Integral equation formulation of piezo-electric diffraction problems

In this paper the integral equation formulation of scattering or diffraction problems associated with wave motions in a piezo-electric solid is investigated. Starting from the basic equations a power balance and a reciprocity relation are derived. In the power balance the mechanical and the electric power flow density occur. The reciprocity relation serves as a starting point for the integral equation formulation of diffraction problems. In the reciprocity relation, two possible but different field situations occur. In one of them the field quantities are chosen to be the actual ones, in the other the field quantities are chosen to be the auxiliary ones that are generated by either an electric point charge or a mechanical point force. In this way, Helmholtz type of integral representations for the electric potential and the mechanical displacement are obtained.For the technically important configuration of a piezo-electric component excited by a finite number of electrodes, the homogeneous, linear relations between the potentials on the electrodes and the currents fed into these are represented through the impedance matrix. It is shown that the elements of this impedance matrix can be obtained from certain integral equations.A method for obtaining the auxiliary fields used in the integral representations for the electric potential and the mechanical displacement is outlined in an appendix.     

A boundary element formulation for natural convection problems

This paper presents a boundary element formulation employing a penalty function technique for two-dimensional steady thermal convection problems. By regarding the convective and buoyancy force terms in Navier-Stokes equations as body forces, the standard elastostatics analysis can be extended to solve the Navier-Stokes equations. In a similar manner, the standard potential analysis is extended to solve the energy transport equation. Finally, some numerical results are included, for typical natural convection problems, in order to demonstrate the efficiency of the present method.     

On boundary integral equation formulation for axisymmetric problems of transversely isotropic media

Summary In this paper we deal with the formulation of the axisymmetric boundary integral equation for the transversely isotropic elastic body. We assume that the axis of elastic symmetry is coincident with the axis of rotation and that the elastic body is subjected to arbitrary axisymmetric loading without torsion. Numerical calculations are carried out and compared with results by the finite element method.

---

Zur Beschreibung von axialsymmetrischen, elastisch transversal-isotropen Problemen durch Randintegralgleichungen

Übersicht Behandelt wird die Formulierung der Randintegralgleichung für den axialsymmetrischen, elastisch transversal-isotropen Körper. Es wird angenommen, daß die Achse der elastischen Symmetrie mit der Rotationsachse zusammenfällt und der Körper einer beliebigen axialsymmetrischen Belastung ohne Torsion unterworfen ist. Ausgeführte numerische Beispiele werden mit Ergebnissen der Finite-Element-Methode verglichen.

     

An optimizing reduced PLSMFE formulation for non‐stationary conduction–convection problems

In this paper, proper orthogonal decomposition (POD) is combined with the Petrov–Galerkin least squares mixed finite element (PLSMFE) method to derive an optimizing reduced PLSMFE formulation for the non‐stationary conduction–convection problems. Error estimates between the optimizing reduced PLSMFE solutions based on POD and classical PLSMFE solutions are presented. The optimizing reduced PLSMFE formulation can circumvent the constraint of Babu?ka–Brezzi condition so that the combination of finite element subspaces can be chosen freely and allow optimal‐order error estimates to be obtained. Numerical simulation examples have shown that the errors between the optimizing reduced PLSMFE solutions and the classical PLSMFE solutions are consistent with theoretical results. Moreover, they have also shown the feasibility and efficiency of the POD method. Copyright © 2008 John Wiley & Sons, Ltd.     

A new integral equation formulation of two-dimensional inclusion–crack problems

A new integral equation formulation of two-dimensional infinite isotropic medium (matrix) with various inclusions and cracks is presented in this paper. The proposed integral formulation only contains the unknown displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks. In order to solve the inclusion–crack problems, the displacement integral equation is used when the source points are acting on the inclusion–matrix interfaces, whilst the stress integral equation is adopted when the source points are being on the crack surfaces. Thus, the resulting system of equations can be formulated so that the displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks can be obtained. Based on one point formulation, the stress intensity factors at the crack tips can be achieved. Numerical results from the present method are in excellent agreement with those from the conventional boundary element method.     

Stability of buoyancy-driven convection in an Oldroyd-B fluid-saturated anisotropic porous layer

The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffusivity is investigated with the perturbation method. A modified Darcy-Oldroyd model is used to describe the flow in a layer of an anisotropic porous medium. The results of the linear instability theory are delineated. The thresholds for the stationary and oscillatory convection boundaries are established, and the crossover boundary between them is demarcated by identifying a codimension-two point in the viscoelastic parameter plane. The stability of the stationary and oscillatory bifurcating solutions is analyzed by deriving the cubic Landau equations. It shows that these solutions always bifurcate supercritically. The heat transfer is estimated in terms of the Nusselt number for the stationary and oscillatory modes. The result shows that, when the ratio of the thermal to mechanical anisotropy parameters increases, the heat transfer decreases.     

Effect of anisotropy of porous media on the onset of buoyancy-driven convection

A theoretical analysis of buoyancy-driven instability under transient basic fields is conducted in an initially quiescent, fluid-saturated, horizontal porous layer. Darcy’s law is used to explain characteristics of fluid motion and the anisotropy of permeability is considered. Under the Boussinesq approximation and the principle of exchange of stabilities, the stability equations are derived by using the linear stability theory and the energy method. The linear stability equations are analyzed numerically by using the frozen-time model and the linear amplification theory and the global stability limits are obtained numerically from the energy method. For the various anisotropic ratios, the critical times are predicted as a function of the Darcy–Rayleigh number and the critical Darcy–Rayleigh number is also obtained. The present predictions are compared each another and with existing theoretical ones.     

Interferometric study of buoyancy-driven convection in a differentially heated circular fluid layer

The present study is concerned with buoyancy-driven convection experiments in a circular horizontal differentially heated layer of air. The radius-to-height ratio of 14, and Rayleigh numbers of 5,861 and 12,124 have been considered. A Mach–Zehnder interferometer has been used to visualize the convection patterns in the fluid layer. The fluid layer has been imaged at view angles of 0, 45 and 90°. Results obtained show that fringe patterns appropriate for a cavity square in plan are seen in the fluid layer during the early stages of the experiments. After the passage of the initial transients, steady fringes have been observed in the fluid layer for a Rayleigh number of 5,861. At Ra=12,124, a dominant pattern was detectable combined with mild unsteadiness. The steady thermal field at Ra=5,861 displayed symmetry with respect to the viewing angle. A stronger three dimensionality was seen at the higher Rayleigh number. The average steady state Nusselt numbers were found to vary with view angle from 1.91 to 2.04 at Ra=5,861 and 2.28 to 2.43 at Ra = 12,124. The cavity-averaged Nusselt numbers are in good agreement with the available correlations.     

An integral equation formulation of three dimensional anisotropic elastostatic boundary value problems

An approximate solution capability is developed to handle three dimensional anisotropic elastostatic boundary value problems. The method depends crucially on the existence and explicit definition of a fundamental solution to the governing partial differential equations. The construction of this solution for the anisotropic elastostatic problem is presented as is the derivation of the expression for the surface tractions necessary to maintain the fundamental solution in a bounded region. After the fundamental solution and its associated surface tractions are determined, a real variable boundary integral formula is generated which can be solved numerically for the unknown surface tractions and displacements in a well-posed boundary value problem. Once all boundary quantities are known, the field solution is given by a Somigliana type integral formula. Techniques for numerically solving the integral equations are discussed.

---

Zusammenfassung Es wird eine Näherungslösung entwickelt die es dreidimensionale anisotrope elastostatische Randwertprobleme zu lösen. Die Methode hängt entscheidend vom Vorhandensein und expliziten Bestimmung einer Grundlösung der zugeehörigen partiellen Differentialgleichungen. Die Êntwicklung dieser Lösung fur den Fall eines anisotropen elastostatischen Problems wird gegeben wie auch die Ableitung einer Formel fur Oberflächenspannung die erforderlich ist, um die Gultigkeit der Grundlösung im Randgebiet fortbestehen zu lassen. Nachdem die Grundlösung und die mit ihr verbundenen Oberflächenspannungen gefunden worden sind, wird eine Randwertformel in Integralform fur reelle Variablen entwickelt, die numerisch fur unbekannte Oberflächenspannungen und Verschiebungen jedoch mit gut definierten Randwerten gelöst werden kann. Nachdem alle Randwerte bekannt sind ist die Feldlösung durch eine Integralformel vom Typ Somigliana gegeben. Verfahren zur numerischen Lösung der Integralgleichungen werden erörtert.

     

The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium

The onset of buoyancy-driven convection in an initially quiescent ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. The Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-paramagnetic, while the upper paramagnetic boundary is considered to be either rigid or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective–radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved numerically using the Galerkin technique. It is found that increase in the Biot number Bi, porous parameter σ, viscosity ratio Λ, magnetic susceptibility χ, and decrease in the magnetic number M ₁ and non-linearity of magnetization M ₃ is to delay the onset of ferroconvection in a porous medium. Further, increase in M ₁, M ₃, and decrease in χ, Λ, σ and Bi is to decrease the size of convection cells.     

Analytical and numerical study of buoyancy-driven convection in a vertical enclosure filled with nanofluids

This paper presents an analytical and numerical study of natural convection of nanofluids contained in a rectangular enclosure subject to uniform heat flux along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number Ra, the Prandtl number Pr, the aspect ratio of the cavity A and the solid volume fraction of nanoparticles, Φ. Three types of nanoparticles are taken into consideration: Cu, Al₂O₃ and TiO₂. Various models are used for calculating the effective viscosity and thermal conductivity of nanofluids. In the first part of the analytical study, a scale analysis is made for the boundary layer regime situation. In the second part, an analytical solution based on the parallel flow approximation is reported for tall enclosures (A ≫ 1). In the boundary layer regime a good agreement is obtained between the predictions of the scale analysis and those of the analytical solution. Solutions for the flow fields, temperature distributions and Nusselt numbers are obtained explicitly in terms of the governing parameters of the problem. A numerical study of the same phenomenon, obtained by solving the complete system of the governing equations, is also conducted. A good agreement is found between the analytical predictions and the numerical simulations.     

解对流方程的子域精细积分并行算法

基于子域精细积分的思想,针对对流方程初边值问题,首先提出了含参数a＞0的一族三层显格式和一族二层隐格式,它们的局部截断误差分别为O(a△t+△t2+△x2)和O(α△t+△t+△x2).当参数a≥(In△t-ln△x)/2△t时三层显格式是稳定的,而二层隐格式则对所有的参数α＞0都是无条件稳定的.然后,以二层隐格式为基础,设计了一种交替分组显武迭代(AGEI)方法,并证明了该迭代过程的收敛性.由于三层显格式和AGEI方法的整个计算过程都是显式的,所以非常适合于并行计算.文末的数值算例表明,上述方法具有很高的精确度和良好的实用性.     

Integral solution for Forchheimer free convection over a horizontal flat surface

The aim of the present paper is to study the non-Darcy free convection from a horizontal flat surface in a fluid saturated porous medium using integral method for the case when the heat flux from the surface remains constant. The thermal dispersion effects are taken into consideration. The linear relation between the dispersion thermal diffusivity and the streamwise velocity component has been adopted. Exponential profiles are choosen for the velocity and temperature distributions. The Nusselt number results are in good agreement with the existing similarity solution.     

Integral relations for rogue wave formations of Gardner equation

Nonlinear Dynamics - Rogue wave structures can often be described by a set of rational solutions. These can be derived for physical systems satisfying various well-known nonlinear equations. We...     

Influence of inclination angle on buoyancy-driven convection in triangular enclosure filled with a fluid-saturated porous medium

The present investigation deals with the numerical analysis of steady-state laminar buoyancy-driven convection in an inclined triangular enclosure filled with fluid saturated porous media using the Darcy law equation. One wall of the enclosure is isothermally heated and the other is cooled, while the remaining wall is adiabatic. The effect of inclination angle on natural convection is investigated by varying the angle of inclination (φ) between 0° and 360°. The governing transformed equations are solved numerically using a finite-difference method. Obtained results are shown in the form of streamlines, isotherms, mean Nusselt numbers and dimensionless stream function for different values of the Rayleigh number Ra in the range 100 ≤ Ra ≤ 1,000. It is found that the values of the maximum and minimum mean Nusselt number are reached for φ = 330° and φ = 210° , respectively. However, the lowest flow strength is formed at φ = 240° for all values of Ra.     

Correlation equation for laminar and turbulent natural convection from spheres

Natural convective heat transfer from isothermal spheres to air or water is studied. The limiting equations for the Nusselt numbers at very small and large values of the Rayleigh number respectively are developed using a collection of theoretical and experimental results obtained by several investigators. A single correlation of general form is constructed employing both limiting equations in conjunction with additional values of the Nusselt number at intermediate Rayleigh numbers. This single correlation interpolates the limiting equations and is valid for both laminar and turbulent regimes. The computed results agree very well with those given by individual equations for subintervals of the Rayleigh number spectrum.

---

Eine Beziehung für laminare und turbulente freie Konvektion an einer Kugel

Zusammenfassung Es wird der Wärmeübergang bei freier Konvektion von isothermen Kugeln anLuft oder Wasser untersucht. Die Grenzgleichungen für sehr kleine und sehr große Rayleigh-Zahlen erhält man aus einer Sammlung von Literaturwerten. Unter Benutzung dieser beiden Beziehungen wird eine einheitliche Korrelation sowohl für den laminaren wie für den turbulenten Bereich abgeleitet, die zusätzlich Terme f:ur den mittleren Bereich enthält. Die so berechneten Werte stimmen gut mit jenen überein, die aus Gleichungen für Teilbereiche der Rayleigh-Zahl erhalten werden.

---

Nomenclature d diameter (m) - g acceleration of gravity (m/s²) - G_r Grashof number, g(T_s – T)d³/² - ¯h mean convection coefficient (W/Km²) - k fluid thermal conductivity (W/Km) - n constant - N_u mean Nusselt number, ¯hd/k - asymptotic N_u for R_aO - asymptotic N_u for R_a - P_r Prandtl number, / - R_a Rayleigh number, G_r P_r - T_s surface temperature (K) - T fluid temperature (K) - fluid thermal diffusivity (m²/s) - volumetric coefficient of expansion (l/K) - kinematic viscosity (m²/s)     

Boundary condition formulation for viscous fluid flow problems

In the solution of the Navier-Stokes equations by difference methods in infinite regions, the question arises as to the nature of the approximate boundary conditions at those portions of the computational region boundary where these conditions are not determined directly by the formulation of the basic problem. In certain cases of practical importance, these boundary conditions may be obtained by coupling the N-S equations with equations which are similar to the boundary-layer equations.In the present paper, we propose boundary conditions for the case of viscous incompressible fluid flow. Their application is illustrated for the problem of flow past the leading edge of a semi-infinite flat plate.The author wishes to thank I. Yu. Brailovskaya and L. A. Chudov for helpful suggestions in the course of this investigation.     

Variational formulation of stability problems for thin-walled members

Summary The governing equations for thin-walled members subjected to axial force, biaxial bending and torsion are formulated by variational method. The resulting equations differ slightly with currently accepted governing equations. Numerical examples are solved for simple problems to illustrate some of the features of the proposed equations.

---

Übersicht Die beschreibenden Differentialgleichungen für dünnwandige Bauglieder, die durch achsiale Kräfte, biachsiale Biegung und Torsion beansprucht sind, werden nach einem Variationsverfahren aufgestellt. Die so erhaltenen Gleichungen unterscheiden sich etwas von den zur Zeit üblichen Gleichungen. Um die Eigenschaften der jetzt vorgeschlagenen Gleichungen zu zeigen, werden numerische Ergebnisse für einige einfache Aufgaben angegeben.

---

---

On leave from University of Tokyo, Tokyo Japan.     

An adaptive remeshing technique for laminar natural convection problems

 An adaptive remeshing procedure based on interpolation error has been developed for solving laminar natural convection problems. A simple relation between the Rayleigh number and the equilibration constant is introduced to increase the efficiency of the method. Unstructured meshes have been regenerated based on an initial solution and used in the calculations. This avoids the expensive mesh sensitivity study. Two typical natural convection problems are solved to demonstrate the present technique. Excellent performance has been observed at moderate and high Rayleigh numbers although present method gives good results over the whole range of Rayleigh numbers considered. Received on 27 October 2000 / Published online: 29 November 2001     

Integral transform analysis of natural convection in porous enclosures

A hybrid numerical-analytical solution for steady-state natural convection in a porous cavity is proposed, based on application of the ideas in the generalized integral transform technique. The integral transformation process reduces the original coupled partial differential equations, for temperature and stream function, into an infinite system of non-linear ordinary differential equations for the transformed potentials, which is adaptively truncated and numerically solved through well-established algorithms. The approach is applied to a vertical rectangular enclosure subjected to uniform internal heat generation. The convergence characteristics of the explicit inversion formulae are illustrated and critical comparisons with previously reported purely numerical solutions are performed.