium. A dispersion relation is derived and the rate of the wave decay is computed. The approximate explicit expressions of the decay rate for different water depth are given, where the viscoelasticity is either very large or very small. Compared with the numerical results, our results are very accurate, which can be used by an engineer.

Random heteropolymers in layered fluids

We study a simple model of a random heteropolymer, without excluded volume, in layered fluid systems of the form (i) ABA or ABC, where A, B, C denote three immiscible fluids, (ii) AB+hard wall, (iii) ABABA... . The quenched randomness is contained in the affinities {ξj; j=1,2,...,N} of the N links of the chain with the different fluids: these affinities are taken as random independent variables. Case (i) may be of interest for the localization of proteins in membranes, and (ii) pertains to the adsorption onto a wall. Using a replica-symmetric theory with a Hartree variational procedure, we study the behavior of the chain as a function of temperature. Various transitions [localization in case (i), adsorption in (ii)] are then exhibited. In the periodic situation (iii), the quenched and annealed randomness lead to the same results.

Stability of magnetic fluid penetration through a porous medium with uniform magnetic field oblique to the interface

Recent work has shown that the fingering instability, which develops when a more viscous fluid is pushed through the voids of a porous medium or through a Hele-Shaw cell by a less viscous fluid, can be prevented if a magnetic field is applied tangential to a flat fluid interface separating magnetizable and non-magnetizable fluids. This earlier work is extended here by considering equilibrium magnetic field components both perpendicular and parallel to the flat interface. The tangential field component stabilizes those waves traveling along the field lines while the normal field is destabilizing. The analysis is developed through a general set of relations for perturbation field and flow interfacial variables defined for a prototype magnetizable fluid layer which can be used to describe the small signal stability characteristics of layered fluid systems. In a uniform tangential magnetic field geometry, experimental results of the most unstable wavelength in a Hele-Shaw cell are shown to agree well with theory.

Hydrodynamic instabilities of a layered elastic medium in the presence of a flow field

The stability of a finite thickness layered medium being excited by a flow field is examined and results with experiments presented. The dynamics of the compliant layer is assumed to be governed by the basic elasticity equations. On the lower surface, the layer is assumed to be fixed to a rigid surface. On the upper surface, a potential flow field is assumed to provide the exciting force. A solution to the linearized set of elasticity and fluid mechanics equation is obtained and when all of the boundary conditions are satisfied, a comparability equation is obtained: F(ω, α, h, σ, ρf, ρe, Fe, U∞) = 0. Complex or imaginary values of ω and/or α indicate either a temporal or a spatial instability of the compliant layer fluid System. Roots of this exact equation for different thickness, Poisson's ratio, Young's modulus, and flow velocity have been obtained. In the limit as the plate thickness goes to infinity and as the fluid density goes to zero, the classical Rayleigh surface waves are recovered. The results have been compared with experiments conducted in a rotating cylinder facility. Different compliant coatings have been tested and the experimental results have been compared with the theory.

Mass transport in layered fluid systems

The method of matched asymptotic expansions is employed to calculate the mass transport velocity due to small amplitude oscillatory waves propagating in conditions of density and viscosity discontinuities. For progressive waves in a two-layer system, it is found that the velocity at the interface is in the direction of wave propagation; when the uppermost surface is free, the velocity there is in the direction opposite to that at the interface. If the difference in the densities is small, the calculated transport velocity associated with an internal wave can be of more importance than that associated with the surface wave as obtained from the work of Longuet-Higgins (1953).