Stress-free Boundary Conditions Research Articles

Spin-down in a rapidly rotating cylinder container with mixed rigid and stress-free boundary conditions

A comprehensive study of the classical linear spin-down of a constant-density viscous fluid (kinematic viscosity $\unicode[STIX]{x1D708}$) rotating rapidly (angular velocity $\unicode[STIX]{x1D6FA}$) inside an axisymmetric cylindrical container (radius $L$, height $H$) with rigid boundaries, which follows the instantaneous small change in the boundary angular velocity at small Ekman number $E=\unicode[STIX]{x1D708}/H^{2}\unicode[STIX]{x1D6FA}\ll 1$, was provided by Greenspan & Howard (J. Fluid Mech., vol. 17, 1963, pp. 385–404). For that problem $E^{1/2}$ Ekman layers form quickly, triggering inertial waves together with the dominant spin-down of the quasi-geostrophic (QG) interior flow on the $O(E^{-1/2}\unicode[STIX]{x1D6FA}^{-1})$ time scale. On the longer lateral viscous diffusion time scale $O(L^{2}/\unicode[STIX]{x1D708})$, the QG flow responds to the $E^{1/3}$ sidewall shear layers. In our variant, the sidewall and top boundaries are stress-free, a set-up motivated by the study of isolated atmospheric structures such as tropical cyclones or tornadoes. Relative to the unbounded plane layer case, spin-down is reduced (enhanced) by the presence of a slippery (rigid) sidewall. This is evidenced by the QG angular velocity, $\unicode[STIX]{x1D714}^{\star }$, evolution on the $O(L^{2}/\unicode[STIX]{x1D708})$ time scale: spatially, $\unicode[STIX]{x1D714}^{\star }$ increases (decreases) outwards from the axis for a slippery (rigid) sidewall; temporally, the long-time $(\gg L^{2}/\unicode[STIX]{x1D708})$ behaviour is dominated by an eigensolution with a decay rate slightly slower (faster) than that for an unbounded layer. In our slippery sidewall case, the $E^{1/2}\times E^{1/2}$ corner region that forms at the sidewall intersection with the rigid base is responsible for a $\ln E$ singularity within the $E^{1/3}$ layer, causing our asymptotics to apply only at values of $E$ far smaller than can be reached by our direct numerical simulation (DNS) of the linear equations governing the entire spin-down process. Instead, we solve the $E^{1/3}$ boundary layer equations for given $E$ numerically. Our hybrid asymptotic–numerical approach yields results in excellent agreement with our DNS.

Post buckling analysis of shallow composite shells based on the third order shear deformation theory

In the present paper, a higher-order shear deformation theory of elastic shells was developed for geometrically nonlinear analysis of laminated shells of orthotropic layers. The theory can better represent the kinematics, may not require shear correction factors and can yield more accurate interlaminar stress distributions. The nonlinear mathematical model is derived using the Green–Lagrange type geometric nonlinearity in the framework of the higher order shear deformation theory. The developed model accounts for parabolic distribution of the transverse shear strains through the thickness and tangential stress-free boundary conditions on the boundary surfaces of the shell. Therefore, shear correction factors of the usual shear deformation theory are not required in the present theory. The principle of the virtual work forms the basis to derive the nonlinear finite element equations. The nonlinear equilibrium equations are solved using an incremental iterative technique based on the arc length method.

Asymptotic theory for torsional convection in rotating fluid spheres

This paper is concerned with the classical, well-studied problem of convective instabilities in rapidly rotating, self-gravitating, internally heated Boussinesq fluid spheres. Sanchez et al. (J. Fluid Mech., vol. 791, 2016, R1) recently found, unexpectedly via careful numerical simulation, that non-magnetic convection in the form of axially symmetric, equatorially antisymmetric torsional oscillation is physically preferred in a special range of small Prandtl number for rapidly rotating fluid spheres with the stress-free boundary condition. We derive an asymptotic solution describing convection-driven torsional oscillation – whose flow velocity and pressure are fully analytical and in closed form – that confirms the result of the numerical analysis and is in quantitative agreement with the numerical solution. We also demonstrate, through the derivation of a different asymptotic solution, that convection-driven torsional oscillation cannot occur in rapidly rotating fluid spheres with the no-slip boundary condition.

Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone

In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional B\'enard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress free boundary condition on the walls, and fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

Free boundary value problem to 3D spherically symmetric compressible Navier–Stokes–Poisson equations

In the paper, we consider the free boundary value problem to 3D spherically symmetric compressible isentropic Navier-Stokes-Poisson equations for self-gravitating gaseous stars with $\gamma$-law pressure density function for $6/5 <\gamma \leq 4/3$. For stress free boundary condition and zero flow density continuously across the free boundary, the global existence of spherically symmetric weak solutions is shown, and the regularity and long time behavior of global solution are investigated for spherically symmetric initial data with the total mass smaller than a critical mass.

Mechanical behaviour of FGM sandwich plates using a quasi-3D higher order shear and normal deformation theory

This paper presents an original hyperbolic (first present model) and parabolic (second present model) shear and normal deformation theory for the bending analysis to account for the effect of thickness stretching in functionally graded sandwich plates. Indeed, the number of unknown functions involved in these presents theories is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of ail displacements across the thickness and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. It is evident from the present analyses; the thickness stretching effect is more pronounced for thick plates and it needs to be taken into consideration in more physically realistic simulations. The numerical results are compared with 3D exact solution, quasi-3-dimensional solutions and with other higher-order shear deformation theories, and the superiority of the present theory can be noticed.

Dynamic Response of a Circular Tunnel in an Elastic Half Space

The vibration of a circular tunnel in an elastic half space subjected to uniformly distributed dynamic pressure at the inner boundary is studied in this paper. For comparison purposes, two different ground materials (soft and hard soil) are considered for the half space. Under the assumption of plane strain, the equations of motion for the tunnel and the surrounding medium are reduced to two wave equations in polar coordinates using Helmholtz potentials. The method of wave expansion is used to construct the displacement fields in terms of displacement potentials. The boundary conditions associated with the problem are satisfied exactly at the inner surface of the tunnel and at the interface between the tunnel and surrounding medium, and they are satisfied approximately at the free surface of the half space. A least-squares technique is used for satisfying the stress-free boundary conditions at the half space. It is shown by comparison that the stresses and displacements are significantly influenced by the properties of the surrounding soil, wave number (i.e., the frequency), depth of embedment, and thickness of the tunnel wall.

Effects of thickness stretching in FGM plates using a quasi-3D higher order shear deformation theory

In this paper, a higher order shear and normal deformation theory is presented for functionally graded material (FGM) plates. By dividing the transverse displacement into bending, shear and thickness stretching parts, the number of unknowns and governing equations for the present theory is reduced, significantly facilitating engineering analysis. Indeed, the number of unknown functions involved in the present theory is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of ail displacements across the thickness and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamiltonas principle. Analytical solutions for the bending and free vibration analysis are obtained for simply supported plates. The obtained results are compared with three-dimensional and quasi- three-dimensional solutions and those predicted by other plate theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of functionally graded plates.

The break-up of free films pulled out of a pure liquid bath

In this paper, we derive a lubrication model to describe the non-stationary free liquid film that is created when a vertical frame is pulled out of a liquid reservoir at a given velocity. We here focus on the case of a pure liquid, corresponding to a stress-free boundary condition at the liquid/air interfaces of the film, and thus employ an essentially extensional description of the flow. Taking into account van der Waals interactions between the interfaces, we observe that film rupture is well defined in time as well as in space, which allows us to compute the critical thickness and the film height at the moment of rupture. The theoretical predictions of the model turn out to be in quantitative agreement with experimental measurements of the break-up height of silicone oil films in a wide range of pulling velocities and supporting fibre diameters.

Free vibration of refined higher-order shear deformation composite laminated beams with general boundary conditions

Abstract In this paper, a unified formulation which is based on a general refined shear deformation beam theory is presented to conduct free vibration analysis of composite laminated beams subjected to general boundary conditions. In the refined theory model, the displacement fields are chosen by including the high-order variation of transverse shear strain through the thickness of the beam and meeting the stress-free boundary conditions on both the top and bottom surfaces. With considering the material couplings and the Poisson's effect, the governing equations and appropriate boundary conditions are derived from the Hamilton's principle. Exact solutions are obtained by employing the method of reverberation ray matrix (MRRM). In order to implement general boundary conditions, the artificial spring boundary technique is introduced in the MRRM to make it suitable for different boundary cases. The present solutions are compared with those available in the literature to confirm their validity. A systematic parameter study for composite beams with various boundary conditions, fiber orientations, lamina numbers and orthotropic ratios is also performed. New results for free vibration involving composite laminated beams with various boundary constraints are also presented for the first time and they may be served as benchmark for researchers in this field.

Analytical solution for nonlocal buckling characteristics of higher-order inhomogeneous nanosize beams embedded in elastic medium

In this paper, buckling characteristics of nonhomogeneous functionally graded (FG) nanobeams embedded on elastic foundations are investigated based on third order shear deformation (Reddy) without using shear correction factors. Third-order shear deformation beam theory accounts for shear deformation effects by a parabolic variation of all displacements through the thickness, and verifies the stress-free boundary conditions on the top and bottom surfaces of the FG nanobeam. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam in deformation, which acts in tension as well as in compression. The material properties of FG nanobeam are supposed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. Nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. Comparison between results of the present work and those available in literature shows the accuracy of this method. The obtained results are presented for the buckling analysis of the FG nanobeams such as the effects of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

Global strong solutions to radial symmetric compressible Navier–Stokes equations with free boundary

In this paper, we consider the two-dimensional barotropic compressible Navier–Stokes equations with stress free boundary condition imposed on the free surface. As the viscosity coefficients satisfies μ(ρ)=2μ, λ(ρ)=ρβ, β>1, we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary. In particular, we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive, and the free boundary propagates along the particle path and expand outwards at an algebraic rate.

A note on response of tunnels to incident SH-waves near hillsides

In this study, diffraction of plane SH waves by a cylindrical tunnel embedded in homogeneous, isotropic and linear elastic quarter-space is investigated. Analytical solution techniques are used to solve the two dimensional wave propagation problem. When the excitation is assumed to be harmonic, the governing equation would be the Helmholtz equation. By applying separation of variables method to Helmholtz equation for cylindrical coordinates, general solutions are obtained in terms of Fourier-Bessel series. Unknown complex constants of the Fourier-Bessel series are to be determined from boundary conditions. Boundary conditions at inner and outer side of the tunnel are satisfied directly since they are defined in cylindrical coordinates. Stress-free boundary conditions at the ground and the hillside surfaces are satisfied in closed form via imaging method and addition theorems. Numerical examples are compared with earlier studies and the effect of the tunnel is discussed.

A novel computational approach for functionally graded isotropic and sandwich plate structures based on a rotation-free meshfree method

This paper investigates a rotation-free moving Kriging (MK) meshfree approach for isotropic and sandwich functionally graded material (FGM) plates based on a refined plate theory (RPT). The present formulation makes certain that the tangential stress-free boundary conditions at the top and bottom surfaces of the plate are satisfied with any nonlinear distribution functions. The basic idea of computational method behind this work is to exploit a quartic spline correlation function for establishment of MK basic shape functions. Our finding yields an improved meshfree formulation whose solution becomes certainly stable and no longer depends on a correlation parameter θ. In particular, it is interesting that boundary conditions related to the slopes can be eliminated by a simple rotation-free technique as addressed in isogeometric analysis. Our method uses only four variables for each node and hence the computational cost is completely well-controlled. Several numerical examples are provided to validate high reliability of the present method in comparison with other numerical methods. The present results also show that no shear locking is observed.

Instabilities and chaos in low-Prandtl number rayleigh-Bénard convection

We investigate instabilities and chaos in Rayleigh-Bénard convection (RBC) of low Prandtl-number (Pr) fluids with stress-free boundary conditions. Three dimensional direct numerical simulations (DNS) of the governing equations of RBC are done for investigation. DNS shows the existence of multiple solutions near the onset of convection. To understand the origin of these solutions and their bifurcations, we construct a low dimensional model from the DNS data. The low dimensional model unfolds a rich bifurcation structure in RBC of low Prandtl-number fluids. The bifurcation structure involves various bifurcations including pitchfork, Neimark-Sacker, Homoclinic gluing, Hopf bifrucations, and attractor merging crisis near the onset of convection. The results of the model are consistent with that of the DNS.

Transition to geostrophic convection: the role of the boundary conditions

Rotating Rayleigh–Bénard convection, the flow in a rotating fluid layer heated from below and cooled from above, is used to analyse the transition to the geostrophic regime of thermal convection. In the geostrophic regime, which is of direct relevance to most geo- and astrophysical flows, the system is strongly rotating while maintaining a sufficiently large thermal driving to generate turbulence. We directly simulate the Navier–Stokes equations for two values of the thermal forcing, i.e. $Ra=10^{10}$ and $Ra=5\times 10^{10}$, at constant Prandtl number $Pr=1$, and vary the Ekman number in the range $Ek=1.3\times 10^{-7}$ to $Ek=2\times 10^{-6}$, which satisfies both requirements of supercriticality and strong rotation. We focus on the differences between the application of no-slip versus stress-free boundary conditions on the horizontal plates. The transition is found at roughly the same parameter values for both boundary conditions, i.e. at $Ek\approx 9\times 10^{-7}$ for $Ra=1\times 10^{10}$ and at $Ek\approx 3\times 10^{-7}$ for $Ra=5\times 10^{10}$. However, the transition is gradual and it does not exactly coincide in $Ek$ for different flow indicators. In particular, we report the characteristics of the transitions in the heat-transfer scaling laws, the boundary-layer thicknesses, the bulk/boundary-layer distribution of dissipations and the mean temperature gradient in the bulk. The flow phenomenology in the geostrophic regime evolves differently for no-slip and stress-free plates. For stress-free conditions, the formation of a large-scale barotropic vortex with associated inverse energy cascade is apparent. For no-slip plates, a turbulent state without large-scale coherent structures is found; the absence of large-scale structure formation is reflected in the energy transfer in the sense that the inverse cascade, present for stress-free boundary conditions, vanishes.

Ballistic thermal transport properties at low temperatures in semiconductor nanowires-based heterojunctions

In this paper, we study ballistic thermal transport properties at low temperatures in semiconductor nanowires-based heterojunctions under hard-wall boundary conditions (HWBCs) and stress-free boundary conditions (SFBCs). Here, the numerical calculations for the asymmetric heterojunction (ASHJ) and symmetric heterojunction (SHJ) are done. When SFBCs are employed, the transmission coefficient exhibits different behaviors between ASHJ and SHJ especially at low frequency, but when HWBCs are employed, the transmission coefficient displays similar smooth platforms in both heterojunctions. In low temperature limit, the quantized thermal conductance can be observed in SHJ under SFBCs regardless of the structural details. However, this quantization cannot be observed in ASHJ under SFBCs, and the thermal conductance is strongly sensitive to the transverse width ratio rather than the slant angle. With increasing the transverse width ratio, the thermal conductance in both heterojunctions gradually increases especially, and such the increasing degree is more evident at higher temperatures. A brief analysis of these results is given.

Effect of local velocity on diffusion-induced stress in large-deformation electrodes of lithium-ion batteries

In this work, the contribution of local velocity to the resultant flux of lithium in lithium-ion battery is introduced into the diffusion equation to describe the migration of lithium in the active material of electrodes. The effect of the local velocity on the stress evolution in a spherical electrode made of silicon is analyzed, using the derived diffusion equation and nonlinear theory of elasticity. Two boundary conditions at the surface of the electrode, which represent two extreme conditions of real electrode materials, are used in the stress analysis: one is stress-free, and the other is immobile. The numerical results with the stress-free boundary condition suggest that the effect of the local velocity on the distribution of radial stress and hoop stress increases with the increase of time and the effect of the local velocity on the distribution of lithium is relatively small. In comparison with the results without the effect of the local velocity, the effect of the local velocity is negligible for the immobile boundary condition. The numerical result shows that the use of the immobile boundary condition leads to the decrease of von-Mises stress, which likely will retard the mechanical degradation of electrode and improve the electrochemical performance of lithium-ion battery.

An asymptotic study of instability to Langmuir circulation in shallow layers

The rôle of boundary conditions applied to perturbation equations describing the instability of wavy shear flows to counter-rotating vortical structures aligned with the flow in shallow water layers is considered. In the ocean context the structures are described by the Craik-Leibovich equations and are known as Langmuir circulation (LC); they arise via an instability mechanism known as CL2, which requires the presence of shear and differential drift of the same sense and that a threshold Rayleigh number be exceeded. While Neumann or stress-free boundary conditions applied at the free surface in deep water act to yield a physically plausible spanwise wavenumber l at onset, that is not the case in shallow layers, where non-zero l at onset is ensured only when mixed boundary conditions are applied. This paper questions why that is so and goes on to determine how onset is affected by distributions of and . In order to proceed analytically, results are computed by a small-l asymptotic approximation for which an algorithm is generated enabling computation to, essentially, any order. It is found that the extra stress induced by the perturbed motion, which is reflected in mixed but not Neumann boundary conditions, acts to suppress the growth rate of the instability over all wavenumbers, thereby restricting instability to non-zero l; further details are seen to depend on the distributions of and . Specifically, while the critical wavenumber is not overly sensitive to precise distributions of and , the critical Rayleigh number is sensitive. Minor variations in and can also lead to multiple rather than single layers of LC and even contrive to ensure non-zero l at onset with Neumann boundary conditions. Finally, although crafted as an asymptotic approximation for , the results concur closely with numerical calculations at much larger l.

3D graphical dynamic responses of FGM plates on Pasternak elastic foundation based on quasi-3D shear and normal deformation theory

Abstract A five-variable quasi-three-dimensional (quasi-3D) theory for power law and sigmoid Functionally Graded Material (P- and S-FGM) plates based on a refined higher-order shear and normal deformation approach is presented. The theory accounts for a displacement field in which the in-plane displacement is a third-degree function of the thickness coordinates while the out-of-plane displacement varies parabolically through the plate thickness. The theory satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor; moreover, it has fewer unknowns and equations of motion than other quasi-3D theories. In the present study, the displacement field of the four-variable plate theory was modified by considering the thickness stretching effect, and the equations of motion were derived from Hamilton's principle. Numerical results of natural frequencies and transient analysis are presented herein for two-phase graded material with a power-law and sigmoid through the plate thickness variation of the volume fractions. It was assumed that the elastic medium is modeled as the Pasternak elastic foundation. The accuracy of the obtained numerical results was verified by comparing them with those derived from first-order shear deformation theory and with other higher-order shear deformation theories. The important conclusions that emerge suggest that the proposed five-variable quasi-3D theory produces results as good as those of higher-order shear deformation theories.