Geometric Length Scales Research Articles

Highlighting the effects of gravity in a compact thermoacoustic heat pump

We present a numerical study of gravity impact on thermoacoustic heat pumping effects in a compact cavity used as a refrigeration device. More precisely, we show how gravity acts on the temperature distribution, on the mean temperature heat flux along the stack area and on flow dynamics. Studies are performed in a two-dimensional geometrical configuration which is simplified with respect to common experimental devices but sufficiently detailed to take into account the main physical mechanisms. Fluid motion, heat and mass transfer as well as compression/expansion effects are estimated from the computation of the Navier-Stokes equations in the low-Mach number hypothesis coupled with the heat equation in solid parts. The low-Mach number hypothesis is here relevant because the characteristic scale of the acoustic wavelength is much smaller than the geometrical length scales of the cavity. The operating mode is characterized by a moderate drive ratio, a fluid oscillating motion of moderate amplitude and fixed low frequency, typical of such devices. It is shown that when gravity is included, buoyancy effects modify the temperature and velocity fields as well as heat pumping efficiency.

Drift-flux model for upward dispersed two-phase flows in a vertical rod bundle

The present study investigates the drift-flux model for upward dispersed two-phase flows in a vertical rod bundle. The drift-flux model is critical to formulating the interfacial drag force in one-dimensional simulation codes. A rod bundle includes three length scales: a large outer casing width, a medium hydraulic equivalent diameter of a subchannel, and a small gap between rods. Existing research does not provide comprehensive information on an appropriate geometrical length scale for a rod bundle. The present study calculates the evolution process of the drift velocity using the two-bubble-group approach. The two-bubble-group-based formulation is simplified to calculate the drift velocity behavior at the transition between bubbly and beyond-bubbly flows. A scheme is proposed to explicitly calculate the distribution parameter and drift velocity with operating parameters. The new drift-flux correlation is validated by 317 data collected for the pressure from 0.1 to 12 MPa. The extensive evaluation identifies that an appropriate length scale characterizing a two-phase flow in a rod bundle is an outer casing width of a rod bundle. The prediction bias of the new drift-flux correlation was negligibly small, and the random error was 0.0571 in an absolute value measure and 16.1 % in a relative value measure.

Minimum length (scale) in quantum field theory, generalized uncertainty principle and the non-renormalisability of gravity

The notions of minimum geometrical length and minimum length scale are discussed with reference to correlation functions obtained from in-in and in-out amplitudes in quantum field theory. Whereas the in-in propagator for metric perturbations does not admit the former, the in-out Feynman propagator shows the emergence of the latter. A connection between the Feynman propagator of quantum field theories of gravity and the deformation parameter δ0 of the generalised uncertainty principle (GUP) is then exhibited, which allows to determine an exact expression for δ0 in terms of the residues of the causal propagator. A correspondence between the non-renormalisability of (some) theories (of gravity) and the existence of a minimum length scale is then conjectured to support the idea that non-renormalisable theories are self-complete and finite. The role played by the sign of the deformation parameter is further discussed by considering an implementation of the GUP on the lattice.

The deflection platform phenomenon of functionally graded flexoelectric simply supported nanobeam

Direct flexoelectricity appears in dielectrics as a kind of coupling effect of electric field and elastic strain gradient. Its strong size dependence makes it effective in nanostructures. In this paper, based on the flexoelectric theory and mixed finite element method (FEM), the mechanical properties of the two-dimensional functionally graded (FG) flexoelectric nanobeam are studied, and the phenomenon of the deflection platform is found. By applying various types of loads to FG nanobeams, the influence of geometric parameters, length scale, and material distribution on the phenomenon of the deflection platform is discussed. According to the simulation results, it is found that the aspect ratio and the length scale of the beam will change the shape of the deflection platform, making it change regularly among various shapes. The geometric size will greatly increase the height of the platform, but it has little effect on the dimensionless deflection. Reasonable material distribution can enhance the internal coupling effect, increase the height of the platform and reduce the required load. Our results suggest that when studying nano-scale flexoelectric models, the deflection platforms or similar phenomena should be considered.

Low temperature quantum bounds on simple models

In the past few years, there has been considerable activity around a set of quantum bounds on transport coefficients (viscosity) and chaos (Lyapunov exponent), relevant at low temperatures. The interest comes from the fact that Black-Hole models seem to saturate all of them. The goal of this work is to gain physical intuition about the quantum mechanisms that enforce these bounds on simple models. To this aim, we consider classical and quantum free dynamics on curved manifolds. These systems exhibit chaos up to the lowest temperatures and -- as we discuss -- they violate the bounds in the classical limit. First of all, we show that the quantum dimensionless viscosity and the Lyapunov exponent only depend on the de Broglie length and a geometric length-scale, thus establishing the scale at which quantum effects become relevant. Then, we focus on the bound on the Lyapunov exponent and identify three different ways in which quantum effects arise in practice. We illustrate our findings on a toy model given by the surface of constant negative curvature — a paradigmatic model of quantum chaos — glued to a cylinder. By exact solution and numerical investigations, we show how the chaotic behaviour is limited by the quantum effects of the curvature itself. Interestingly, we find that at the lowest energies the bound to chaos is dominated by the longest length scales, and it is therefore a collective effect.

Correlation between slip precursors and topological length scales at the onset of frictional sliding

Understanding the interplay between concurrent length scales is a fundamental issue in many problems involving friction between sliding interfaces, from tribology to the study of earthquakes and seismic faults. On the one hand, a macroscopic sliding event is preceded by slip precursors with a characteristic propagation length scale. On the other hand, the emergent frictional properties can be modified by surface patterning depending on their geometric length scale. This suggests that macroscopic sliding of structured surfaces is governed by the interplay between the length scale of the slip precursors and those characterizing the geometric features. In this paper, we investigate these aspects by means of numerical simulations using a two-dimensional spring-block model. We discuss the influence of the geometric features on the occurrence and localization of slip precursors, extending the study to interfaces characterized by two geometric length scales. We find that different types of detachment sequences are triggered by specific surface structures, depending on their scales and relation to sliding direction, leading to a macroscopically smooth transition to sliding in the case of hierarchical and/or anisotropic features. These concepts could be exploited in devices switching from static to dynamic sliding, and can contribute to an improvement in the understanding and interpretation of seismic data.

Pumping Rate and Size of Demosponges—Towards an Understanding Using Modeling

Filter-feeding sponges pump large amounts of water and contribute significantly to grazing impact, matter transport and nutrient cycling in many marine benthic communities. For ecological studies it is therefore of interest to be able to estimate the pumping rate of different species from their volume size or osculum cross-sectional area by means of experimentally determined allometric correlations. To help understand allometric data correlations and observed large variations of volume-specific pumping rate among species we developed a model that determines the pumping rate as a function of the size (volume) of a tubular-type demosponge described by 4 geometric length scales. The model relies on a choanocyte-pump model and standard pressure loss relations for flow through the aquiferous system, and density and pumping rate per choanocyte is assumed to be constant. By selecting different possibilities for increase of the length scales, which may also simulate different growth forms, we demonstrate that the model can imitate the experimental allometric correlations. It is concluded that the observed dependence of pumping rate on size is primarily governed by the hydraulics of pump performance and pressure losses of the aquiferous system rather than, e.g., decreasing density of choanocytes with increasing sponge size.

Maximizing and minimizing the boundary scattering mean free path in diameter-modulated coaxial cylindrical nanowires

The thermal conductivity (k) of semiconducting nanomaterials is influenced by the geometry-dependent phonon boundary scattering mean free path (ΛBdy). Although prior work has calculated ΛBdy of periodically corrugated rectangular nanowires and used these results to study phonon backscattering in nanomaterials, ΛBdy remains unknown for recently fabricated periodic coaxial cylindrical nanowires. Here, we use phonon ray tracing simulations to comprehensively study the effect of geometric parameters on ΛBdy in coaxial cylindrical nanowires. We find that for a fixed smaller cylinder diameter (D1) and cylinder length ratio, ΛBdy of periodic nanowires can be maximized or minimized via geometric control of the pitch (p) and larger cylinder diameter (D2). Our simulations show that saturated phonon backscattering for small pitch ratio (pr) nanowires gives rise to a minimum in ΛBdy/D1 at pr near unity, while the maximum in ΛBdy/D1 for large pr nanowires can be understood using a simple thermal resistor model for two individual nanowires in series. Combining our ΛBdy calculations with analytical phonon dispersion and bulk scattering models, we predict that k of periodic silicon nanowires with fixed D1can be tuned by up to 34% in the boundary scattering dominated regime by modifying D2 and p and that variations as large as 135% can be observed in the normalized thermal conductance. Our results provide insight into geometry-dependent phonon backscattering and can be used to predict k of periodic cylindrical nanowires over a range of temperatures and geometric lengthscales.

A theoretical and computational investigation of mixed mode creep crack growth along an interface

In this paper, we propose a theoretical framework for studying mixed mode (I and II) creep crack growth under steady state creep conditions. In particular, we focus on the problem of creep crack growth along an interface, whose fracture properties are weaker than the bulk material, located either side of the interface. The theoretical framework of creep crack growth under mode I, previously proposed by the authors, is extended. The bulk behaviour is described by a power-law creep, and damage zone models that account for mode mixity are proposed to model the fracture process ahead of a crack tip. The damage model is described by a traction-separation rate law that is defined in terms of effective traction and separation rate which couple the different fracture modes. Different models are introduced, namely, a simple critical displacement model, empirical Kachanov type damage models and a micromechanical based model. Using the path independence of the C^{*}-integral and dimensional analysis, analytical models are developed for mixed mode steady-state crack growth in a double cantilever beam specimen (DCB) subjected to combined bending moments and tangential forces. A computational framework is then implemented using the Finite Element method. The analytical models are calibrated against detailed Finite Element models and a scaling function (C_{k}) is determined in terms of a dimensionless quantity phi _{0} (which is the ratio of geometric and material length scales), mode mixity chi and the deformation and damage coupling parameters. We demonstrate that the form of the C_{k}-function does not change with mode mixity; however, its value depends on the mode mixity, the deformation and damage coupling parameters and the detailed form of the damage zone. Finally, we demonstrate how parameters within the models can be obtained from creep deformation, creep rupture and crack growth experiments for mode I and II loading conditions.

ZigZagZ: Improving mechanical performance in extrusion additive manufacturing by nonplanar toolpaths

This study investigates the effect of novel non-planar deposition methods in fused filament fabrication (FFF) additive manufacturing. A range of non-planar geometries were developed including a ZigZagZ sequence in which filaments were deposited as the nozzle moved in the X or Y direction while simultaneously zigzagging up-and-down i.e. in the Z direction. As a result, repeating non-planar layers were generated throughout the specimen’s geometry. The use of the ZigZagZ toolpath to deposit the material significantly improved the mechanical performance of parts manufactured by FFF in the Z-direction by up to 62% in strength, 123% in strain-at-fracture and 245% in toughness compared to an optimised conventional planar geometry. All specimens in the study had only a single filament through their thickness; they were specially developed to enable precise mechanical characterisation. This is the first work to have developed and analysed nonplanar deposition with cyclic nonplanar nozzle movement of a geometric length scale similar to the nozzle diameter. Three novel toolpath designs were developed for this study: (i) zigzag (ZZ), based on the aforementioned ZigZagZ deposition; (ii) up-down (UD) involving vertically deposited nonplanar bulges with interconnecting planar sections; (iii) forward-back (FB) employing the nozzle’s movement forward and backward during planar deposition to enhance the nozzle’s contact and promote the ploughing of the deposited filament. These designs - along with the conventional planar toolpath designs (original (OR)) - were characterised and their mechanical properties compared to generate new understanding about the impact of deposition with the Z coordinate varying along the path on performance. The geometrical outcomes of these different deposition strategies were analysed microscopically to assess the effects of nonplanar toolpaths on the filament-scale geometry. It was established that the ZZ strategies resulted in higher extruded-filament thickness compared to OR. Additionally, various ZZ designs were developed to understand the impact of the zigzag height-to-width ratio and size not only on mechanical properties, but also geometry and fracture path. Fractographic analysis indicated that nonplanar FFF extrusion promoted through-filament fracture, suggesting a reduced concentration of stresses at interlayer bonds by redirecting the load into the filament; this could contribute to the increased level of toughness observed in ZZ specimens. The understanding developed in this study is readily adaptable for the use in both single-wall and infill geometries to provide their improved mechanical performance. A broad range of potential industrial applications and research relevance resulting from the findings is discussed in addition to future development opportunities. • Zigzag interfaces improved load-bearing capacity, strain-at-fracture and toughness. • Interfacial weakness is caused by filament-scale geometry, not the bond. • Nonplanar geometry overcame geometric vulnerabilities of the interface in FFF. • Zigzag nonplanar unit can be used for both thin-walled and infilled structures. • Zigzag geometry redirected the load into the filament, increasing plasticity.

A Levy solution for bending, buckling, and vibration of Mindlin micro plates with a modified couple stress theory

Based on the modified couple stress and first-order shear deformation plate theories, a Levy-type solution is presented for bending, buckling, and vibration analyses of rectangular isotropic micro plates with simple supports at opposite edges and different boundary conditions at the other two ones. The governing equations are derived using the Hamilton's principle, and then solved by a single Fourier series expansion and the state-space method, which its implementation has not been straightforward. The results are verified with the existing ones for a fully simply supported micro plate in the literature. Finally, the effect of geometric parameters and length scale parameter on bending, buckling, and vibration behaviors of micro plates is studied. Since, there are no analytical solutions for bending, buckling loads, and natural frequencies of Mindlin micro plates with different boundary conditions in the literature and the Navier method is the only available analytical solution for the Mindlin and higher order shear deformable micro plates within the modified couple stress theory, the results presented here can be used as a benchmark in future studies. In addition, it is shown that the difference between results of the Kirchhoff and the Mindlin plate models depends not only on the plate thickness but also on the length scale parameter to thickness ratio (\(l/h)\) as well as the boundary supports. This result emphasizes the significance of analytical solutions for shear deformation models of micro plates with different boundary supports.

A novel nonlocal strain gradient Quasi-3D bending analysis of sigmoid functionally graded sandwich nanoplates

This manuscript aims to examine and investigate the bending deflection and stress distribution of sandwich functionally graded nanoplates rested on variable Winkler elastic foundation based on new quasi 3D hyperbolic shear theory in conjoint with nonlocal strain gradient theory. The sandwich sigmoidal function graded material (SFGM) is proposed with three different configurations through thickness direction. New 3D hyperbolic shear theory is exploited to satisfy parabolic variation of shear through thickness direction and zero shear at the bottom and top surfaces. Modified continuum nonlocal strain gradient theory is used to include the material and geometrical nanosize length scales. Variable Winkler foundation model is proposed to the first time to include the variable elastic environmental under the nanoplate. The comprehensive model and governing equilibrium equations of SFG nanoplates is derived in detail with principle of virtual work and solved analytically by Galerkin method. The developed model is verified with previous work and parametric studies are presented to illustrate influences of the elastic foundation models, sigmoidal distribution index constant, configuration of sandwich plate, material and length nanoscales, boundary conditions on the static deflection and stress distributions of FG sigmodal nanoplate. The proposed model can be applied in design and analysis of NEMS structure under static load, such as nanocapacitor and nanoswitch.

Experimental investigation of the impact of porous parameters on trailing-edge noise

This paper presents a sensitivity and parametric study of the sound generation at the non-tortuous and wall-normal permeable trailing edge of an aerofoil. Design parameters for the porous properties include the porosity, pore-size and porous-coverage. For a combination of large pore-size, small porosity and large porous-coverage, wake vortex shedding is likely to be triggered, and either sharp tone or broadened tone will dominate the radiated field. Using the appropriate hydrodynamic and geometrical length scales, the radiated spectra for the tones are found to follow the Strouhal number relationship, thus allow a reasonably accurate prediction of the primary tone frequency. These extraneous tones can potentially undermine the current porous trailing edge concept. Still, they can also be avoided if the porous parameters are mostly of small pore-size (sub-millimetre), medium to large porosity or small porous-coverage. Under these porous settings, better spatially distributed permeable air will seep through the surface and disrupt the generation mechanism of the turbulent boundary layer, which then translate into a lower level of turbulent broadband noise radiation. The most optimised non-tortuous, wall-normal permeable trailing edge tested in the current study can achieve a maximum of 7 dB reduction for the turbulent broadband noise. Considering that the primary trailing edge noise source is situated very near to the edge, a targeted approach (i.e. small porous-coverage) is already sufficient to achieve significant trailing edge broadband noise reduction.

Torsional dynamics of bi-directional functionally graded small-scale tubes possessing a variable length scale parameter

The aim of the current study is to investigate torsional vibrations of bi-directional functionally graded (FG) small-scale tubes. The size effect is captured using the modified couple stress theory (MCST). Through-the-thickness distribution profile of material properties are characterized by employing a power-law function, which incorporates a gradation index, β, and longitudinal varation in material properties is featured by an exponential function which employs a gradient parameter, α. Similar to other material properties, continuous spatial variation of the length scale parameter is also incorporated into the analysis of two-dimensionally FG tubes. The model, including governing equations of motion and boundary conditions, is developed through the utilization of Hamilton’s principle. By employing differential quadrature method (DQM) numerical results regarding torsional vibrations are provided. Accuracy of the proposed model and procedures are verified through verification comparisons made to limiting cases available in the literature. Numerical results reveal effects of various material and geometric parameters on natural frequencies and indicate that torsional free vibration characteristics of small-sized tubes are sensitive to distribution profile of constituents, geometric aspect ratios and effective length scale parameter. Results also demonstrate that length scale parameter variation has significant influence on natural frequencies which justifies the necessity of its consideration in analyses.

Assessment of Grid Anisotropy Effects on Large-Eddy-Simulation Models with Different Length Scales

The effect of the eddy viscosity length scale on the accuracy of large-eddy simulation (LES) is investigated for several commonly used subgrid models. Multiple types of length scales are considered, both purely geometric length scales (including the cube root of the cell volume and the maximum cell dimension) and length scales that depend on both the grid and the flow. The assessment focuses on how the LES prediction changes for increasingly anisotropic grids in forced isotropic turbulence and channel flow at high Reynolds number. Although some length scales perform better than others, the main conclusion is that no single length scale produces LES results that are completely independent of the grid anisotropy. Because every model and length scale tested produces accurate results on at least one grid and inaccurate results on at least one other grid, the study shows clearly the need to assess LES models on multiple grids with multiple different anisotropy ratios.

Towards a new concept of structural integrity

This paper offers a novel and forward-looking approach to the concept of structural integrity, comprising and covering all length scales in the conceptual framework of the innovative concepts of mega-, macro-, micro- and nano-structural integrity, and thus the science of fracture mechanics & structural integrity can be considered as a branch of material science & engineering, and the novel concept of material integrity can also be coined, since crack-like defects can affect material behavior. In this conceptual framework, any material can be considered as a structure (a quite more general concept with a broader meaning and comprising also, e.g., the structure of a Bach’s fugue or of a Beethoven’s symphony), such a structure being analyzable at different geometric levels or length scales: mega-, macro-, micro- and nano-levels.

Nonlinear microrheology of active Brownian suspensions.

The rheological properties of active suspensions are studied via microrheology: tracking the motion of a colloidal probe particle in order to measure the viscoelastic response of the embedding material. The passive probe particle with size R is pulled through the suspension by an external force Fext, which causes it to translate at some speed Uprobe. The bath is comprised of a Newtonian solvent with viscosity ηs and a dilute dispersion of active Brownian particles (ABPs) with size a, characteristic swim speed U0, and a reorientation time τR. The motion of the probe distorts the suspension microstructure, so the bath exerts a reactive force on the probe. In a passive suspension, the degree of distortion is governed by the Péclet number, Pe = Fext/(kBT/a), the ratio of the external force to the thermodynamic restoring force of the suspension. In active suspensions, however, the relevant parameter is Ladv/l = UprobeτR/U0τR∼Fext/Fswim, where Fswim = ζU0 is the swim force that propels the ABPs (ζ is the Stokes drag on a swimmer). When the external forces are weak, Ladv≪l, the autonomous motion of the bath particles leads to "swim-thinning," though the effective suspension viscosity is always greater than ηs. When advection dominates, Ladv≫l, we recover the familiar behavior of the microrheology of passive suspensions. The non-Newtonian behavior for intermediate values of Ladv/l is determined by l/Rc = U0τR/Rc-the ratio of the swimmer's run length l to the geometric length scale associated with interparticle interactions Rc = R + a. The results in this manuscript are approximate as they are based on numerical solutions to mean-field equations that describe the motion of the active bath particles.

Permeability estimation from induced polarization: an evaluation of geophysical length scales using an effective hydraulic radius concept

ABSTRACTGeophysical length scales defined from induced‐polarization measurements can be used in models of permeability (k) prediction. We explore the relative merit of different induced‐polarization parameters as proxies of an effective hydraulic radius (reff) that can be used to predict permeability from a modified Hagen–Poiseuille equation. Whereas geometrical measures of the hydraulic radius are good proxies of reff, the induced‐polarization measures are not well correlated with reff. However, a new proxy of reff that considers both imaginary conductivity and formation factor shows an improved correlation with reff. The resulting model enables a better quality of permeability prediction compared with the other geophysical length scales, but does not reach the predictive quality of the models based on geometrical length scales. The specific polarizability defined when incorporating the effect of the formation factor on imaginary conductivity appears to be independent of pore geometry, indicating that it is the correct parameter representing the role of the surface electrochemistry on the induced‐polarization effect. However, the joint dependence of induced‐polarization measurements on both the pore radius and the tortuosity and porosity of the interconnected pore network is a limitation to the widely explored use of induced‐polarization measurements to isolate surface properties from volumetric properties of the interconnected pore network.

Numerical Modeling of Frictional Contact Between a Blunt Tool and Quasi-Brittle Rock

Frictional contact is an important component of rock cutting with a blunt cutter. There has been a gap in knowledge of frictional contact between experimental tests conducted on quasi-brittle rocks and previous numerical and analytical analyses on elastoplastic rocks. The average contact stress between a sliding wear flat and a rock is generally overestimated in previous studies without considering material length scales. This study analyzes the contact stress using an elasto-plastic-damage model that accounts for the material length scales of quasi-brittle materials. The constitutive model is developed for quasi-brittle materials by incorporating fracture energy, and it is implemented in the commercial finite element software ABAQUS as a user-defined material model. The implemented constitutive model is verified under uniaxial tension and compression. The average contact stress in frictional contact is mainly governed by a dimensionless elastoplastic parameter $$ \eta $$ for an elastoplastic rock, while it is predominantly governed by two dimensionless parameters: an elastoplastic parameter $$ \eta $$ and a brittleness number $$ \xi $$ for a quasi-brittle rock. The dimensionless elastoplastic parameter $$ \eta $$ contrasts the magnitude of a characteristic elastic contact stress to the yield strength of a material, and the brittleness number $$ \xi $$ is the ratio of a geometrical length scale to a material length scale. The tool–rock interface can be approximately divided into an elastic zone and a damage process zone. The contact stress generally increases in the elastic zone and then decreases in the damage process zone towards the tip of the slider. The average contact stress generally increases with the dimensionless elastoplastic parameter $$ \eta $$ and then levels off at a limit value, and it generally decreases with the brittleness number $$ \xi $$ . This study reduces the gap in previous studies to some extent by introducing the brittleness number $$ \xi $$ in frictional contact. The current numerical results of the average contact stress are generally consistent with typical experimental results conducted on quasi-brittle rocks, despite limitations in the finite element modeling.

On the Stability of Density Stratified Flow Below a Ponded Surface

Flooding of coastal areas with seawater often leads to density stratification. The stability of the density-depth profile in a porous medium initially saturated with a fluid of density rho _mathrm{f} after flooding with a salt solution of higher density rho _mathrm{s} is analyzed. The standard convection/diffusion equation subject to the so-called Boussinesq approximation is used. The depth of the porous medium is assumed to be infinite in the analytical approaches and finite in the numerical simulations. Two cases are distinguished: the laterally unbounded {{{mathbf {{small {uppercase {case~A}}}}}}} and the laterally bounded {{{mathbf {{small {uppercase {case~B}}}}}}}. The ratio of the diffusivity and the density difference (rho _mathrm{s} - rho _mathrm{f}) induced gravitational shear flow is an intrinsic length scale of the problem. In the unbounded {{{mathbf {{small {uppercase {case~A}}}}}}}, this geometric length scale is the only length scale and using it to write the problem in dimensionless form results in a formulation with Rayleigh number R = 1. In the bounded {{{mathbf {{small {uppercase {case~B}}}}}}}, the lateral geometry provides another length scale. Using this geometrical length scale to write the problem in dimensionless form results in a formulation with a Rayleigh number R given by the ratio of the geometric and intrinsic length scales. For both {{{mathbf {{small {uppercase {case~A}}}}}}} and {{{mathbf {{small {uppercase {case~B}}}}}}}, the well-known Boltzmann similarity solution provides the ground state. Three analytical approaches are used to study the stability of this ground state, the first two based on the linearized perturbation equation for the concentration and the third based on the full nonlinear equation. For the first linear approach, the surface spatial density gradient is used as an approximation of the entire background density profile. This results in a crude estimate of the L^2-norm of the concentration showing that the perturbation at first grows, but eventually decays in time. For the other two approaches, the full ground-state solution is used, although for the second linear approach subject to the restriction that the ground state slowly evolves in time (the so-called frozen profile approximation). Just like the ground state, the resulting eigenvalue problems can be written in terms of the Boltzmann variable. The linearized stability approach holds only for infinitesimal small perturbations, whereas the nonlinear, variational energy approach is not subject to such a restriction. The results for all three approaches can be expressed in terms of Boltzmann sqrt{t} transformed relationships between the system Rayleigh number and perturbation wave number. The results of the linear and nonlinear approaches are surprisingly close to each other. Based on the system Rayleigh number, this allows delineation of systems that are unconditionally stable, marginally stable, or transiently unstable. These analytical predictions are confirmed by direct two-dimensional numerical simulations, which also show the details of the transient instabilities as function of the wave number for {{{mathbf {{small {uppercase {case~A}}}}}}} and the wave number and Rayleigh number for {{{mathbf {{small {uppercase {case~B}}}}}}}. A numerical example of the effect of a layer with low permeability is also shown. Using typical values of the physical parameters, the analytical and numerical results are interpreted in terms of dimensional length and time scales. In particular, an explicit stability criterion is given for vertical column experiments.