Regular Perturbation Approach Research Articles

Buckling and post-buckling of non-uniform non-linearly elastic rods

The post-buckling solutions of non-uniform linearly and non-linearly elastic rods are constructed via a higher-order perturbation approach. For rods with continuously varying flexural stiffness, the onset of buckling and the post-buckling branches are determined employing a suitable set of admissible functions within the context of Galerkin's method combined with a regular perturbation approach. On the other hand, for piece-wise continuous rods, closed-form eigenvalue solutions are exploited as a basis for the non-linear post-buckling description. Closed-form conditions on the non-linearly elastic constitutive function for the bending moment are obtained ensuring supercritical or subcritical divergence bifurcations. Several illustrative examples are discussed highlighting the effects of the stiffness non-uniformity on the onset of buckling and the post-buckling non-linear regime as well as the influence of the non-linear elasticity on the latter.

Bistable Resonance In Gravity‐driven Film Flows

AbstractWe study the flow of a viscous liquid down an inclined plane with a sinusoidal bottom profile. Applying the integral boundarylayer method leads to a nonlinear ordinary differential equation for the film thickness. Linear resonance between the free surface and the bottom contour is observed via a regular perturbation approach. Considering nonlinear contributions, the resonance curve shows a foldover effect which is typical for nonlinear oscillators. In order to understand the origin of this bistable behaviour we derive a simple model equation and show an analogy to the Duffing oscillator. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

TRANSIENT LINEAR AND NONLINEAR HEAT CONDUCTION IN WEAKLY MODULATED DOMAINS

Heat conduction in two-dimensional domains with spatially periodic boundary is addressed in this study. The periodic modulation is assumed to be weak, but is of arbitrary shape. A regular perturbation approach is implemented to determine the temperature and heat flux throughout the domain. It is observed that the validity of the perturbation approach extends to include geometries of practical importance. Transient linear as well as steady nonlinear heat conduction problems are examined. The periodic domain is mapped onto the rectangular domain. For both steady and transient linear heat conduction, a fully analytical spectral solution becomes possible. The nonlinear problem is shown to reduce to a set of ordinary differential equations of the two-point-boundary-value type, which is solved using a variable-step-size finite-difference scheme. The perturbation approach is validated upon comparison with conventional methods; excellent agreement is obtained against the boundary- and finite-element methods.

Convective instability of the thermovibrational flow of binary mixture in the presence of the Soret effect

The convective instability of the thermovibration flow in a plane horizontal layer filled with an incompressible binary gaseous mixture is investigated. The study takes into account the effect of thermal diffusion or the Ludwig-Soret effect. Several instability mechanisms are discussed. To determine the instability threshold with respect to cell and long-wave perturbations, the Floquet theory was applied to the linearized equations of convection formulated in the Boussinesq approximation. We found that regime parametric instability can occur owing to the finite frequency vibrations. The evolution of plane, spiral and three-dimensional disturbances is studied. We demonstrated that, because of the properties of the system, the subharmonic response of plane disturbances to the external periodic action cannot be observed. The instability can be associated only with synchronous or quasiperiodic modes. Depending on the vibration parameters, modulations can stabilize or destabilize the base state. For spiral perturbations the stability boundary does not depend on the amplitude and frequency of vibrations. In the case of long-wave instability we apply the regular perturbation approach with the wavenumber as a small parameter in power expansions. The stability boundaries are found.

Improved Techniques for the Development of Quadratic Perturbation Models

In this paper, the quadratic model developed by Patwardhan and Madhavan (1993) using the regular perturbation approach has been further refined to produce a better estimate of second-order effects. As an alternate to the regular perturbation model, a structurally better form of the quadratic model has also been derived that does not require any of the simplifying assumptions that were used in our previous work (Patwardhan and Madhavan, 1993). The coefficients of this model are computed using the solutions of the first- and second-order sensitivity equations. A recursive form of quadratic perturbation models is also developed for the systems with a nonlinear output map. The computational advantages of the proposed quadratic models have been presented. The improved prediction capabilities of the quadratic models obtained using the proposed modifications have been demonstrated using a simulation example. The analysis of the model structure and the simulation results indicate that the quadratic model developed using the sensitivity equation approach is a better choice for approximating the local behavior of a highly nonlinear system and, consequently, for the development of an nonlinear model predictive control scheme.

Dynamics of axisymmetric slender liquid bridges in isorotation

The dynamics of axisymmetric slender liquid bridges in isorotation under μg conditions is studied. Slendernesses of the bridge very close to the static stability limit for cylindrical bridges are considered. The inviscid three-dimensional problem is solved in the neighborhood of that stability limit following a regular perturbation approach. A nonlinear model where the pressure and velocity fields are coupled with the free-surface deformation is obtained. The free-surface deformation in the case of stable configurations is analyzed. The effect of Coriolis forces in the dynamics of the bridge turns out to be very important: for large values of the rotation speed the movement of the free surface is very irregular, and its amplitude is much smaller than that in the case of small rotation.

Steady flows of non‐Newtonian fluids past a porous plate with suction or injection

AbstractThe problem of the steady flow of three classes of non‐linear fluids of the differential type past a porous plate with uniform suction or injection is studied. The flow which is studied is the counterpart of the classical ‘asymptotic suction’ problem, within the context of the non‐Newtonian fluid models. The non‐linear differential equations resulting from the balance of momentum and mass, coupled with suitable boundary conditions, are solved numerically either by a finite difference method or by a collocation method with a B‐spline function basis. The manner in which the various material parameters affect the structure of the boundary layer is delineated. The issue of paucity of boundary conditions for general non‐linear fluids of the differential type, and a method for augmenting the boundary conditions for a certain class of flow problems, is illustrated. A comparison is made of the numerical solutions with the solutions from a regular perturbation approach, as well as a singular perturbation.

An asymptotic approach to compressible boundary-layer flow

A regular perturbation approach is applied to take into account variable property effects. Of special interest is the effect of the pressure-dependent density. In the framework of an asymptotic theory, compressibility effects are considered as variable property effects. By means of this asymptotic theory the deviation of skin friction and heat transfer results from their incompressible, isothermal values are determined for laminar Falkner-Skan boundary layers. As far as laminar flow is concerned there is no need for any empirical information.

A regular perturbation approach to the problem of diffusion towards a growing mercury drop electrode

The problem of diffusion of an electroactive species towards a growing mercury drop electrode is treated. An approximate solution for the complete differential equation is given using a regular perturbation technique.

A regular perturbation approach to surface tension driven flows

Surface tension induced convection in a liquid bridge held between two parallel, coaxial, solid disks is considered. The surface tension gradient is produced by a small temperature gradient parallel to the undisturbed surface. The study is performed by using a mathematical regular perturbation approach based on a small parameter, ϵ, which measures the deviation of the imposed temperature field from its mean value. The first order velocity field is given by a Stokes-type problem (viscous terms are dominant) with relatively simple boundary conditions. The first order temperature field is that imposed from the end disks on a liquid bridge immersed in a non-conductive fluid. Radiative effects are supposed to be negligible. The second order temperature field, which accounts for convective effects, is split into three components, one due to the bulk motion, and the other two to the distortion of the free surface. The relative importance of these components in terms of the heat transfer to or from the end disks is assessed.

Resonant Oscillations: A Regular Perturbation Approach

The resonant oscillations of a gas in a closed-end tube are treated. The time-periodic motion is given by a regular perturbation method in which the expansion parameter is the Mach number of the resulting flow. The solution at the first order exhibits a shock traveling in the tube, but, to this order, the waves do not interact. The interaction of the waves is calculated at the second order. The perturbation procedure proceeds systematically by an application of the Fredholm alternative at each stage.