Rotation of an axisymmetric particle in a plane flow

The rotation of an inertialess ellipsoidal particle in a shear flow of a Newtonian fluid has been firstly analyzed by Jeffery [1]. He found that the particle rotates such that the end of its symmetry axis describes a closed periodic orbit. Based on the balance equation of the angular momentum we derived the equation of rotational motion of a cylindrical particle, that is suspended in a plane shear flow field of a viscous fluid, and solved numerically. The rotary inertia is taken into account. The solution is compared with the rotation of a slender particle. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The forces and couples acting on two nearly touching spheres in low-Reynolds-number flow

When two unequal spheres are very close, the low-Reynolds-number flow in the narrow gap between them can be analysed using lubrication approximations, and asymptotic formulae for the forces and couples acting on the spheres deduced. The expressions for the forces and couples have previously been regarded as independent, but it is shown here that they are linked by simple physical considerations. The new formulae can be used to improve the accuracy of companion calculations which apply to cases in which the spheres are not close.

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Zusammenfassung Es wird die Strömung bei kleinen Reynoldszahlen untersucht in einem Spalt zwischen zwei ungleichen Kugeln, die sich nahezu berühren. Asymptotische Formeln für die Kräfte und Momente an den Kugeln werden hergeleitet. Es wird gezeigt daß die Ausdrücke für die Kräfte und Momente, die bis jetzt als unabhängig betrachtet wurden, tatsächlich durch einfache physikalische Betrachtungen verknüpft werden können. Die neuen Formeln können zur Verbesserung der Genauigkeit von Rechnungen für größere Kugelabstände benützt werden.

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During this work, D. J. Jeffrey was supported by the Natural Environment Research Council of Great Britain.     

The resistive and separative forces acting on parallel plates in a viscous flow

Zusammenfassung Die Strömung einer zähen Flüssigkeit längs zweier paralleler Platten wird im Rahmen der Oseenschen Theorie behandelt. Für kleine Reynoldszahlen wurden Näherungslösungen der Bewegungsgleichungen erhalten.     

Mixed problems of axisymmetric elasticity theory for some bodies of revolution

We consider the problem of axisymmetric elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on the cavity surface and the main mixed problem of axisymmetric elasticity theory for a hyperboloidal layer formed by the two surfaces of a two-cavity hyperboloid of revolution symmetrical about the plane z = O. The problems are solved by the method of p-analytical functions. The solution of the first problem is reduced to solving a Fredholm integral equation of the second kind. We investigate the behavior of the normal stress near the boundary lines. The solution of the second problem is reduced to solving a system of two Fredholm integral equations of the second kind. Existence and uniqueness of the solution is proved for this system.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 88–101, 1989.     

High-frequency projection method for plane problems of wave diffraction on dielectric bodies

A version of the Galerkin incomplete projection method is described for plane problems of wave diffraction on dielectric bodies of arbitrary shape. The proposed method generalizes the Sommerfeld method, which constructs diffraction series rapidly converging at high frequencies for circular and spherical bodies. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 58–67.     

The method of successive approximation of a free boundary in problems of cavitation flow

We discuss a numerical method of “recursion on the shape of the cavity” for solving problems of cavitation flow past planar and axisymmetric objects according to the Ryabushinskii scheme. We give a comparison of a closed-form solution with the numerical results. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Stokes flow for the axisymmetric motion of several spherical particles perpendicular to a plane wall

The creeping flow around several spherical particles moving on a line perpendicular to a plane wall is calculated numerically using the boundary integral method. The locations of the point forces on the surfaces of the spheres are chosen so as to describe precisely the lubrication regions when the surfaces are close to one another. Earlier results are recovered for the cases of a single sphere and a wall and of two equal spheres far from a wall. New results are presented for two (equal or unequal) spheres close to a plane wall and several equal spheres far from a wall.     

A method of solving the integral equation of plane contact problems for semi-bounded bodies

A typical integral equation, which arises when solving linear plane contact problems for semi-bounded bodies, is considered. By using a special representation of the kernel of this equation, an approximate method is developed for solving it that is effective over a wide range of variation of the dimensionless geometrical parameter occurring in the kernel. The method is tested on the problem of the symmetrical compression of an elastic strip along its boundaries by two similar punches.     

Reduced convex bodies in the plane

A convex bodyR of Euclideand-spaceE ^d is called reduced if there is no convex body properly contained inR of thickness equal to the thickness Δ(R) ofR. The paper presents basic properties of reduced bodies inE ². Particularly, it is shown that the diameter of a reduced bodyR?E ² is not greater than √2Δ(R), and that the perimeter is at most (2+½π)Δ(R). Both the estimates are the best possible.     

Some comments on floating and centroid bodies in the plane

Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let \(K_w\) be the wet part of K. In this paper we study some properties of the locus of the centroid of \(K_w\) and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane.     

On a method of solving two-dimensional integral equations of axisymmetric contact problems for bodies with complex rheology

A two-dimensional integral equatin appearing in axisymraetric contact problems for bodies with complex rheology is studied. A method of constructing the solution of this equation in proposed, based on inspecting the non-classical spectral properties of an integral operator. A contact problem for a non-uniformly aging viscoelastic foundation is solved as an example.     

Generalized inertia forces of rigid and flexible bodies

In the paper a method for deriving the dynamic equations of motion of rigid and flexible multibody systems is presented. Three dimensional matrices are applied for working out the configuration space dynamic equations. Finite elements are used for modeling of flexible systems. Novel generalized Newton‐Euler equations and relations for inertia forces in nodes of flexible elements are proposed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of flow-induced motion of floating bodies

A computational procedure for the prediction of motion of rigid bodies floating in viscous fluids and subjected to currents and waves is presented. The procedure is based on a coupled iterative solution of equations of motion of a rigid body with up to six degrees of freedom and the Reynolds-averaged Navier–Stokes equations describing the two- or three-dimensional fluid flow. The fluid flow is predicted using a commercial CFD package which can use moving grids made of arbitrary polyhedral cells and allows sliding interfaces between fixed and moving grid blocks. The computation of body motion is coupled to the CFD code via user-coding interfaces. The method is used to compute the 2D motion of floating bodies subjected to large waves and the results are compared to available experimental data, showing favorable agreement.     

Boundary element formulation for elastoplastic analysis of axisymmetric bodies

The complete formulation of B.E.M. applied to the analysis of axisymmetric bodies acting in the plastic range is presented in this paper. The concept of derivative of a singular integral given by Mikhlin has been used in order to calculate the stresses in internal points.Also a semianalytical approach is proposed to compute the matrix coefficients, presenting the way in which it can be done and the results obtained.     

Double-lattice packings of convex bodies in the plane

Mahler [7] and Fejes Tóth [2] proved that every centrally symmetric convex plane bodyK admits a packing in the plane by congruent copies ofK with density at least 3/2. In this paper we extend this result to all, not necessarily symmetric, convex plane bodies. The methods of Mahler and Fejes Tóth are constructive and produce lattice packings consisting of translates ofK. Our method is constructive as well, and it produces double-lattice packings consisting of translates ofK and translates of–K. The lower bound of 3/2 for packing densities produced here is an improvement of the bounds obtained previously in [5] and [6].     

Solving nonlinear multicommodity flow problems by the analytic center cutting plane method

The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra’s d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities). This research has been supported by the Fonds National de la Recherche Scientifique Suisse, grant #12-34002.92, NSERC-Canada and FCAR-Quebec. This research was supported by an Obermann fellowship at the Center for Advanced Studies at the University of Iowa.     

Unsteady axisymmetric stagnation flow on a circular cylinder

Unsteady, axisymmetric stagnation flow about a circular cylinderis examined when the far-field flow is a periodic function oftime with a fixed time average and an oscillatory part of prescribedamplitude and frequency. Solutions are computed for arbitraryvalues of the Reynolds number, quantifying the effects of surfacecurvature, and a frequency parameter based on the period ofthe far-field flow. It is found that solutions remain regularand periodic provided that the far-field amplitude lies belowa critical value. Above this value, solutions terminate in afinite-time singularity. The blow-up time is delayed by increasingthe curvature of the surface. These results are corroboratedby asymptotic predictions valid in the limits of small and largeamplitude and frequency. For large Reynolds number, the problemreduces to the two-dimensional stagnation-point flow againsta plane wall studied by previous authors.     

On plane problems in magneto-thermo-elasto-statics

The well-known complex variable method of solving plane problems in classical elasticity has been extended to plane problems of magnetothermo-elasticity. The method is illustrated by considering the plane deformation of an infinite body containing a cylindrical cavity submitted to uniform heat flow and uniform magnetic field.