Dimensionless Length Scale Research Articles

Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory

On the basis of the modified strain gradient elasticity theory, the free vibration characteristics of curved microbeams made of functionally graded materials (FGMs) whose material properties vary in the thickness direction are investigated. A size-dependent first-order shear deformation beam model is developed containing three internal material length scale parameters to incorporate small-scale effect. Through Hamilton’s principle, the higher-order governing equations of motion and boundary conditions are derived. Natural frequencies of FGM curved microbeams corresponding to different mode numbers are evaluated for over a wide range of material property gradient index, dimensionless length scale parameter and aspect ratio. Moreover, the results obtained via the present non-classical first-order shear deformation beam model are compared with those of degenerated beam models based on the modified couple stress and the classical theories. It is found that the difference between the natural frequencies predicted by the various beam models is more significant for lower values of dimensionless length scale parameter and higher values of mode number.

Size-dependent buckling analysis of functionally graded third-order shear deformable microbeams including thermal environment effect

Presented herein is the prediction of buckling behavior of size-dependent microbeams made of functionally graded materials (FGMs) including thermal environment effect. To this purpose, strain gradient elasticity theory is incorporated into the classical third-order shear deformation beam theory to develop a non-classical beam model which contains three additional internal material length scale parameters to consider the effects of size dependencies. The higher-order governing differential equations are derived on the basis of Hamilton’s principle. Afterward, the size-dependent differential equations and related boundary conditions are discretized along with commonly used end supports by employing generalized differential quadrature (GDQ) method. A parametric study is carried out to demonstrate the influences of the dimensionless length scale parameter, material property gradient index, temperature change, length-to-thickness aspect ratio and end supports on the buckling characteristics of FGM microbeams. It is revealed that temperature change plays more important role in the buckling behavior of FGM microbeams with higher values of dimensionless length scale parameter.

Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates

In the present study, the dynamic pull-in instability and free vibration of circular microplates subjected to combined hydrostatic and electrostatic forces are investigated. To take size effects into account, the strain gradient elasticity theory is incorporated into the Kirchhoff plate theory to develop a nonclassical plate model including three internal material length scale parameters. By using Hamilton’s principle, the higher-order governing equation and the corresponding boundary conditions are obtained. Afterward, a generalized differential quadrature (GDQ) method is employed to discritize the governing differential equations along with simply supported and clamped edge supports. To evaluate the pull-in voltage and vibration frequencies of actuated microplates, the hydrostatic-electrostatic actuation is assumed to be calculated by neglecting the fringing field effects and utilizing the parallel plate approximation. Also, a comparison between the pull-in voltages predicted by the strain gradient theory and the degenerated ones is presented. It is revealed that increasing the dimensionless internal length scale parameter or decreasing the applied hydrostatic pressures leads to higher values of the pull-in voltage. Moreover, it is found that the value of pull-in hydrostatic pressure decreases corresponding to higher dimensionless internal length scale parameters and applied voltages.

Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory

Size-dependent dynamic stability response of higher-order shear deformable cylindrical microshells made of functionally graded materials (FGMs) and subjected to simply supported end supports is investigated. Material properties of the microshells vary in the thickness direction according to the Mori–Tanaka scheme. The modified couple stress elasticity theory in conjunction with the classical higher-order shear deformation shell theory is utilized to develop non-classical shell model containing additional internal length scale parameter to interpret size effect. The differential equations of motion and boundary conditions are derived by using Hamilton’s principle. The governing equations are then written in the form of Mathieu–Hill equations and then Bolotin’s method is employed to determine the instability regions. Selected numerical results are given to indicate the influences of internal length scale parameter, material property gradient index, static load factor and axial wave number on the dynamic stability behavior of FGM microshells. It is found that the width of the instability region for an FGM microshell increases with the decrease of the value of dimensionless length scale parameter. Moreover, it is shown that the classical shell model has an overestimated prediction for the width of instability region corresponding to the FGM microshells especially with lower values of material property gradient index.

On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory

The prime aim of the present study is to predict the free vibration behavior of microplates made of functionally graded materials (FGMs). The material properties of FGM microplates are assumed to be varied across the thickness of the microplates according to the Mori–Tanaka homogenization technique. On the basis of strain gradient elasticity theory, a non-classical higher-order shear deformable plate model containing three material length scale parameters is developed which can effectively capture the size dependencies. By using Hamilton’s principle, the size-dependent governing differential equations of motion and associated boundary conditions are derived. To evaluate the natural frequencies of FGM microplates, a Navier-type closed-form solution is carried out in which the generalized displacements are stated as multiplication of undetermined functions with known trigonometric functions so as to satisfy identically the simply-supported boundary conditions at all edges. Selected numerical results are presented to reveal the influences of dimensionless length scale parameter, material property gradient index and aspect ratio on the free vibration characteristics of FGM microplates. It is found that by approaching the thickness of microplates to the value of internal material length scale parameter, the natural frequency increases considerably.

Study of Small Scale Effects on the Nonlinear Vibration Response of Functionally Graded Timoshenko Microbeams Based on the Strain Gradient Theory

In the current study, the nonlinear free vibration behavior of microbeams made of functionally graded materials (FGMs) is investigated based on the strain gradient elasticity theory and von Karman geometric nonlinearity. The nonclassical beam model is developed in the context of the Timoshenko beam theory which contains material length scale parameters to take the size effect into account. The model can reduce to the beam models based on the modified couple stress theory (MCST) and the classical beam theory (CBT) if two or all material length scale parameters are taken to be zero, respectively. The power low function is considered to describe the volume fraction of the ceramic and metal phases of the FGM microbeams. On the basis of Hamilton’s principle, the higher-order governing differential equations are obtained which are discretized along with different boundary conditions using the generalized differential quadrature method. The dimensionless linear and nonlinear frequencies of microbeams with various values of material property gradient index are calculated and compared with those obtained based on the MCST and an excellent agreement is found. Moreover, comparisons between the various beam models on the basis of linear and nonlinear types of strain gradient theory (SGT) and MCST are presented and it is observed that the difference between the frequencies obtained by the SGT and MCST is more significant for lower values of dimensionless length scale parameter.

Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer

Abstract In this work we develop improved asymptotic solutions to one-dimensional Fredholm integral equations of the first kind using linear regression. For the cases under consideration the unknown function is the flux distribution along a strip, and the integral equation depends on a parameter or a number of parameters, i.e. the Peclet number, the Biot number, the dimensionless length scales etc. It is assumed that asymptotic solutions, with respect to the parameters, are available. We show that the asymptotic solutions can be improved and extended by relaxing the coefficients associated with them and applying regression analysis to yield best-fit coefficients. The asymptotic solutions may even be combined to obtain a matched asymptotic expansion. Explicit expressions for the coefficients, which can depend on a number of parameters, are obtained using regression analysis, i.e. by creating a variational principle for the Fredholm Integral Equation and employing the least squares method. The resulting expression, although it provides an approximate solution to the flux distribution, it is explicit and estimates accurately the overall transport rate.

Instabilities in the homogeneous cooling of a granular gas: A quantitative assessment of kinetic-theory predictions

Previous work has indicated that inelastic grains undergoing homogeneous cooling may be unstable, giving rise to the formation of velocity vortices, which may also lead to particle clustering. In this effort, molecular dynamics (MD) simulations are performed over a wide parameter space to determine the critical system size demarcating the stable and unstable regions. Specifically, a system of monodisperse, frictionless, inelastic hard spheres is simulated for restitution coefficients e ≥ 0.6 and solids fractions φ ≤ 0.4. Simulations for each e, φ pairing are then carried out over a range of system sizes to determine the critical dimensionless length scale LC/d (L is the system length and d is the particle diameter), above which velocity vortices appear (unstable system) and below which they are suppressed (stable system). The results show excellent agreement with the theoretical predictions obtained by Garzó [Phys. Rev. E 72, 021106 (2005)] using a linear stability analysis of kinetic-theory-based (continuum) equations that were derived from the Enskog equation. Finally, the time required for onset of the unstable behavior is also explored via MD and found to be a universal function of the ratio of L/d to LC/d.

Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory

Investigated herein is the free vibration characteristics of microbeams made of functionally graded materials (FGMs) based on the strain gradient Timoshenko beam theory. The material properties of the functionally graded beams are assumed to be graded in the thickness direction according to the Mori–Tanaka scheme. Using Hamilton’s principle, the equations of motion together with corresponding boundary conditions are obtained for the free vibration analysis of FGM microbeams including size effect. A detailed parametric study is performed to indicate the influences of beam thickness, dimensionless length scale parameter, and slenderness ratio on the natural frequencies of FGM microbeams. Moreover, a comparison between the various beam models on the basis of the classical theory (CT), modified couple stress theory (MCST), and strain gradient theory (SGT) is presented for different values of material property gradient index. It is observed that the value of gradient index play an important role in the vibrational response of the microbeams of lower slenderness ratios. It is further observed that by increasing the length-to-thickness ratio of the microbeam, the value of dimensionless natural frequency tends to decrease for all amounts of the gradient index.

Constructal solar chimney configuration

In this study, we describe the constructal-theory search for the geometry of a solar chimney. The objective is to increase the power production over the area occupied by the plant. The ratio height/radius, maximum mass flow rate and maximum power under the constraints of a fixed area and volume are determined. We find that the power generated per unit of land area is proportional to the length scale of the power plant. The analysis is validated by a detailed mathematical model. Pressure losses are reported in terms of the dimensionless length scale of the system, and are illustrated graphically. They indicate that the pressure drop at the collector inlet and at the transition section between the collector and chimney are negligible, and the friction loss in the collector can be neglected when the svelteness (Sv) of the entire flow architecture is greater than approximately 6.

Factors that control the angle of shear bands in geodynamic numerical models of brittle deformation

Numerical models of brittle deformation on geological timescales typically use a pressure-dependent (Mohr–Coulomb or Drucker–Prager) plastic flow law to simulate plastic failure. Despite its widespread usage in geodynamic models of lithospheric deformation, however, certain aspects of such plasticity models remain poorly understood. One of the most prominent questions in this respect is: what are the factors that control the angle of the resulting shear bands? Recent theoretical work suggest that both Roscoe (45°), Coulomb angles (45 +/− ϕ/2, where ϕ is the angle of internal friction) and Arthur angles (45° +/− ( ϕ + Ψ/4) where Ψ is the dilation angle), as well as all intermediate angles are possible. Published numerical models, however, show a large range of shear band angles with some codes favoring Arthur angles, whereas others yield Coulomb angles. In order to understand what causes the differences between the various numerical models, here I perform systematic numerical simulations of shear localization around an inclusion of given length scale. Both numerical (element type), geometrical and rheological (viscoplastic versus viscoelastoplastic) effects are studied. Results indicate that the main factor, controlling shear band angle, is the non-dimensional ratio between the length scale of the heterogeneity d and the size of the numerical mesh Δx. Coulomb angles are observed only in cases where the inclusion is resolved well ( d/ Δx > 5–10), and in which it is located sufficiently far from the boundary of the box. In most other cases, either Arthur or Roscoe orientations are observed. If heterogeneities are one element in size, Coulomb angles are thus unlikely to develop irrespective of the employed numerical resolution. Whereas differences in element types and rheology do have consequences for the maximum obtainable strain rates inside the shear bands, they only have a minor effect on shear band angles. Shear bands, initiated from random noise or from interactions of shear bands with model boundaries or other shear bands, result in stress heterogeneities with dimensionless length scales d/Δx ~ 1–2. Such shear bands are thus expected to form Roscoe or Arthur orientations, consistent with the findings in previous numerical models.

Turbulent length–time scales distributions in hydraulic jumps

Air–water flow measurements were performed in hydraulic jump flows for a range of inflow Froude numbers. The experiments were conducted in a large-sized facility using phase-detection intrusive probes. The void fraction measurements showed the presence of an advective diffusion shear layer where the air concentration vertical distributions were successfully compared with an analytical solution of the advective diffusion equation for air bubbles. In the air–water shear layer, a new empirical relationship between the maximum air concentration decay as a function of both the distance from the jump toe and the inflow Froude number was derived. Air–water turbulent time and length scales were deduced from auto- and cross-correlation analyses based on the method of Chanson (2007). The result provided some characteristic transverse time and length scales of the eddy structures advecting the air bubbles in the developing shear layer. The turbulence time scale data showed an increase with the relative elevation above the bed, as well as some decrease with increasing distance from the toe. The dimensionless integral turbulent length scale Lxz/d1was closely related to the inflow depth.

Nonlinear alternating electric field dipolophoresis of spherical nanoparticles

We consider the nonlinear electrokinetic problem of a freely suspended conducting (infinitely polarized) spherical micro- or nanosize particle surrounded by an unbounded electrolyte solution. The uncharged particle is exposed to an alternating (ac), nonuniform, and axisymmetric ambient electric field. As a result, the particle acquires a dipolophoretic (DIP) mobility of magnitude, which is quadratic in the amplitude of the applied electric field. The resulting phoretic velocity is driven by two independent nonlinear mechanisms. One is the common dielectrophoretic effect, whereby the nonuniform field exerts an electrostatic force on the image multipole singularity system within the particle. The other is the so-called “induced-charge electrophoresis” resulting from the action of the electric field on the excess charge around the particle induced in the diffused layer by the field itself. Both effects are quadratic in the amplitudes of the electric field and depend on the forcing frequency and on the dimensionless Debye screening length scale. It is demonstrated in the sequel that the two generally act in opposite directions which may result in mutual cancellation. Under the assumptions of a “weak” electric field and the neglect of surface conductance, we present a concise analysis of the resulting nonlinear streaming (dc) velocity (averaged over a period) for a spherical metalic particle that is exposed to a time-harmonic oscillating (ac) electric field. The analysis of this fundamental nonlinear DIP problem is provided for arbitrary forcing frequencies and for any Debye thickness. Numerical simulations are given for the case of a “two-mode” interaction consisting of a uniform-gradient electric field combined with a uniform field, where the two modes are either “in” or “out” of phase.

Experimental investigation of single particle settling in turbulence generated by oscillating grid

Abstract In this study, turbulence influence on the settling behavior of solid particles was investigated experimentally in confined turbulent aquatic environment generated by oscillating grid. An enhanced PIV system was employed to conduct simultaneous velocity measurements of individual settling particles and ambient fluid. Grains varying in shape (spherical and cylindrical) and diameter (2.78–7.94 mm) were tested with different turbulent conditions. The results showed clearly that the settling behavior of particles subjected to turbulence is significantly modified. First, the settling velocity modification is closely correlated to the mean vertical velocity of the fluid zone (very close to the settling particle), which in size is in the order of a few particle diameters. Second, the relative settling velocity is smaller than the still water terminal velocity for the most cases. Lastly, the fluctuation in the settling velocity is significantly increased, as compared to the still water conditions, and clearly dependent on the turbulence intensity. The experimental data were also analyzed with dimensional considerations. By comparing to literature, turbulence effects on the relative settling velocity were discussed with regard to Stokes number, Richardson number and dimensionless turbulence length scale. Finally, a simple analytical model was proposed for estimating the turbulence-modified settling velocity.

Turbulence measurements in the bubbly flow region of hydraulic jumps

A hydraulic jump is characterized by a highly turbulent flow with macro-scale vortices, some kinetic energy dissipation and a bubbly two-phase flow structure. New air–water flow measurements were performed in a large-size facility using two types of phase-detection intrusive probes: i.e. single-tip and double-tip conductivity probes. These were complemented by some measurements of free-surface fluctuations using ultrasonic displacement meters. The void fraction measurements showed the presence of an advective diffusion shear layer in which the void fractions profiles matched closely an analytical solution of the advective diffusion equation for air bubbles. The free-surface fluctuations measurements showed large turbulent fluctuations that reflected the dynamic, unsteady structure of the hydraulic jumps. The measurements of interfacial velocity and turbulence level distributions provided new information on the turbulent velocity field in the highly-aerated shear region. The velocity profiles tended to follow a wall jet flow pattern. The air–water turbulent integral time and length scales were deduced from some auto- and cross-correlation analyses based upon the method of Chanson [H. Chanson, Bubbly flow structure in hydraulic jump, Eur. J. Mech. B/Fluids 26 (3) (2007) 367–384], providing the turbulent scales of the eddy structures advecting the air bubbles in the developing shear layer. The length scale L xz is an integral air–water turbulence length scale which characterized the transverse size of the large vortical structures advecting the air bubbles. The experimental data showed that the dimensionless integral turbulent length scale L xz / d 1 was closely related to the inflow depth: i.e. L xz / d 1 = 0.2–0.8, with L xz increasing towards the free-surface.

A pedagogical look at Jeans' density scale

We illustrate the derivations of Jeans' criteria for the gravitational instabilities in a static homogeneous Newtonian system for pedagogical objectives. The critical Jeans density surface is presented in terms of dimensionless sound speeds and (characteristic) length scales.

On the flame height definition for upward flame spread

Flame height is defined by the experimentalists as the average position of the luminous flame and, consequently is not directly linked with a quantitative value of a physical parameter. To determine flame heights from both numerical and theoretical results, a more quantifiable criterion is needed to define flame heights and must be in agreement with the experiments to allow comparisons. For wall flames, steady wall flame experiments revealed that flame height may be defined by a threshold value on the wall heat flux. From steady wall flame measurements, three definitions of flame height from wall heat flux are retained: the first is based on the continuous flame while the two others are based on threshold values of 4 kW/m 2 and 10 kW/m 2. These definitions are applied to determine flame heights from a two-dimensional time-dependent CFD model used to describe flame spread on a vertical slab of PMMA. Results show that the predicted flame heights are consistent with the available data of the literature. Defining flame height by threshold values on the wall heat flux of 4 and 10 kW/m 2 allows the correlation of the wall heat flux in terms of ( x− x p)/( x fl− x p), which is the dimensionless characteristic length scale for upward flame spread. It is also found that the continuous flame is not a characteristic length for the heat transfer to the unburnt fuel and is not really appropriate to define flame height in upward flame spread.

The spatial relationships between dissipation and production rates and vortical structures in turbulent boundary and mixing layers

A novel approach for studying the spatial relationship between the production and dissipation rates of turbulent kinetic energy and vortical structures is presented. Two turbulent flows were investigated: the zero pressure gradient boundary layer and the two-stream mixing layer. In both flows, a multisensor hot-wire probe was used to measure the velocity components in all three coordinate directions, as well as six components of the velocity gradient tensor. The remaining three velocity gradients were determined using Taylor’s hypothesis. With these data, the “instantaneous” production and dissipation rates, defined by P=−(∂U¯i∕∂xj)uiuj and D=−ν[(∂ui∕∂xj)2+(∂ui∕∂xj)(∂uj∕∂xi)], respectively, were determined. Cross-correlating the fluctuations of these two signals reveals that they are not randomly distributed in time with respect to each other; rather they display significant levels of correlation. Plotting the cross-correlation coefficients versus a dimensionless length scale, defined as L′=sgn(τ)∣τ∣∕νU¯, reveals an asymmetric pattern that persists at several cross-stream locations for both flows. Furthermore, correlating both the dissipation and production rates with a vortex identifier, ωxy=[(ωx)2+(ωy)2]1∕2, also reveals consistent cross-stream patterns. The magnitude of these correlations and their persistent shapes across the flows suggest that the spatial separation between regions of concentrated dissipation and production rates is associated with the presence of quasistreamwise vortices in both of these flows. More specifically, they imply that regions of concentrated rates of dissipation are primarily in the cores of the vortices, whereas regions of rates of production are more concentrated on their periphery.

Diffusion of Mesoscopic Probes in Aqueous Polymer Solutions Measured by Fluorescence Recovery after Photobleaching

Fluorescence recovery after photobleaching (FRAP) has been used to follow the diffusion of mesoscopic probes (1 nm < R < 20 nm) in aqueous poly(ethylene oxide) (PEO) and guar galactomannan solutions. We define “mesoscopic” as the regime for which the size of the diffusing species is of the same order as the screening length ξ in the polymer matrix solution. We show that diffusion depends not only on the dimensionless length scale R/ξ but also on the dimensionless time scale corresponding to the relaxation of the polymer mesh by “constraint release” vs the time for motion of the probe species over the length ξ. Two different FRAP techniques were used: fringe pattern bleaching and recovery (FPBR) and confocal scanning laser microscopy (CSLM). The effect of probe structure on diffusion through polymer matrices was investigated by measurements on probes with differing fractal dimensions (df): proteins and polystyrene latex particles behave as rigid spheres (df = 3); dextrans are slightly branched polymers w...

The scattering phase shift due to Bragg resonance in one-dimensional fluctuation reflectometry

An explicit integral representation is derived for the tangent of the phase shift due to one-dimensional (1D) scattering of an S-polarized, O-mode, electromagnetic field from a localized wavepacket sitting on top of an inhomogeneous plasma. A Green function technique is used in the derivation together with the Born approximation. The integral representation is evaluated using asymptotic techniques and Bragg resonance is seen to be the dominant mechanism producing the phase shifts due to fluctuations with wavelengths that are short compared to the Airy length. By suitably normalizing the governing differential equation, we have identified the two dominant parameters that control the approximations in our analysis. These are delta n/n0 and kf. The first is the magnitude of the maximum density fluctuation multiplied by the square of the dimensionless length scale that characterizes both the background plasma density profile (with scalelength L) and the incoming microwave field (with vacuum wavenumber k0): delta n/n0 identical to ( delta n/n0)*(k0L)23/. The second is the fluctuation wavenumber normalized to k0 and scaled by the similarly normalized Airy wavenumber: kf identical to (kf/k0)*(k0L)13/. The Born approximation is expected to be valid as long as delta n/n0<1, and the Bragg resonance picture dominates as long as kf>1.