Instability of three-dimensional boundary layers due to streamline curvature

Investigated are the fundamental roles of the curvature of external streamlines in the stability of three-dimensional boundary layers, the partial differential equations governing small disturbances superimposed on the basic flows are modeled by a simple system of ordinary differential equations, which include a nondimensional parameter denoting magnitude of the streamline curvature. The eigenvalue problem posed by the model equations is numerically solved to evaluate effects of this parameter on critical Reynolds numbers of the Falkner-Skan-Cooke family of three-dimensional velocity profiles. Computational results predict the possibility of a new instability, essentially due to the streamline curvature, of the centrifugal type similar to the Taylor-Görtler instability caused by a concave curvature of the wall.     

Mechanical model for fiber-laden membranes

An integrated mechanical model for fiber-laden membranes is presented and representative predictions of relevance to cellulose ordering and orientation in the plant cell wall are presented. The model describes nematic liquid crystalline self-assembly of rigid fibers on an arbitrarily curved fluid membrane. The mechanics of the fluid membrane is described by the Helfrich bending-torsion model, the fiber self-assembly is described by the 2D Landau-de Gennes quadrupolar Q-tensor order parameter model, and the fiber-membrane interactions (inspired by an extension of the 2D Maier-Saupe model to curved surfaces) include competing curvo-philic (curvature-seeking) and curvo-phobic (curvature-avoiding) effects. Analysis of the free energy reveals three fiber orientation regimes: (a) along the major curvature, (b) along the minor curvature, (c) away from the principal curvatures, according to the competing curvo-philic and curvo-phobic interactions. The derived shape equation (normal stress balance) now includes curvature-nematic ordering contributions, with both bending and torsion renormalizations. Integration of the shape and nematic order equations gives a complete model whose solution describes the coupled membrane shape/fiber order state. Applications to cylindrical membranes, relevant to the plant cell wall, shows how growth decreases the fiber order parameter and moves the fibers’ director from the axial direction towards the azimuthal orientation, eventually leading to a state of stress predicted by pure membranes. The ubiquitous 54.7° cellulose fibril orientation with respect to the long axis in a cylindrical plant cell wall is shown to be predicted by the preset model when the ratio of curvo-phobic and curvo-philic interactions is in the range of the cylinder radius.     

Dynamic buckling of a shallow arch under shock loading considering the effects of the arch shape

In this paper, the influence of the initial curvature of thin shallow arches on the dynamic pulse buckling load is examined. Using numerical means and a multi-dof semi-analytical model, both quasi-static and non-linear transient dynamical analyzes are performed. The influence of various parameters, such as pulse duration, damping and, especially, the arch shape is illustrated. Moreover, the results are numerically validated through a comparison with results obtained using finite element modeling. The main results are firstly that the critical shock level can be significantly increased by optimizing the arch shape and secondly, that geometric imperfections have only a mild influence on these results. Furthermore, by comparing the sensitivities of the static and dynamic buckling loads with respect to the arch shape, non-trivial quantitative correspondences are found.     

Optimization of elastic bars in torsion

The paper considers the problem of optimization of mechanical systems described by partial differential equations. The shape of the region of integration of these equations is not specified beforehand but is to determined from the condition that a certain integral functional attains an extremal value. The mathematical optimization problem is reduced to a variational one having no differential constraints and the necessary optimality conditions are derived. The latter are used for seeking the cross-sectional shape of elastic bars of maximum torsional rigidity. Exact and approximate analytical solutions are given and the effectiveness of the optimal solutions is estimated.     

Linear and geometrically nonlinear analysis of non-uniform shallow arches under a central concentrated force

In this paper an integral equation solution to the linear and geometrically nonlinear problem of non-uniform in-plane shallow arches under a central concentrated force is presented. Arches exhibit advantageous behavior over straight beams due to their curvature which increases the overall stiffness of the structure. They can span large areas by resolving forces into mainly compressive stresses and, in turn confining tensile stresses to acceptable limits. Most arches are designed to operate linearly under service loads. However, their slenderness nature makes them susceptible to large deformations especially when the external loads increase beyond the service point. Loss of stability may occur, known also as snap-through buckling, with catastrophic consequences for the structure. Linear analysis cannot predict this type of instability and a geometrically nonlinear analysis is needed to describe efficiently the response of the arch. The aim of this work is to cope with the linear and geometrically nonlinear problem of non-uniform shallow arches under a central concentrated force. The governing equations of the problem are comprised of two nonlinear coupled partial differential equations in terms of the axial (tangential) and transverse (normal) displacements. Moreover, as the cross-sectional properties of the arch vary along its axis, the resulting coupled differential equations have variable coefficients and are solved using a robust integral equation numerical method in conjunction with the arc-length method. The latter method allows following the nonlinear equilibrium path and overcoming bifurcation and limit (turning) points, which usually appear in the nonlinear response of curved structures like shallow arches and shells. Several arches are analyzed not only to validate our proposed model, but also to investigate the nonlinear response of in-plane thin shallow arches.     

Closed form solutions for free vibrations of rectangular Mindlin plates

A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.     

Jeffrey fluid flow due to curved stretching surface with Cattaneo-Christov heat flux

The two-dimensional (2D) motion of the Jeffrey fluid by the curved stretching sheet coiled in a circle is investigated. The non-Fourier heat flux model is used for the heat transfer analysis. Feasible similarity variables are used to transform the highly nonlinear ordinary equations to partial differential equations (PDEs). The homotopy technique is used for the convergence of the velocity and temperature equations. The effects of the involved parameters on the physical properties of the fluid are described graphically. The results show that the curvature parameter is an increasing function of velocity and temperature, and the temperature is a decreasing function of the thermal relaxation time. Besides, the Deborah number has a reverse effect on the pressure and surface drag force.     

Analyzing the stress–strain state of thick cylindrical shells

Two approaches to the analysis of the stress–strain state of thick cylindrical shells are elaborated. The shell is divided by concentric cross-sectional circles into several coaxial cylindrical shells. Each of these shells has its own curvature determined on its midline. The stress–strain state of the original shell is described by satisfying the interface conditions between the component shells. The distribution of unknown functions throughout the thickness is determined by finding the analytic solution of a system of differential equations in the first approach and is approximated by polynomial functions in the second approach. The stress–strain state of thick shells is analyzed. It is revealed that the effect of reduction becomes stronger with increasing curvature     

Optimization of body shape at small reynolds numbers

The problem of the optimization of the shape of a body in a stream of viscous liquid or gas was treated in [1–5]. The necessary conditions for a body to offer minimum resistance to the flow of a viscous gas past it were derived in [1], The necessary optimality conditions when the motion of the fluid is described by the approximate Stokes equations were derived in [2], The shape of a body of minimum resistance was found numerically in [3] in the Stokes approximation. The optimality conditions when the motion of the fluid is described by the Navier—Stokes equations were derived in [4, 5], and in [4] these conditions were extended to the case of a fluid whose motion is described in the boundary-layer approximation. The necessary optimality conditions when the motion of the fluid is described by the approximate Oseen equations were derived in [5] and an asymptotic analysis of the behavior of the optimum shape near the critical points was performed for arbitrary Reynolds numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp, 87–93, January–February, 1978.     

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper.A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given.Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass.The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.     

Explicit solutions for depressed masonry arches loaded until collapse—Part I: a one-dimensional nonlinear elastic model

In this paper the equilibrium problem for masonry arches is formulated in terms of a suitable set of nonlinear ordinary differential equations. We show that by making a small number of simple hypotheses it is possible to find the explicit expressions for the displacements and rotations of the cross-sections of an in-plane loaded masonry arch. To this end, the masonry arch is schematised as a curved, one-dimensional nonlinear elastic beam made of a material that is by hypothesis incapable of withstanding significant tensile stresses. In this first part of the two-part paper, the one-dimensional model and the explicit expressions for the displacements and rotations, obtained by integrating the set of differential equations, are presented. In particular, the formal expressions for displacement, stress and strain fields are illustrated in full detail for an explicit, albeit approximate, solution for a statically determinate depressed arch subjected to a uniform vertical load.     

Nonlinear flexural-flexural-torsional interactions in beams including the effect of torsional dynamics. I: Primary resonance

Nonlinear coupling between torsional and both in-plane and out-of-plane flexural motion is examined for inextensional beams (or beam-like structures) whose torsional and flexural eigenfrequencies are of the same order. The analysis presented here is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities, and account for torsional dynamics. Response characteristics, including stability, are determined for cantilever beams subjected to a lateral periodic excitation. The beam's response in the presence of a one-to-one internal resonance involving a torsional frequency and an in-plane bending frequency is investigated in detail.     

On the parametrization of three-point nonlinear boundary-value problems

A three-point boundary-value problem for a system of nonlinear differential equations is reduced to a family of two-point problems, whose solutions are investigated by using the numerical-analytic method.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 395–413, July–September, 2004.     

Harmonic non-linear response of beck's column to a lateral excitation

The non-linear response of a column with a follower force (Beck's column) subjected to a distributed periodic lateral excitation, or to a support excitation, is determined. An analytical solution for the response amplitude in terms of the loading and system parameters is obtained by a perturbation analysis of the differential equations of motion. Non-linear inertia and non-linear curvature terms are taken into account in the formulation of the differential equations.     

等截面抛物线无铰拱挠度影响线实用解析解

为解决弧长微分表达式复杂导致抛物线拱挠度解析计算繁琐的难题,采取有简单弧长微分表达式的近似曲线拟合抛物线的方法,通过以沿曲线的曲率和为量化指标评价3种曲线(直线)的拟合度,最终得出悬索线拟合度最优的结论。在此基础上,利用基本力学原理,推导出抛物线无铰拱挠度影响线解析计算公式,最后应用于实例,并与有限元分析结果进行对比。算例结果表明,公式计算与有限元分析结果最大相差不超过1%,具有非常高的工程精度,可用于工程实践。     

Flow-induced deformation of an elastic membrane adhering to a wall

The flow-induced deformation of a two-dimensional membrane with a circular unstressed shape clamped at the two ends on a plane wall at an arbitrary contact angle is considered. Working under the auspices of generalized shell theory, the membrane is allowed to develop in-plane tensions, transverse tensions, and bending moments determined by the curvature of the resting and deformed shapes. A system of ordinary differential equations governing the membrane shape is formulated, and the associated boundary-value problem is solved by numerical methods. Numerical results are presented to illustrate the deformation of a clamped membrane due to gravity or a negative transmural pressure. The shell formulation is coupled with a boundary-integral formulation for Stokes flow, and an efficient iterative scheme is developed to describe deformed equilibrium shapes of a membrane attached to a plane wall in the presence of an overpassing shear flow. Computations for different contact angles and shear rates reveal a wide variety of profiles and illustrate the distribution of the membrane tension developing due to the flow-induced deformation.     

Issue Information

This paper describes a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear partial differential equation, whose solutions are called hydrostatic equilibria. We present a well‐balanced method, meaning that besides discretizing the complete equations, the method is also able to maintain all hydrostatic equilibria. The method is a finite volume method, whose Riemann solver is approximated by a so‐called relaxation Riemann solution that takes all hydrostatic equilibria into account. Relaxation ensures robustness, accuracy, and stability of our method, because it satisfies discrete entropy inequalities. We will present numerical examples, illustrating that our method works as promised. Copyright © 2015 John Wiley & Sons, Ltd.     

Viscous dissipation and thermal radiation effects on the magnetohydrodynamic (MHD) flow and heat transfer over a stretching slender cylinder

An axisymmetric magnetohydrodynamic (MHD) boundary layer flow and heat transfer of a fluid over a slender cylinder are investigated numerically. The effects of viscous dissipation, thermal radiation, and surface transverse curvature are taken into account in the simulations. For this purpose, the governing partial differential equations are transformed to ordinary differential equations by using appropriate similarity transformations. The resultant ordinary differential equations along with appropriate boundary conditions are solved by the fourth-order Runge–Kutta method combined with the shooting technique. The effects of various parameters on the velocity and temperature profiles, local skin friction coefficient, and Nusselt number are analyzed.     

Effect of longitudinal surface curvature on heat transfer

The effect of longitudinal surface curvature on heat transfer has been analysed for laminar forced convection by the method of matched asymptotic expansions. Utilizing the classical solution of boundary layer equations as the first order approximations, the second order perturbation for the velocity and temperature field has been calculated by a similarity analysis. The analysis permits the wall temperature to vary as a power function of distance from the stagnation point. Numerical solutions have been obtained for the resulting coupled ordinary differential equations. The results for the variation in the second order temperature profile and the second order wall temperature gradient due to surface curvature parameter, Prandtl number, wall temperature distribution parameter, and pressure gradient parameters are presented graphically. The variation in a typical temperature profile due to curvature, and percentage variations from the first order theory due to longitudinal surface curvature are also presented graphically.     

Analysis of in-plane 1:1:1 internal resonance of a double cable-stayed shallow arch model with cables' external excitations

The nonlinear dynamic behaviors of a double cable-stayed shallow arch model are investigated under the one-to-one-to-one internal resonance among the lowest modes of cables and the shallow arch and external primary resonance of cables. The in-plane governing equations of the system are obtained when the harmonic excitation is applied to cables. The excitation mechanism due to the angle-variation of cable tension during motion is newly introduced. Galerkin's method and the multi-scale method are used to obtain ordinary differential equations(ODEs) of the system and their modulation equations, respectively. Frequency-and force-response curves are used to explore dynamic behaviors of the system when harmonic excitations are symmetrically and asymmetrically applied to cables. More importantly, comparisons of frequency-response curves of the system obtained by two types of trial functions, namely, a common sine function and an exact piecewise function, of the shallow arch in Galerkin's integration are conducted.The analysis shows that the two results have a slight difference; however, they both have sufficient accuracy to solve the proposed dynamic system.