Set Of Stability Equations Research Articles

An analytical approach for elastic and non-elastic buckling of pipes under external/internal pressures

Abstract An analytical model is presented to evaluate the buckling pressure of pipes subjected to external/internal pressures. At firstly, a set of stability equations was solved mathematically and the elastic buckling solution was established. By introducing the inner plastic section and the middle non-linear elastic section, the analytical approach was then extended to the critical pressure evaluation of the thick pipes in non-elastic buckling and the effect of the initial ovality on the critical pressure is considered. Details of the analytical formulations and the required parameters are defined. The results are presented to show that the initial ovality has a significant influence on the bifurcation pressure and, however, its effect becomes trivial as the pipes become thicker. Compared to the plastic hinge approach, the proposed solution represents a lower critical pressure, which indicates that the pipes would collapse before the formation of the plastic hinges. Furthermore, some factors that affect the pressure capacity of the pipes are discussed, the results from the proposed analytical solution are compared with those of other solutions and experiments.

The onset of Görtler vortices in laminar boundary layer flow over a slightly concave wall

The onset of convective instability in the laminar boundary layer over the slightly curved wall is analyzed theoretically and compared with the existing experimental data. A new set of stability equations are derived by the propagation theory considering the relative instability under the linear stability theory. In this analysis the disturbances are assumed to have the form of longitudinal vortices and also to grow themselves in streamwise direction. The critical position to mark the onset of Görtler instability is obtained as a function of the Görtler number, where disturbances at the critical state are mainly confined to the hydrodynamic boundary layer. Comparing the theoretical predictions with available experimental and other theoretical results, the present predictions follow experimental trends fairly well with slightly higher critical Görtler numbers than those from the local stability theory. The propagation theory commanding the local eigenvalue analysis is successful to obtain stability conditions reasonably in Görtler vortex problems, relaxing the limitations by the conventional analyses.

Onset of Soret convection in a nanoparticles-suspension heated from above

The onset of Soret-driven convection in a nanoparticles suspension heated from above is analyzed theoretically based on linear theory and relative instability concept. A new set of stability equations are derived and solved by using the dominant mode method. The dimensionless critical time τ c to mark the onset of instability is presented here as a function of the Rayleigh number, the Lewis number and the separation ratio. Available experimental data indicate that for large Rayleigh number convective motion is detected starting from a certain time τ≈3 τ c . This means that the growth period of initiated instabilities is needed for convective motion to be detected experimentally. It seems evident that during τ c ≤ τ ≤ 3 τ c convective motion is relatively very weak and the primary diffusive transfer is dominant.

Onset of solutal marangoni convection in a suddenly desorbing liquid layer

The onset of surface-tension-driven convection in an initially quiescent liquid layer experiencing sudden desorption from a free surface is analyzed using the linear stability theory. The propagation theory was developed for the basic time-dependent concentration profile, which is strongly nonlinear. Based on this theory, a new set of stability equations is derived in considering the effect of Gibbs adsorption and by neglecting the surface diffusion. It is found that the liquid layer becomes more stable by decreasing the Schmidt and Biot numbers. It is interesting that, during an initial period, the system is stable, depending on the Gibbs effect. The predictions agree favorably with the existing experimental results of triethylamine desorption in water in a wetted-wall and a liquid-jet column.

Analysis of the secondary stability of compressible boundary layer flows over a two-dimensional plate

This paper used the Floquet's three-dimensional linear stability theory in the analysis of two-dimensional compressible boundary layer, a set of stability equations is constructed, the effect of three dimensional linear small perturbation on the two-dimensional compressible boundary layer transition is studied, and the effect of coming flow Ma number on growth and development of the subharmonics is calculated. It can be seen from the calculations, the effect caused by the interaction of two-dimensional and three-dimensional perturbation waves on the development of two-dimensional compressible laminar boundary layer.

A theoretical analysis of the onset of surface-tension-driven convection in a horizontal liquid layer cooled suddenly from above

The onset of surface-tension-driven convection in a horizontal liquid layer cooled suddenly from above is analyzed by using linear stability theory. To obtain the critical condition to mark the onset of instability in the form of regular cellular motion the time domain is divided into two. With small time the basic temperature profile is strongly nonlinear and therefore the propagation theory we have developed is employed. Based on the propagation theory a new set of stability equations are derived and their scale analysis is discussed. It is found that the fluid layer is more stable with decreasing the Biot number and the Prandtl number. With large time the frozen-time model is applied. In this time domain the stability criteria are independent of the Prandtl number. By connecting the predictions from the above two models the overall stability criteria are constructed. The interesting role of the Biot number on the critical condition is discussed in detail.

Braced, partially braced and unbraced columns: Complete set of classical stability equations

Stability equations that evaluate the elastic critical axial load of columns in any type of construction with sidesway uninhibited, partially inhibited, and totally inhibited are derived in a classical manner. These equations can be applied to the stability of frames (unbraced, partially braced, and totally braced) with rigid, semirigid, and simple connections. The complete column classification and the corresponding three stability equations overcome the limitations and paradoxes of the well known alignment charts for braced and unbraced columns and frames. Simple criteria are presented that define the concept of partially braced columns and frames, as well as the minimum lateral bracing required by columns and frames to achieve non-sway buckling mode. Various examples are presented in detail that demonstrate the effectiveness and accuracy of the complete set of stability equations.

On the influence of prebuckling deformations on the buckling of thin elastic shells

A buckling theory valid for finite prebuckling deformations is presented for thin homogeneous, isotropic and elastic shells. It is subject to the restriction of the Kirchhoff hypothesis. A set of stability equations is derived by decomposing strain and stress components into four classes according to their characteristics. The influence of the prebuckling deformations on the buckling of thin circular cylindrical shells under lateral pressure is investigated with the aid of the basic equations derived above and the results are compared with the solutions of the Flügge equations and those obtained by Yamaki.