Shallow Horizontal Layer Research Articles

Bi-stability Bifurcation in Convection Double Diffusion Through a Shallow Horizontal Layer Saturated with a Complex Fluid

The onset of non-linear convection in a porous layer saturated by a shear-thinning liquid is studied. The Carreau-Yasuda model is utilized for modeling the behavior of the working medium. Constant fluxes of heat and mass are defined on the horizontal surfaces of the cavity, while the vertical sides are assumed adiabatic. The parallel flow approximation and the finite difference approach are used to conduct the investigation analytically and numerically, respectively. The linear stability inspection of the convective and diffusive circumstances is carried out by taking into account an infinitesimal perturbation. The theory of linear stability is employed to determine the critical Rayleigh number for the motion from the rest state, Hopf bifurcation, and the transition from the stationary to oscillatory convection. Overall, the Carreau-Yasuda rheological parameters have a significant impact on the thresholds of convection. The most interesting findings of this study is highlighting the existence of a bi-stability phenomenon, i.e., the existence of two steady-state solutions, which was not observed before in non-Newtonian fluids convection.

Significance of vertical vibration on the stability of thermo-bioconvection in a suspension of oxytactic microorganisms

This current article study the significance of vertical vibration on a dilute suspension containing oxytactic microorganisms heated from below in a shallow horizontal layer. A time-averaged method and linear stability analysis are employed to attain the governing equations describing the mean flow of suspension. Under time-averaged method, it is permissible to break the set of solutions of governing equations into two distinct parts, the first one runs gradually with time while second one runs rapidly with time. A simple Galerkin method is employed to find the eigenvalue equation and the thermal stability curves for onset of thermo-bioconvection. The values of RTcr, Rbcr and ac are compared and found in close agreement with earlier reported articles. It is concluded that vertical vibration of high frequency and low amplitude always stabilizes the suspension of oxytactic microorganisms heated from below in shallow horizontal layer. In addition, the depth parameter ω stabilizes the suspension while, in case of no vertical vibration depth parameter destabilizes the suspension.

Double-diffusive natural convection study in a shallow horizontal porous layer filled with a binary fluid and submitted to destabilized conditions in the presence of Soret effect

This paper reports an analytical and numerical study of thermosolutal convection in a Brinkman horizontal porous layer saturated with a binary mixture, destabilized by submitting it either to both heat and mass fluxes (a=0, with Soret effect) or only to a heat flux to recover the case of pure Soret effect (a=1). The analytical solution is derived based on the parallel flow approximation and validated numerically using a finite difference method. A good agreement is found between the predictions of the parallel flow approximation and the numerical results obtained by solving the full governing equations. The governing parameters of this study are the Lewis number, Le, the Rayleigh thermal, RT, the Soret number, Sr, and the buoyancy ratio, φ. The thresholds for the onset of convection and the generated heat and mass transfer rates are examined both in the presence (a=0) and in the absence (a=1) of an external mass flux. The influence of the governing parameters on the onset of motion and the resulting fluid flow, temperature and concentration fields is discussed. It was found that the onset of convection for a=0 and positive values of Sr is delayed compared to the case a=1 (pure Soret effect) and, for the latter case, the onset of convection is delayed compared to a=0 and negative values of the Soret number.

Departures from isotropy: the kinematics of a larval snail in response to food.

The swimming behavior of invertebrate larvae can affect their dispersal, survival and settlement in the ocean. Modeling this behavior accurately poses unique challenges as behavior is controlled by both physiology and environmental cues. Some larvae use cilia to both swim and create feeding currents, resulting in potential trade-offs between the two functions. Food availability is naturally patchy and often occurs in shallow horizontal layers in the ocean. Also, larval swimming motions generally differ in the horizontal and vertical directions. In order to investigate behavioral response to food by ciliated larvae, we measured their behavioral anisotropy by quantifying deviations from a model based on isotropic diffusion. We hypothesized that larvae would increase horizontal swimming and decrease vertical swimming after encountering food, which could lead to aggregation at food layers. We considered Crepidula fornicata larvae, which are specifically of interest as they exhibit unsteady and variable swimming behaviors that are difficult to categorize. We tracked the larvae in still water with and without food, with a portion of the larvae starved beforehand. On average, larvae in the presence of food were observed higher in the water column, with higher swimming speeds and higher horizontal swimming velocities when compared with larvae without food. Starved larvae also exhibited higher vertical velocities in food, suggesting no aggregation behavior. Although most treatments showed strong anisotropy in larval behavior, we found that starved larvae without food exhibited approximately isotropic kinematics, indicating that behavioral anisotropy can vary with environmental history and conditions to enhance foraging success or mitigate food-poor environments.

Clustering of vertically constrained passive particles in homogeneous isotropic turbulence.

We analyze the dynamics of small particles vertically confined, by means of a linear restoring force, to move within a horizontal fluid slab in a three-dimensional (3D) homogeneous isotropic turbulent velocity field. The model that we introduce and study is possibly the simplest description for the dynamics of small aquatic organisms that, due to swimming, active regulation of their buoyancy, or any other mechanism, maintain themselves in a shallow horizontal layer below the free surface of oceans or lakes. By varying the strength of the restoring force, we are able to control the thickness of the fluid slab in which the particles can move. This allows us to analyze the statistical features of the system over a wide range of conditions going from a fully 3D incompressible flow (corresponding to the case of no confinement) to the extremely confined case corresponding to a two-dimensional slice. The background 3D turbulent velocity field is evolved by means of fully resolved direct numerical simulations. Whenever some level of vertical confinement is present, the particle trajectories deviate from that of fluid tracers and the particles experience an effectively compressible velocity field. Here, we have quantified the compressibility, the preferential concentration of the particles, and the correlation dimension by changing the strength of the restoring force. The main result is that there exists a particular value of the force constant, corresponding to a mean slab depth approximately equal to a few times the Kolmogorov length scale η, that maximizes the clustering of the particles.

ON DOUBLE-DIFFUSIVE HADLEY FLOW IN MAXWELL FLUIDS THROUGH A HORIZONTAL POROUS LAYER

The present study deals with the effect of stress relaxation on the mechanism of instability of the double-diffusive Hadley flow through a shallow horizontal layer of a porous medium induced by thermal and solutal gradients inclined to the vertical by means of linear stability analysis. The Hadley flow which is set up by the horizontal components of these gradients becomes unstable when vertical components are sufficiently large and this instability is analyzed using three-dimensional normal modes. The vertical thermal Rayleigh number is treated as the eigenvalue. The system that constitutes the eigenvalue problem is solved by using a two-term Galerkin approximation for various modes of instability. The detailed analysis is carried out for both the stationary and oscillatory modes. Results are presented for various values of the governing parameters of the flow. It is observed that the stress relaxation parameter has a significant role in convective instability.

Convection in a horizontal fluid layer under an inclined temperature gradient for Prandtl numbers Pr > 1

In this paper calculations are done of the flow instability of a shallow horizontal fluid layer under an inclined temperature gradient for Prandtl numbers of magnitudes Pr>1. In a previous paper (Ortiz-Pérez and Dávalos-Orozco, 2011 [17]), the range of values investigated were Pr⩽1. There, the magnitudes of the horizontal Rayleigh number investigated were far larger than those used in the literature and it was found the new oblique oscillatory mode Obo. In the present paper it is found that this oblique oscillatory mode still appears for Pr=2. Further, it is also shown that some modes become irrelevant when the Prandtl number is increased, and different modes become the first to go unstable. An important result is that above a large enough Prandtl number, the curves of criticality remain almost the same. Therefore, the limit Pr→∞ is employed to calculate useful analytical expressions for the longitudinal modes of criticality which cover almost all the range of horizontal Rayleigh numbers. In contrast to the authors previous results, in this paper it is shown that for Pr>1 the transversal oscillatory mode has a range, in the horizontal Rayleigh number, as the first unstable one which first increases and then decreases with Pr, but never disappears.

Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below

The aim of this paper is to develop a theory describing the onset of convection instability (called here nanofluid bioconvecion) that is induced by simultaneous effects produced by oxytactic microorganisms, nanoparticles, and vertical temperature variation. The theory is developed for the situation when the nanofluid occupies a shallow horizontal layer of finite depth. The layer is defined as shallow as long as oxygen concentration at the bottom of the layer is above the minimum concentration required for the bacteria to be active (to actively swim up the oxygen gradient). The lower boundary of the layer is assumed rigid, while at the upper boundary both situations when the boundary is rigid or stress free are considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. A linear instability analysis is performed, and the resulting eigenvalue problem is solved analytically using the Galerkin method.

Nanofluid bioconvection in water-based suspensions containing nanoparticles and oxytactic microorganisms: oscillatory instability

The aim of this article is to propose a novel type of a nanofluid that contains both nanoparticles and motile (oxytactic) microorganisms. The benefits of adding motile microorganisms to the suspension include enhanced mass transfer, microscale mixing, and anticipated improved stability of the nanofluid. In order to understand the behavior of such a suspension at the fundamental level, this article investigates its stability when it occupies a shallow horizontal layer. The oscillatory mode of nanofluid bioconvection may be induced by the interaction of three competing agencies: oxytactic microorganisms, heating or cooling from the bottom, and top or bottom-heavy nanoparticle distribution. The model includes equations expressing conservation of total mass, momentum, thermal energy, nanoparticles, microorganisms, and oxygen. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. An approximate analytical solution of the eigenvalue problem is obtained using the Galerkin method. The obtained solution provides important physical insights into the behavior of this system; it also explains when the oscillatory mode of instability is possible in such system.

Soret-driven thermosolutal convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium

The stability of Soret-driven thermosolutal convection in a shallow horizontal layer of a porous medium subjected to inclined thermal and solutal gradients of finite magnitude is investigated theoretically by means of a linear stability analysis. The horizontal components of these gradients induce a Hadley circulation, which becomes unstable when vertical components are sufficiently large. We employed a two-term Galerkin approximation for various modes of instability. The effect of the Soret parameter on the mechanism of instability of the thermosolutal convection is investigated. Results are presented for various values of the governing parameters of the flow. It is observed that the Soret parameter has a significant effect on convective instability and this is discussed.

Influence of vertical vibrations on the separation of a binary mixture in a horizontal porous layer heated from below

We study the influence of vertical high-frequency and small-amplitude vibrations on the separation of a binary mixture saturating a shallow horizontal porous layer heated from below. The monocellular flow obtained for a separation ratio ψ > ψ mono > 0 leads to a migration of the species towards the two vertical boundaries of the cell. The 2D direct numerical simulations and the linear stability analysis of the averaged governing equations show that the vertical vibrations delay the transition from monocellular flow to multicellular flow. The vibrations also decrease the value of ψ mono, which allows species separation for a wide variety of binary mixtures.

A NUMERICAL ANALYSIS OF STOP BAND CHARACTERISTICS BY MULTILAYERED DIELECTRIC GRATINGS WITH SINUSOIDAL PROFILE

An effective computational method based on a conven- tional modal expansion approach is presented for handling a multilay- ered dielectric grating whose profiles are multilayered and sinusoidally modulated. This structure fabricated by dielectric material is one of the useful photonic crystals. The method is based on Yasuura's modal expansion, which is known as a least-squares boundary residual method or a modified Rayleigh method. In the extended method, each layer is divided into shallow horizontal layers. The Floquet modal func- tions and approximate solutions are defined in each shallow layer, and the latter are matched with boundary conditions in the least-squares sense. A huge-sized least-squares problem that appears in finding the modal coefficients is solved by the QR decomposition accompanied by sequential accumulation. This procedure makes it possible to treat the case where the groove depths are the same as or a little more than the grating period. As numerical example, we calculate a diffractive char- acteristic by a multilayered deep dielectric grating and confirm that a common band gap occurs for both polarizations.

Investigation of the onset of bioconvection in a suspension of oxytactic microorganisms subjected to high-frequency vertical vibration

This paper investigates the effect of vertical vibration on the stability of a dilute suspension of oxytactic microorganisms in a shallow horizontal fluid layer. For the case of high-frequency vibration, an averaging method is utilized to derive the equations describing the mean flow by decomposing the solutions of governing equations into two components: one that varies slowly with time, and a second that varies rapidly with time. Linear stability analysis is used to investigate the stability of the obtained averaged equations. It is predicted that high-frequency, low-amplitude vertical vibration has a stabilizing effect on a suspension of oxytactic microorganisms confined in a shallow horizontal layer.

Double-diffusive and Soret-induced convection in a shallow horizontal porous layer

This paper reports an analytical and numerical study of the natural convection in a horizontal porous layer filled with a binary fluid. A uniform heat flux is applied to the horizontal walls while the vertical walls are impermeable and adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal walls (double-diffusive convection, $a\,{=}\,0$ ) or by temperature gradients (Soret effects, $a\,{=}\,1$ ). The governing parameters for the problem are the thermal Rayleigh number, $R_T$ , the Lewis number, $Le$ , the solutal Rayleigh number, $R_S$ , the aspect ratio of the cavity, $A$ , the normalized porosity of the porous medium, $\varepsilon$ , and the constant $a$ . The onset of convection in the layer is studied using a linear stability analysis. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined in terms of the governing parameters. For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained by assuming parallel flow in the core of the cavity. The critical Rayleigh numbers for the onset of supercritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sup}}$ , or subcritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sub}}$ , convection are predicted by the present theory. A linear stability analysis of the parallel flow pattern is conducted in order to predict the thresholds for Hopf bifurcation. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical prediction and the numerical simulations.

The Soret effect on convection in a horizontal porous domain under cross temperature and concentration gradients

Natural convection with Soret effect in a binary fluid saturating a shallow horizontal porous layer is studied both numerically and analytically. The vertical walls of the enclosure are heated and cooled by uniform heat fluxes and a solutal gradient is imposed vertically. In the formulation of the problem, we use the Darcy model and the density variation is taken into account by the Boussinesq approximation. The governing parameters of the problem are the aspect ratio, A, the thermal Rayleigh number, RT, the buoyancy ratio, N, the Lewis number, Le and the Soret coefficient, NS. The analytical solution, based on the parallel flow approximation, is found to be in good agreement with a numerical solution of the full governing equations. In the presence of a vertical destabilizing concentration gradient, the existence of both natural and antinatural flows is demonstrated. When the vertical concentration gradient is stabilizing, multiple steady state solutions are possible in a range of buoyancy ratio, N, that depends strongly on the Soret coefficient, NS.

On the stability of thermally driven shear flow of an Oldroyd-B fluid heated from below

The linear stability analysis of convection for Oldroyd-B fluid, in a shallow horizontal layer, with applied temperature gradient inclined to the vertical is presented. The flow is driven by a horizontal temperature gradient and the horizontal boundaries are taken to be rigid conducting. The eigenvalue problem is solved by the Chebyshev Tau–QZ method and results are presented in terms of horizontal and vertical Rayleigh numbers. It is found that consideration of lateral heating induces instabilities both in stationary and oscillatory rolls and also both as transverse and longitudinal modes and that presence of elasticity, in general, accelerates the destabilization.

Nonlinear electrohydrodynamics of free surfaces

Electrical stresses acting at electrified interfaces govern the dynamics of fluids in many situations of practical relevance. In order to understand the electrohydrodynamics of these systems, a knowledge of the forces acting at interfaces is necessary. We will review the balance equations for the forces and the charge density. The relevant nondimensional numbers will be derived and illustrated with some particular cases. The problems of shallow horizontal layers and inclined thin films of perfectly conducting liquids will be examined in some detail.

EFFECT OF AN EXTERNAL MAGNETIC FIELD ON BUOYANCY-DRIVEN FLOW IN A SHALLOW POROUS CAVITY

Abstract A study is made of steady, natural convection in a shallow horizontal porous layer, saturated by an electrically conducting fluid, to which a transverse magnetic field is applied. The enclosure is insulated on the top and bottom walls, while an end-to-end temperature difference between the vertical walls is imposed. The problem is governed by three dimensionless parameters; the Darcy-Rayleigh number Ra, the cavity aspect ratio A, and the Hartmann number Ha. On the basis of Darcy's equation, an approximate analytical solution, valid in the limit of a shallow cavity ( A→0), is developed using matched asymptotic expansions. The solution is given up to 0( A3). A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is also conducted. The study covers the range of Ra from 0 to 500, Ha from Oto5, and A from O.O5 to 1. Results are presented for the velocity and temperature profiles and heat transfer in terms of the governing parameters. Upon comparing th...

On the nonlinear stability of thermally driven shear flow heated from below

The energy method, which gives the sufficient condition for the stability, is developed for the convection problem induced by an applied inclined temperature gradient in a shallow horizontal layer. A nonlinear stability analysis, which is valid for any Prandtl number, is carried out for the case of rigid conducting horizontal boundaries. The eigenvalue problem is solved by compound matrix method and the nonlinear stability results are compared with the results of the linear stability method.

Unicellular natural circulation in a shallow horizontal porous layer heated from below by a constant flux

Natural convection in a saturated horizontal porous layer heated from below and cooled at the top with a constant flux is studied both analytically and numerically. Linear stability analysis indicates that unicellular recirculation remains a stable mode of flow as the aspect ratio (A) of the layer is increased, in contrast to the situation for an isothermally heated and cooled layer. An analytical solution is presented for fully developed counterflow in the infinite-aspect-ratio limit; this flow is found to be linearly stable to transverse disturbances for Rayleigh number (Ra) as high as 506, at which point a Hopf bifurcation sets in; however, further analysis indicates that an exchange of stability due to longitudinal disturbances will occur much sooner at Ra ≈ 311.53. The velocity and temperature profiles of the counterflow solution, whilst not strictly speaking valid in the extreme end regions of the layer, otherwise agree very well with full numerical computations conducted for the ranges 25 [les ] Ra [les ] 1050, 2 [les ] A [les ] 10. However, for sufficiently high Rayleigh number (Ra between 630 and 650 for A = 8 and Ra between 730 and 750 for A = 4, for example), the computations indicate transition from steady unicellular to oscillatory flow, in line with the Hopf bifurcation predicted by the linear stability analysis for infinite aspect ratio.