METHOD–OF–LINES SOLUTION OF TIME–DEPENDENT TWO–DIMENSIONAL NAVIER–STOKES EQUATIONS

A novel approach to the development of a code for the solution of the time-dependent two-dimensional Navier–Stokes equations is described. The code involves coupling between the method of lines (MOL) for the solution of partial differential equations and a parabolic algorithm which removes the necessity of iterative solution on pressure and solution of a Poisson-type equation for the pressure. The code is applied to a test problem involving the solution of transient laminar flow in a short pipe for an incompressible Newtonian fluid. Comparisons show that the MOL solutions are in good agreement with the previously reported values. The proposed method described in this paper demonstrates the ease with which the Navier–Stokes equations can be solved in an accurate manner using sophisticated numerical algorithms for the solution of ordinary differential equations (ODEs).     

Invariants of a Remarkable Family of Nonlinear Equations

In classical literature, invariants of families of differentialequations were considered for linear equations only, e.g. the renownedLaplace invariants for linear hyperbolic partial differential equationsand invariants of linear ordinary differential equations with variablecoefficients. The restriction to linear equations was essential inpioneering works of Cockle, Laguerre, Halphen, andForsyth for tackling the problem of invariants of differentialequations. Lie regretted that these authors did not use advantagesprovided by his theory of infinite continuous groups, but he himself didnot undertake further developments in this direction.Recently, the present author considered the possibility hinted byLie's remark and introduced the infinitesimal technique in thetheory of invariants of families of differential equations thatwas lacking in old methods. In consequence, a simple unifiedapproach was developed for calculation of invariants of algebraicand differential equations independent on the assumption oflinearity of equations. It was employed recently for calculationof Laplace type invariants for parabolic equations. Here, themethod is applied to calculation of invariants for the family ofnonlinear equations appearing in the problem on linearization ofnonlinear ordinary differential equations.     

Application of a non-linear local analysis method for the problem of mixed convection instability

We consider the problem of laminar mixed convection flow between parallel, vertical and uniformly heated plates where the governing dimensionless parameters are the Prandtl, Rayleigh and Reynolds numbers. Using the method based on the centre manifold theorem which was derived from the general theory of dynamical systems, we reduce a three-dimensional simplified model of ordinary differential amplitude equations emanating from the original Navier-Stokes system of the problem in the vicinity of a trivial stationary solution. We have found that when the forcing parameter, the Rayleigh number, increases beyond the critical value Ra_s, the stationary solution is a pitchfork bifurcation point of the system.     

An Analysis of Phase‐Field Alloys and Transition Layers

A phase‐field approach to binary alloys is studied. Formal asymptotics of the system of parabolic differential equations leads to new interface relations as part of a macroscopic model which arises in the limit of vanishing interface thickness. Under suitable conditions we prove that the phase‐field system has a unique solution which converges to the limiting macroscopic solution. The concentration and phase are monotonic across the interface for a simplified system. Transition layers in concentration are induced due to the change in phase and the change in material diffusion across the interface. Excess impurities may be trapped as a consequence of these layers. (Accepted October 28, 1996)     

The effect of radiation on transient natural convection past a doubly infinite plate

The present research note is concerned with the transient (short time) simultaneous free convection and radiation analysis of a viscous fluid along a doubly infinite vertical isothermal flat plate. To simplify a very complicated problem, an incompressible flow field is used in the analysis. Generally, the exact numerical solution of this problem is quite lengthy. However, by considering an optically thick radiating gas, expressed by the Rosseland diffusion approximation, the solution is much simpler. Moreover, this case leads to a complete similarity transformation of the governing partial differential equations into a set of ordinary differential equations. An exact numerical solution is obtained of the resulting ordinary differential equations for a Prandtl number equal to 0.733 and for a wide range of involved parameters.     

A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

Analytical solution of the contact problem for a system of bodies under collective wear

The contact problem is considered for a system of bodies subject to wear on a common base. The deformation properties of the bodies and the base are described by the Winkler model. The problem is reduced to a system of ordinary differential equations and an integral equation of hereditary type with difference kernel. The solution of the problem is constructed with the use of the Laplace transform. The asymptotics of the solution at large times is studied. The continuity conditions for the contact of each of the bodies with the base are derived.     

Dynamic response in a bi-material spherical medium

Summary The elastodynamic problem of a bi-material spherical medium is solved under the condition that the external load applied is spherically symmetric. Exact and explicit formulas are provided for displacements and stresses induced by the propagating, reflected and transmitted waves. The D' Alembert solution is taken as a basic form, thereby reducing the boundary and interface conditions to ordinary differential equations and systems of ordinary differential equations. The integration constants contained in the solutions of the differential equations are fixed by a singularity extraction procedure, which removes from the solution those portions that are inadmissible to the wave motion problem. A number of numerical results are offered, to validate the analysis and to demonstrate the capability of the solution method in solving elastodynamic problems of engineering significance. Received 12 March 1996; accepted for publication 16 December 1996     

Periodic solution to a kind of singularly perturbed differential equations of parabolic type in chemical kinetics

This paper deals with the problem on the periodic solution for the singularly perturbeddifferential equations of parabolic type originating from chemical kinetics in stratifiedmedia,A uniformly valid asymptotic solution is constructed and the related asymptoticestimate is given.     

Similarity Solution of Unsteady Boundary Layer Equations with a Magnetic Field

The unsteady laminar incompressible boundary layer flow of an electricallyconducting fluid in the stagnation region of two-dimensional and axisymmetricbodies with an applied magnetic field has been studied. The boundary layerequations which are parabolic partial differential equations with threeindependent variables have been reduced to a system of ordinary differential equations by using suitable transformations and then solved numerically using a shooting method. Here, we have obtained new solutions which are solutions of both the boundary layer and Navier-Stokes equations.     

Motion of a strong shock front in a two-dimensional gas layer with moving internal boundary

We examine the problem of planar one-dimensional motion of a strong shock wave with moving internal boundary in which the initial position of the front, its intensity, the mass of the gas involved in the motion, and the energy contained in this gas are known. The problem is not self-similar and its exact solution, which involves working with partial differential equations, presents serious difficulties. In the following we determine the law of shock-front motion in this problem via the method of [1], which makes it possible to find a system of ordinary differential equations for the problem. The method is based on an initial specification of the power-law coupling between the dimensionless Lagrangian and Eulerian variables and replacement of the energy equation by this coupling and the energy integral. The solution is sought in the first approximation.     

Dynamic contact problem with adhesion and damage between thermo–electro–elasto-viscoplastic bodies

We study of a dynamic contact problem between two thermo–electro–elasto-viscoplastic bodies with damage and adhesion. The contact is frictionless and is modeled with normal compliance condition. We derive variational formulation for the model and prove an existence and uniqueness result of the weak solution. The proof is based on arguments of evolutionary variational inequalities, parabolic inequalities, differential equations, and fixed point theorem.     

Asymptotic models of solidification in cooling thin-layer flows of a highly viscous fluid

Asymptotic models are constructed for the solidification process in a highly viscous film flow on the surface of a cone with a given mass supply at the cone apex. In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front. For large Péclet numbers, an analytical steady-state solution for the solidification front is found. A nondimensional parameter which makes it possible to distinguish flows (i) without a solid crust, (ii) with a steady-state solid crust, and (iii) with complete solidification is determined. For finite Péclet numbers and large Stefan numbers, an analytical transient solution is found and the time of complete flow solidification is determined. In the general case, when all the governing parameters are of the order of unity, the original system of equations is studied numerically. The solutions obtained are qualitatively compared with the data of field observations for lava flows produced by extrusive volcanic eruptions.     

Buoyancy-affected laminar film condensation

The flow and heat transfer in a laminar condensate flim on an isothermal vertical plate is modelled mathematically. The strict Boussinesq approximation is adopted to account for buoyancy due to local temperature variations within the film. A similarity transformation reduces the governing boundary-layer type equations to a coupled set of ordinary differential equations and the resulting three-parameter twopoint boundary value problem is solved numerically for Prandtl numbers,Pr, ranging from 0.001 to 1000 and Jakob numbers,Ja, between 0.0001 and 1.5. The principal effects of the favourable buoyancy are to reduce the thickness of the condensate film and increase the film velocity at the smooth liquid-vapour interface, whereas the friction and heat transfer at the plate are enhanced. In accordance with the classical Nusselt theory, it is found that the temperature varies nearly linearly across the film. The computed similarity profiles for velocity reveal, however, substantial departures from the parabolic distribution assumed in the simplified Nusselt analysis.     

Quasi-similar solutions for blast waves with internal heat transfer effects

The point explosion problem with internal heat transfer effects taken into account is analysed. The classical inviscid solutions to this problem yield a non-physical phenomenon of infinite temperature and zero density at the center of explosion for all times. With heat transfer fluxes considered, the solution near the center of symmetry is improved and finite values for temperature are obtained. The non-self-similar solution of the problem is based on the quasi-similar approximate technique which reduces the non-linear partial differential equations governing the problem to ordinary differential ones. However, this formulation yields a two-point boundary-value problem. To facilitate the integration, the flow field is first divided into two regions: an outer inviscid region and an inner region where dissipation effects are manifested. This results in two sets of ordinary differential equations expressing the conservation equations in the inner and outer regions which are then solved and matched together to yield the composite solution. Secondly, the problem is then transformed into an initial-value one. Using the results of the composite solution, the governing equations can be integrated directly from the center until the shock front. The structure of the non-self-similar flow fields with internal heat transfer effects is then fully determined for specified values of the heat transfer parameters.     

Stress and strain fields in rotating elastic/plastic annular disks of pressure-dependent material

The Drucker–Prager yield criterion is used in conjunction with its associated flow rule to find the elastic/plastic stress and strain distributions within the rotating annular disks under plane stress conditions. The main distinguished feature of the model, as compared to typical models used for analysis of disks, is that the material is plastically compressible. Using an approach proposed elsewhere, the solution for strain rates is reduced to one nonlinear ordinary differential equation and two linear ordinary differential equations. These equations can be solved one by one, which significantly simplifies the numerical treatment and increases the accuracy of solution.     

扇形截面杆扭转问题的差分线法

导出了扇形截面杆扭转问题偏微分方程的差分线法常微分方程组, 并解析求解了该方程组, 得到了扭转应力函数的半解析解, 计算了扭转应力及扭转刚度. 计算过程中, 用追赶法计算 常微分方程组的特解, 用公式计算三对角矩阵的特征值与特征向量, 利用实对阵矩阵的特征 向量相互正交的特性避免矩阵求逆计算, 利用复化梯形公式计算扭转刚度. 整个求解过程在 角度方向离散微分方程和用复化梯形公式进行面积积分时引入了误差, 其他求解过程是精确 的. 计算结果与已有结果进行了对比, 显示了算法的正确性. 该算法对工程中扇形截面扭 转杆的设计有一定的实用价值.     

Parabolic method of solving problems of the flow of a sonic gas stream around a thin symmetric profile

A parabolic method consisting of replacement of the stream acceleration ?_xx in the non-linear member of (1.1) by a specially chosen constant has been proposed [1] for the solution of the mixed-type transonic equation with boundary conditions on the profile, and the solution of the linear parabolic-type equation obtained can be considered as a certain approximation to the solution of the initial problem. An improvement of the parabolic method is the method of local linearization [2] (see [3] also), in which the acceleration ?_xx fixed from the beginning is replaced by a function of the coordinate x which satisfies some condition. An ordinary first-order differential equation is obtained for the velocity distribution along the profile in [2]. Another method of “defrosting” the acceleration ?_xx “frozen” from the beginning is proposed in this paper; a second-order ordinary differential equation is obtained for the velocity on the profile, which permits getting rid of some disadvantages of the local linearization method. Several solutions of (2.3) are presented in comparison to known exact solutions.     

Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of ordinary differential equations or differential-algebraic equations, if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods. In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a tangent linear model that corresponds to the Newton–Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.     

Thermoelastically coupled axisymmetric nonlinear vibration of shallow spherical and conical shells

The problem of axisymmetric nonlinear vibration for shallow thin spherical and conical shells when temperature and strain fields are coupled is studied. Based on the large deflection theories of yon Ktirrntin and the theory of thermoelusticity, the whole governing equations and their simplified type are derived. The time-spatial variables are separated by Galerkin ‘ s technique, thus reducing the governing equations to a system of time-dependent nonlinear ordinary differential equation. By means of regular perturbation method and multiple-scales method, the first-order approximate analytical solution for characteristic relation of frequency vs amplitude parameters along with the decay rate of amplitude are obtained, and the effects of different geometric parameters and coupling factors us well us boundary conditions on thermoelustically coupled nonlinear vibration behaviors are discussed.