Superposed Fluid Layers Research Articles

Comparative Study on Effects of Thermal Gradient Direction on Heat Exchange between a Pure Fluid and a Nanofluid: Employing Finite Volume Method

This work aims to determine how the temperature gradient orientation affects the heat exchange between two superposed fluid layers separated by zero wall thickness. The finite volume method (FVM) has been developed to solve the governing equations of both fluid layers. To achieve the coupling between the two layers, the heat flow continuity with the no-slip condition at the interface was adopted. The lower part of the space is filled with a nanofluid while the upper part is filled with a pure fluid layer. We have explored two cases of temperature gradient orientation: parallel gradient to gravity forces of our system and perpendicular gradient to gravity forces. We took a set of parameters, Ri and ϕ, to see their influence on the thermal and hydrodynamic fields as well as the heat exchange rate between the two layers. The main applications of this study related to biological systems such as the cytoplasm and the nucleoplasm are phase-separated solutions, which can be useful as models for membranelles organelles and can serve as a cooling system application using heat exchange. The Richardson number and the volume of nanosolid particles have a big impact on the rate of change of heat transmission. When a thermal gradient is perpendicular to gravity forces, total heat transmission improves with increasing solid volume percentage, but when the thermal gradient is parallel to gravity forces, overall heat transfer decreases significantly.

Effect of Periodic Oscillation on the Interfacial Instability of Two Superposed Fluid Layers in a Fully Saturated Porous Media

We investigate the effect of horizontal periodic oscillation on the interfacial instability of two immiscible and viscous fluids of different densities in a fully saturated porous media. A linear stability analysis of the viscous and time-dependent basic flow leads to a periodic oscillator describing the evolution of the interfacial perturbation amplitude. The horizontal oscillation leads to the occurrence of two types of instability, the Kelvin–Helmholtz’s instability and the parametric resonance. These instabilities appear at the frontier between water and petroleum and have a practical interest in oil reservoir engineering. The results show that, an increase of the oscillation frequency destabilizes the Kelvin–Helmholtz instability and displaces the parametric instability regions toward the short wavelength perturbation. Also, we examine mainly how the other physical parameters of the system affect the instabilities for various permeability and porosity values of the porous medium as well as for relative heights of the two fluid layers.

Stability of two-layer miscible convection

We study the onset of convection in two superposed fluid layers that have different properties. Depending on the parameters of the problem, the most unstable mode can take one of three distinct forms: whole layer solutions, localized solutions, and segregated solutions.

The role of soluble surfactants in the linear stability of two-layer flow in a channel

The linear stability of Couette–Poiseuille flow of two superposed fluid layers in a horizontal channel is considered. The lower fluid layer is populated with surfactants that appear either in the form of monomers or micelles and can also get adsorbed at the interface between the fluids. A mathematical model is formulated which combines the Navier–Stokes equations in each fluid layer, convection–diffusion equations for the concentration of monomers (at the interface and in the bulk fluid) and micelles (in the bulk), together with appropriate coupling conditions at the interface. The primary aim of this study is to investigate when the system is unstable to arbitrary wavelength perturbations, and in particular, to determine the influence of surfactant solubility and/or sorption kinetics on the instability. A linear stability analysis is performed and the growth rates are obtained by solving an eigenvalue problem for Stokes flow, both numerically for disturbances of arbitrary wavelength and analytically using long-wave approximations. It is found that the system is stable when the surfactant is sufficiently soluble in the bulk and if the fluid viscosity ratio $m$ and thickness ratio $n$ satisfy the condition $m<n^{2}$. On the other hand, the effect of surfactant solubility is found to be destabilising if $m\geqslant n^{2}$. Both of the aforementioned results are manifested for low bulk concentrations below the critical micelle concentration; however, when the equilibrium bulk concentration is sufficiently high (and above the critical micelle concentration) so that micelles are formed in the bulk fluid, the system is stable if $m<n^{2}$ in all cases examined.

Influence of Vertical Magnetic Field on the Onset of Rayleigh-Benard-Marangoni Convection in Superposed fluid and Porous Layers with Deformable Free Surface

Influence of Vertical Magnetic Field on the Onset of Rayleigh-Benard-Marangoni Convection in Superposed fluid and Porous Layers with Deformable Free Surface

Effects of Internal Heat Generation and Variable Viscosity on Marangoni Convection in Superposed Fluid and Porous Layers

Effects of Internal Heat Generation and Variable Viscosity on Marangoni Convection in Superposed Fluid and Porous Layers

Numerical prediction of turbulent thermocapillary convection in superposed fluid layers with a free interface

The present paper introduces a new numerical method for predicting the characteristics of thermocapillary turbulent convection in a differentially-heated rectangular cavity with two superposed and immiscible fluid layers. The unsteady Reynolds form of the Navier–Stokes equations and energy equation are solved by using the control volume approach on a staggered grid system using SIMPLE algorithm. The turbulence quantities are predicted by applying the standard k – ε turbulence model. The level set formulation is applied for predicting the topological changes of the interface separating the two fluid layers and to provide an accurate and robust modeling of the interfacial normal and tangential stresses. The computational results obtained showed good agreement when compared with the previous experimental, numerical and analytical benchmark data for different validation cases in both laminar and turbulent regimes. The present numerical method is then applied to predict the velocity and temperature distribution in two immiscible liquid layers with undeformable interface for a wide range of Marangoni numbers. The laminar-turbulent transition is demonstrated by obtaining the turbulence features at high interfacial temperature gradient which is characterized by high Marangoni number. The effect of increasing Marangoni number on the interface dynamics in turbulent regime is also investigated.

The influence of viscosity on the frozen wave instability: theory and experiment

We present the results of an experimental and linear stability study of the influence of viscosity on the frozen wave (FW) instability, which arises when a vessel containing stably stratified layers of immiscible liquids is oscillated horizontally. Our linear stability model consists of two superposed fluid layers of arbitrary viscosities and infinite lateral extent, subject to horizontal oscillation. The effect of the endwalls of the experimental vessel is simulated by enforcing the conservation of horizontal volume flux, so that the base flow consists of counterflowing layers.We perform experiments with four pairs of fluids, keeping the viscosity of the lower layer (ν1) constant, and increasing the viscosity of the upper layer (ν2), so that 1.02 × 102 ≤ N1 = ν2/ν1 ≤ 1.21 × 104. We find excellent quantitative agreement between theory and experiment despite the simple model geometry, for both the critical onset parameter and wavenumber of the FW. We show that the model of lyubimov:1987 (Fluid Dyn. vol. 86, 1987, p. 849), which is valid in the limit of inviscid fluids, consistently underestimates the instability threshold for fluids of equal viscosity, but generally overestimates the threshold for fluids of unequal viscosity. We extend the experimental parameter range numerically to viscosity contrasts 1 ≤ N1 ≤ 6 × 104 and identify four regions of N1 where qualitatively different dynamics occur, which are reflected in the non-monotonic dependence of the most unstable wavenumber and the critical amplitude on N1. In particular, we find that increasing the viscosity contrast between the layers leads to destabilization over a wide range of N1, 10 ≤ N1 ≤ 8 × 103. The intricate dependence of the instability on viscosity contrast is due to considerable changes in the time-averaged perturbation vorticity distribution near the interface.

Effect of pulsations on two-layer channel flow

The effect of vertical wall vibrations on two-phase channel flow is examined. The basic flow consists of two superposed fluid layers in a channel whose walls oscillate perpendicular to themselves in a prescribed, time-periodic manner. The solution for the basic flow is presented in closed form for Stokes flow, and its stability to small periodic perturbations is assessed by means of a Floquet analysis. It is found that the pulsations have a generally destabilizing influence on the flow. They tend to worsen the Rayleigh–Taylor instability present for unstably stratified fluids; the larger the amplitude of the pulsations, the greater the range of unstable wave numbers. For stably stratified fluids, the pulsations raise the growth rate of small perturbations, but are not sufficient to destabilize the flow. In the latter part of the paper, the basic flow for arbitrary Reynolds number is computed numerically assuming a flat interface, and the motion of the interface in time is predicted. The existence of a time-periodic flow is demonstrated in which the ratio of the layer thicknesses remains constant throughout the motion.

Instabilities in two-liquid layers subject to a horizontal temperature gradient

We study theoretically the stability of two superposed fluid layers heated laterally. The fluids are supposed to be immiscible, the interface undeformable and of infinite horizontal extension. Combined thermocapillary and buoyancy forces give rise to a basic flow when a temperature difference is applied. The calculations are performed for a melt of GaAs under a layer of molten B2O3, a configuration of considerable technological importance. Four different flow patterns and five temperature configurations are found for the basic state in this system. A linear stability analysis shows that the basic state may be destabilized by oscillatory motions leading to the so-called hydrothermal waves. Depending on the relative height of the two layers these hydrothermal waves propagate parallel or perpendicular to the temperature gradient. This analysis reveals that these perturbations can alter significantly the liquid flow in the liquid-encapsulated crystal growth techniques.

Time-dependent fluxes across double-diffusive interfaces

A simple, time-dependent model is proposed to predict the heat and salt fluxes transferred between a hot, salty fluid layer and an overlying colder, fresher fluid layer when the overall density of the lower layer is greater than that of the upper layer. The interface separating the two layers, known as a ‘diffusive’ interface, consists of a purely diffusing core sandwiched between vigorously convecting boundary layers. This type of interface occurs whenever the density of superposed fluid layers depends on two diffusing agents with different diffusivities such that the faster diffusing agent causes a statically unstable density variation while the slower diffusing agent causes a statically stable variation. Early experimental and theoretical studies sought a single relationship between the heat flux , and both links and puts into context two previous theories of diffusive interfaces.

Buoyancy-driven convection in two superposed fluid layers in an octagonal cavity

Buoyancy-driven convection in a differentially heated cavity containing superposed layers of air–water, air–silicone oil and silicone oil–water has been experimentally studied. The apparatus is octagonal in plan. It has a nominal diameter of 130.6 mm and a height of 50 mm. The individual fluid layer heights have been kept equal to 25 mm in all the experiments. The cavity temperature difference has been varied over a range of 0.4 to 18 K, depending on the combination of fluid layers studied. Measurement of the thermal field in the octagonal cavity has been carried out using laser interferometry. Four different view angles namely 0, 45, 90, and 135° have been considered. The interface shapes have been recorded in the form of shadowgraphs. In the air–water experiments, the thermal fields in both fluid layers show a loss of axisymmetry, while progressing from a lower to a higher Rayleigh number. In air–oil experiments, a steady axisymmetric thermal field is observed in the oil layer, while a sufficient number of fringes did not form in air. In oil–water experiments, the thermal field in oil is close to axisymmetric. In water, the transition sequence to three dimensionality and unsteadiness is similar that of a single fluid layer. In most experiments, the cavity temperature difference is found to split across the fluid layers inversely in proportion to the effective thermal conductivity of the fluid layers. The experimental interface temperature matches the estimated value from single fluid correlations fairly well. Discrepancy with correlations is observed when the fluid layers are mechanically, rather than thermally coupled.

Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation.

In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential and the dimensionless parameters are the ratio between densities rho = rho(2)/rho(1) and lambda = gh/c(2). We study special values of the parameters such that lambda(1 - rho) is near 1(-), where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with, in addition, a double eigenvalue in 0, a pair of simple imaginary eigenvalues +/-ilambda at a distance O(1) from 0, and for lambda(1 - rho) above 1, another pair of simple imaginary eigenvalues tending towards 0 as lambda(1 - rho) --> 1(+). When lambda(1 - rho) <or= 1, this pair disappears into the essential spectrum. The rest of the spectrum lies at a distance at least O(1) from the imaginary axis. We show in this paper that for lambda(1 - rho) close to 1(-) there is a family of periodic solutions similar to that in the Lyapunov-Devaney theorem (despite the resonance due to the point 0 in the spectrum). Moreover, showing that the full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillation, we also prove the existence of a family of homoclinic connections to these periodic orbits, provided that these ones are not too small.

Non–existence of truly solitary waves in water with small surface tension

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.

Gravity and Capillary-Gravity Periodic Travelling Waves for Two Superposed Fluid Layers, One Being of Infinite Depth

The mathematical study of 2D travelling waves in the potential flow of two superposed layers of perfect fluid, with free surface and interfaces (with or without surface tensions) and with the bottom layer of infinite depth, is set as an ill-posed reversible evolution problem, where the horizontal space variable plays the role of a “time”. We give the structure of the spectrum of the linearized operator near equilibrium. This spectrum contains a set of isolated eigenvalues of finite multiplicities, a small number of which lie near or on the imaginary axis, and the entire real axis constitutes the essential spectrum, where there is no eigenvalue, except 0 in some cases. We give a general constructive proof of bifurcating periodic waves, adapting the Lyapunov-Schmidt method to the present (reversible) case where 0 (which is “resonant”) belongs to the continuous spectrum. In particular we give the results for the generic case and for the 1 : 1 resonance case

Self-gravitating instability of triple superposed fluids of different densities with plane interfaces

The problem of two superposed fluids is formulated and the self-gravitating hydrodynamic basic equations are solved. Upon appropriate boundary conditions, a general eigenvalue relation is derived and discussed. The stability states are identified as ρ' < ρ, ρ' = ρ and ρ' > ρ where ρ' and ρ are the densities of the self-gravitating superposed fluids. The analytical results are confirmed numerically and it is found that ρ'/ρ is stabilizing according to restrictions. A physical interpretation has been declared to some new parameters.In part B the governed equations of the perturbed and unperturbed states are solved. Their solutions satisfy certain conditions across the fluids plane interfaces. A cumbersome stability criterion is derived based on the linear perturbation technique of normal mode analysis. Several limiting cases could be recovered with some simplifications. The (in-)stability restrictions are identified for several different values of the densities ratios of the three superposed fluid layers.

The global hotspot distribution and instability of D″

We present a new spherical harmonic expansion of the global hotspot distribution, with the contribution of each hotspot weighted by its buoyancy flux. The resulting power spectrum, normalized to that expected for a random distribution, has statistically significant peaks only at degrees 1 and 2. To explain this, we study a simple model for the origin of mantle plumes in the D″ layer at the base of the mantle: the Rayleigh‐Taylor instability of two superposed fluid layers with different viscosities and densities in a spherical shell. The wavelength of the fastest growing instability greatly exceeds the thickness of D″ and increases as the viscosity contrast γ across this layer increases. These results are consistent with the hotspot spectrum if γ ∼ 106.

Internal waves in an elliptical channel

Abstract Internal waves in open channels of various depths are studied, in this paper, and a wetted cross section with an elliptical bottom is considered. The frequencies of the first three sloshing modes of internal waves in two superposed fluids contained in an elliptical channel are calculated for various ratios of the depths of the two layers. Numerical solutions converge to analytical solutions based on the shallow water theory as the depth of the thin lower layer approaches zero. Also, solutions for the frequencies of the longitudinal modes of progressive internal waves in two superposed fluid layers contained in an elliptical channel are calculated for various ratios of the depths of the two layers and for two different wave numbers k.

Thermal imaging system with a two-phase ternary mixture of liquids

A new infrared camera operating between 8 and 12 μm is described. The thermo-optical properties of two superposed fluid layers are used. It is shown that the modulation of the fluid-fluid interface, due to the infrared image, of a two-phase ternary mixture can be made much larger than with a two-phase binary one. Making this modulation visible, is the principle of a sensitive IR-visible image converter.

Interface deflections induced by the Marangoni effect: an application to infrared-visible image conversion

Considers a system of two superposed fluid layers F1 and F2 and studies the linear response of the air-F1 and F1-F2 interfaces to a modulation of the surface tension at the air-F1 interface due to inhomogeneous heating by infrared radiation. The deflection of the F1-F2 interface is shown to be strongly enhanced relative to that of the air-F1 interface. Making this deflection visible is the principle of a very sensitive IR-visible image converter.