Boundary integral equations in quasisteady problems of capillary fluid mechanics Part 2: Application of the stress-stream function

In this paper planar viscous flows with a free boundary are further studied using the quasisteady approximation [1]. The introduction of the bianalytical stress-stream function provides an opportunity to adopt the theory of analytical functions. The mode of construction of the Fredholm boundary integral equations is here proposed through the explicit solutions of two Hilbert problems for holomorphic functions with the application of the conformal mappings. The stabiligy of the equilibrium of the annulus liquid layer is investigated by way of example.

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Sommario Si prosegue lo studio di flussi piani viscosi con frontiera libera applicando l'approssimazione quasistazionaria [1]. L'introduzione della funzione stress-stream bianalitica consente l'uso della teoria delle funzioni olomorfe. La costruzione delle equazioni integrali di Fredholm al contorno proposta qui si basa sulla risoluzione esplicita di due problemi di Hilbert per funzioni analitiche mediante applicazione della tecnica delle trasformazioni conformi. Come esempio si studia la stabilità dell'equilibrio di uno strato liquido anulare.

     

Boundary integral equations for inverse problems in the elasticity theory

The paper is concerned with the application of the boundary integral equation method for the holomorphic vector to nonclassical problems in the static theory of elasticity. Problems are considered for the case when on part of a body conditions are overvalued, i.e. the vectors of displacements and loads are prescribed, and on the other part of the body conditions are unknown (so-called (u, p) problem). It is shown that the method proposed is efficient and can be applied to the problems of computer defectoscopy in stationary potential fields.     

Boundary integral equations of unique solutions in elasticity

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Some problems in metallurgical fluid mechanics

Application of computational fluid mechanics in two areas of metallurgy is considered: solidification of liquid alloys and MHD turbulence. In the first class of problems, where the technical issue is to obtain a reasonably homogeneous composition of the cast, one has to consider not only the completely molten and solidified regions but also the “mushy zone” that is made up of small-scale dendrites, which appear between the first two regions. In the completely molten region, the composition is practically constant and the fluid is set into motion due to the inhomogeneous temperature field. In the mushy zone, on the other hand, solutal convection often dominates strongly over thermal convection. It is shown that laminar convection is of prime importance for the composition of the solidified alloy. In the second class of problems, two cases of turbulent MHD flows in cylindrical containers are considered: an electromagnetic furnace and an electromagnetic stirrer. In the electromagnetic furnace, the mean flow consists of two toroidal vortices. The mean motion in the electromagnetic stirrer is a swirling motion that is accompanied by a weak meridional circulation, which is reminiscent of that occuring in spin down phenomena. The MHD flows are computed by using large eddy simulation methodology with a new subgrid model of the Smagorinsky type that accounts for a variable mesh. Predictions from the all model computations are compared with experimental observations. In general, the agreement between theory and experiments is satisfactory.     

Boundary integral equations of fluid flow and electrical and heat conduction in a medium composed of blocks

For many applications in the theory of flow through porous media, diffusion, convective transport, and electrical and heat conduction it is important to consider problems involving systems of blocks separated by boundaries (in particular, cracks or conductors), the transport process taking place both through the blocks themselves and along the separating boundaries. Such problems can conveniently be solved by the method of boundary integral equations [1–4]. The object of this study is to propose a form for these equations and methods of solving them specially designed to take into account the particularities of the problems in question. Firstly, the boundary integral equation is given in a form that contains only the net inflow (of fluid, heat or electricity) to a unit area of the boundary and not the individual inflows from each of the blocks separated from that boundary. This almost halves the number of unknowns subject to determination. Secondly, the principal aspects associated with the specifics of the flow along the boundaries and their intersections and discontinuities are discussed. Thirdly, a numerical experiment to realize the proposed form of the equations on a computer using algorithms with different structures leads to quite general conclusions which may be useful for further developing the method of boundary integral equations in relation to the class of applied problems considered. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 72–78, November–December, 1986. The authors are grateful to their hydrogeologist colleagues at the All-Union Institute of Mining Geomechanics and Surveying (VNIMI) and the Leningrad Mining Institute for taking a stimulating interest in their work over a period of three years and for discussing its various applications.     

Free surface problems in rheological fluid mechanics

Summary In this paper we review our work on the theory of domain perturbations of the rest state of a viscoelastic fluid and its applications to the science of rheometry. To explain the principle behind the domain perturbation analysis we begin the paper with a discussion of a model problem, free of rheological complications, in which the principles involved may be clearly demonstrated. In Chapter two we formulate the analysis for rheological problems and we present some new, previously unpublished, simplifications of the theory. The canonical forms for stress tensors which perturb the state of rest are given in Chapter three. Chapter four is devoted to the problem of steady rod climbing. There we discuss the main physical features of the motion, the phenomenon of the critical radius, the effects of surface tension and temperature, secondary motions, the applicability of theory and experiment for rheological measurements, and an interesting new normal stress anplifier. Unsteady problems of rod climbing are considered in Chapter five. We review recent results on the free surface induced by torsional oscillations of a rod, and we show how to use these results to find the form of the material functions in the canonical forms of the stress. The breathing instability of steady axisymmetric rod climbing (aHopf bifurcation) and the flower instability of time-periodic climb induced by the oscillating rod (a symmetry-breakingPoincaré bifurcation) are also described in Chapter five. In Chapter six we consider other free surface problems: the free surface on a fluid between oscillating planes, die swell and edge effects in rheometers.

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Zusammenfassung Es wird ein Überblick über die Untersuchungen der Verfasser zum Thema Gebietsstörungen des Ruhezustandes (domain perturbations of the rest state) einer viskoelastischen Flüssigkeit und ihre Anwendung auf die Rheometrie gegeben. Zur Erläuterung der Grundlagen der Gebietsstörungs-Analyse wird in Kap. 1 die Diskussion eines Modellproblems vorangestellt, das keinerlei rheologische Komplikationen enthält, so daß die betreffenden Prinzipien in einfacher Weise erläutert werden können. In Kap. 2 wird die Analyse auf rheologische Probleme angewendet, und es werden einige bisher unveröffentlichte Vereinfachungen der Theorie vorgestellt. In Kap. 3 werden die kanonischen Formen der die Störung des Ruhezustandes hervorrufenden Spannungstensoren angegeben. In Kap. 4 wird das Problem des Hochkletterns einer Flüssigkeit am stationär rotierenden Zylinder behandelt. Dabei werden die Hauptmerkmale der Bewegung, das Phänomen des kritischen Radius, die Wirkung von Oberflächenspannung und Temperatur, Sekundärbewegungen, die Anwendbarkeit von Theorie und Experiment auf rheologische Messungen, sowie ein interessanter neuer Normalspannungs-Verstärker diskutiert. In Kap. 5 wird das Problem des Hochkletterns bei instationärer Bewegung des Zylinders betrachtet. Es werden neuere Ergebnisse über die durch eine Torsionsschwingung erzeugte freie Oberfläche referiert, und es wird gezeigt, wie diese Ergebnisse zur Bestimmung der Stoff-Funktionen in der kanonischen Darstellung der Spannungen verwendet werden können. Die atmende Instabilität beim stationären Hochklettern (eine Verzweigungslösung vomHopfschen Typ) und die blütenförmige Instabilität beim zeitlich-periodischen Hochklettern infolge der Zylinder-Oszillation (eine die Symmetrie brechende Verzweigungslösung vomPoincaréschen Typ) werden hier ebenfalls beschrieben. In Kap. 6 schließlich werden weitere an freien Oberflächen auftretende Probleme behandelt: nämlich die freie Oberfläche einer Flüssigkeit zwischen oszillierenden Platten, die Strahlaufweitung und die Randeffekte in Rheometern.

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With 13 figures     

BOUNDARY INTEGRAL FORMULA OF ELASTIC PROBLEMS IN CIRCLE PLANE

IntroductionConcerningtheelasticplaneprobleminaunitcircle ,ZhengShenzhouandZhengXueliangdevelopedaboundaryintegralformulaofthestressfunction[1]:Φ(r,θ) =-( 1 -r2 ) 24π ∫2π0ν( φ)1 -2rcos(θ-φ) r2 dφ　　 12π∫2π011 -2rcos(θ-ω) r2 dω∫2π0μ( φ)1 -cos(ω-φ) dφ　　 1 -r22π∫2π0μ( φ)1 -2rcos(θ -φ) r2 dφ　　 ( 0 ≤r <1 ) ,( 1 )whereμ(θ) =Φ(r,θ) ｜r=1,ν(θ) = Φ n r=1= Φ r r=1.Intheformula ( 1 )theseconditemisastrongsingularintegral,itshouldbeunderstoodasanintegra…     

热弹性平面问题的规则化边界积分方程

对于热弹性平面问题,过去广泛集中在直接变量边界元法研究,本文研究间接变量规则化边界元法,建立了间接变量规则化边界积分方程。和直接边界元法相比,间接法具有降低密度函数的连续性要求、位移梯度方程中的热载荷体积分具有较弱奇异性等优点。数值实施中,用精确单元描述边界几何,不连续插值函数逼近边界量。算例表明,本文方法效率高,所得数值结果与精确解相当吻合。     

Some inverse problems in rheology leading to integral equations

Two inverse problems of the integral type, which are of the general importance to rheology, are discussed. The first of them is the possibility of calculating the molecular weight distribution (MWD) from the flow curve and the second one is the interrelation between creep and relaxation functions. It was shown that the first problem is incorrect and any solution is unstable in respect to minor experimental errors. It means that the general solution of this problem is impossible in principle and only estimations of the width of unimodal MWD can be received from the curvature of the flow curve. The possibility of the correct calculation of the creep function exists in case the relaxation curve being approximated by the sum of exponential members. But the approximation of the relaxation curve within the experimental accuracy is the necessary, but not sufficient condition for the correct solution of this integral inverse problem, because not every mode of approximation leads to the satisfactory prediction of creep function.Delivered as the Courtaulds Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Finite elements for exterior problems using integral equations

We present some integral methods for exterior problems for the Laplace equation. Then we give finite element approximations for these equations and some errors estimates. Finally, we indicate how these integral equations can be coupled with a usual finite element method on a bounded domain to solve an exterior non-linear problem which is linear far away.     

Solutions of matrix equations in problems of mechanics and control

The paper considers algorithms for solving linear matrix equations related to problems of mechanics and control, namely, the Lyapunov and Sylvester matrix equations and Riccati-type nonlinear matrix equations. These algorithms are capable of solving both linear equations and linear matrix inequalities. Algorithms based on the Bass relations are used to solve Riccati-type nonlinear matrix equations in so-called special cases where some eigenvalues of the matrix pencil are on a unit circle. These algorithms are compared with those of other authors by way of examples. It is shown that the algorithms can be implemented in symbolic computing routings, which allows solving these equations with high accuracy     

Boundary integral/spectral element approaches to the Navier-Stokes equations

Numerical algorithms are presented which combine spectral expansions on elemental subdomains with boundary integral formulations for solving viscous flow problems. Three distinct algorithms are described. The first demonstrates the use of spectral elements for the classic boundary integral method for steady Stokes flow. The second extends this algorithm to include domain integrals for solution of the unsteady Navier-Stokes equations. The third algorithm explores the use of boundary integrals as a means of consolidating uncoupled elemental solutions in a domain decomposition approach. Numerical results demonstrating high-order convergence are presented in each case.     

Application of local co-ordinate systems to the solution of integral equations referred to plane fluid flow problems

Two-dimensional fluid flow problems expressed in terms of velocity potentials or stream functions are often summarized as boundary-value problems for the Laplace or Poisson equations, or the homogeneous or non-homogeneous biharmonic equations. Simple local co-ordinate systems have been applied to the solution of integral equations associated with these boundary-value problems. This procedure has been shown to be an efficient technique in the numerical solution of fluid flow problems.     

Singular integral equations in Hilbert space applied to crack problems

Crack problems are solved by application of a system of singular integral equations in Hilbert space. The method consists of replacing the singular integral equation by a system of linear algebraic equations for the values of the unknown function at specially chosen points within the range of integration. Obtained is a solution for a nonsymmetric cross-shaped crack in an infinite and isotropic solid subjected to a constant pressure, the symmetric configuration being a special case.     

NOVEL REGULARIZED BOUNDARY INTEGRAL EQUATIONS FOR POTENTIAL PLANE PROBLEMS

The universal practices have been centralizing on the research of regulariza-tion to the direct boundary integal equations (DBIEs). The character is elimination of singularities by using the simple solutions. However, up to now the research of regular ization to the first kind integral equations for plane potential problems has never been found in previous literatures. The presentation is mainly devoted to the research on the regularization of the singular boundary integral equations with indirect unknowns. A novel view and idea is presented herein, in which the regularized boundary integral equations with indirect unknowns without including the Cauchy principal value (CPV) and Hadamard-finite-part (HFP) integrals are established for the plane potential problems. With some numerical results, it is shown that the better accuracy and higher efficiency, especially on the boundary, can be achieved by the present system.