Viscosimetric flow of an incompressible elastoviscoplastic material under the presence of a lubricant on the boundary surfaces

Generalizing the model of large deformations by accounting for the viscous properties of materials, we obtain the analytical solutions of some quasistatic boundary value problems concerning the viscosimetric flows of an elastoviscoplastic material in the gap between the rigid coaxial cylindrical surfaces when, in the neighborhood of one of the rigid cylinders (either internal or external), there is a layer of an elastic non-Newtonian lubricant, and the rigid adhesion conditions are satisfied on boundary surfaces. The conditions are studied of origination of a flow in the lubricant layer and in the basic material. The values of the maximum velocity are specified under which the flow does not tresspass the lubricant layer.     

The dynamic behaviour of elastic coaxial cylindrical shells conveying fluid

The dynamic behaviour of elastic coaxial cylindrica shells, which interact with a flow of compressible fluid in the inner shell or in the annular gap between the shells when both flows are present, is investigated by the finite element method. A number of test problems is considered in the case of cantilevered coaxial shells. The effect on the stability limit of the gap between the two shells is investigated for different values of the stiffness parameters of the outer shell and the fluid flow. An important difference is found with existing solutions in cases when the loss of stability of the coaxial shells occurs at higher oscillation modes. It is established that, for a certain value of the gap between the shells, the elasticity of the outer shell may have a stabilizing effect. It is shown that the presence of internal and annular flows simultaneously has a considerable stabilizing effect, while a loss of stability when the flow rates increase occurs at extremely high oscillation modes.     

The stability of flow between concentric cylindrical surfaces with a circular magnetic field

Zusammenfassung Das Stabilitätsproblem für Taylorströmungen und Deanströmungen mit kreisförmigem magnetischem Feld wird formuliert. Bekannte Ergebnisse werden kurz zusammengefasst und beschrieben. Neue Ergebnisse betreffend die Stabilität von Deanströmungen, mit leitenden und mit nichtleitenden Wänden, werden gefunden. Das Eigenwertproblem wird mittels der Galerkinschen Methode gelöst.

---

---

Prepared under support from the Office of Naval Research, Department of Navy, USA, Contract Nonr-3731 (OO) (FBM).     

Nonisothermal deformation of an elastoviscoplastic flat heavy layer

We give a solution to the thermoplasticity problem on the slide of a heavy layer from an inclined plane under heating. The reason for the creep effect is the development of a viscoplastic flow due to the dependence of the yield strength of the layer material on temperature. In the framework of large deformation theory we indicate a law of propagation for the elastoplastic boundary, calculate stresses, deformations, and deformation velocities both in the thermoplastic deformation region and the flow region.     

Logarithmic transformations of rigid analytic elliptic surfaces

We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.     

The load transfer between two rigid spherical inclusions in an elastic medium

The paper is concerned with the load transfer problem between two rigid spherical inclusions in an elastic matrix. A reflection-type formula is developed which is accurate up to and including terms ofO(a/R)⁴, wherea is the maximum radius of the inclusions, andR is the centre-to-centre distance between the two inclusions. Asymptotic results are derived at near touching showing a weak logarithmic singularity in the load transfer. The results are verified by a direct numerical calculation using a boundary collocation method. The numerical method uses Kelvin's general solution as the basis functions for the approximate solution and is highly accurate and efficient, even at near touching.     

A characteristic property of multidimensional cylindrical surfaces

Translated from Ukrainskii Geometricheskii Sbornik, No. 31, pp. 19–26, 1988.     

Steady-state flow of an incompressible viscoelastic polymer fluid between two coaxial cylinders

A boundary value problem for a quasi-linear equation determining the velocity profile of a flow of a polymer fluid in a pipe formed by two coaxial cylinders is considered. On the basis of methods of approximation without saturation, a computational algorithm of increased accuracy is developed, making it possible to solve the problem in a wide range of parameters, including record-low values of r ₀, the radius of the inner cylinder.     

Fluid-structure interaction of an elastic pipe immersed in a coaxial rigid pipe: a model for the Tullio Phenomenon

An axisymmetric, elastic pipe is filled with an incompressible fluid and is immersed in a second, coaxial rigid pipe which contains the same fluid. A pressure pulse in the outer fluid annulus deforms the elastic pipe which invokes a fluid motion in the fluid core. It is the aim of this study to investigate streaming phenomena in the core which may originate from such a fluid-structure interaction. This work presents a numerical solver for such a configuration. It was developed in the OpenFOAM software environment and is based on the Arbitrary Lagrangian Eulerian (ALE) approach for moving meshes. The solver features a monolithic integration of the one-dimensional, coupled system between the elastic structure and the outer fluid annulus into a dynamic boundary condition for the moving surface of the fluid core. Results indicate that our configuration may serve as a mechanical model of the Tullio Phenomenon (sound-induced vertigo). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On an extremal property of the Poiseuille flow between two coaxial cylinders

Some extremal property is established of the axial stationary flow of a Newtonian fluid between two concentric cylinders. It is shown that the ratio of the flow rate through the cross-section of an eccentric gap to the pressure drop (along the axis) reaches its minimum when the eccentricity equals zero; i.e., the minimum of the ratio is realized for the Poiseuille flow.     

Immersion ofn-dimensional cylindrical metrics in the form of cylindrical surfaces into Lobachevsky spaces

A definition of l -dimensional k-cylindrical metrics is introduced and it is proved that k-cylindrical metrics admit an immersion in the form of k-cylindrical surfaces into Lobachevsky space L^l+p, where codimension p of the immersion coincides with the codimension of the immersion of the base of the cylindrical metric into a Euclidean or Lobachevsky space.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 66–69, 1991.     

Nonreflecting distortions of an isotropic strip clamped between rigid punches

The sides of a homogeneous and isotropic strip are considered that are in ideal contact with infinite rigid punches. Smooth distortions of the sides are constructed so that, at one or several given frequencies, an elastic wave propagating through waveguide gives rise to only exponentially decaying perturbations and, possibly, to a small phase shift.     

A criterion on instability of cylindrical rotating surfaces

We consider a column of a stationary rotating surface in Euclidean space. In this paper we obtain a value l ₀ > 0 in such a way that if the length l of the column satisfies l > l ₀, then the surface is instable. This extends previous results due to Plateau and Rayleigh for columns of surfaces with constant mean curvature.     

Diffraction of sound waves by a rigid cylindrical cavity of finite length with an internal impedance surface

A rigorous solution is presented for the problem of diffraction of plane harmonic sound waves by a cavity formed by a terminated rigid cylindrical waveguide of finite length whose interior surface is lined by an acoustically absorbent material. The solution is obtained by a modification of the Wiener-Hopf technique and involve an infinite series of unknowns, which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.Received: December 12, 2001     

Diffraction of sound waves by a rigid cylindrical cavity of finite length with an internal impedance surface

A rigorous solution is presented for the problem of diffraction of plane harmonic sound waves by a cavity formed by a terminated rigid cylindrical waveguide of finite length whose interior surface is lined by an acoustically absorbent material. The solution is obtained by a modification of the Wiener-Hopf technique and involve an infinite series of unknowns, which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.     

A note on the rigid core of affine surfaces

During recent decades, $$\mathbb {G}_a$$-actions, especially certain invariants of $$\mathbb {G}_a$$-actions, have been important tools in the study of affine varieties. The $$\mathbb {G}_a$$-actions are usually studied through locally nilpotent derivations in characteristic zero and exponential maps (see Definition 1.1) in arbitrary characteristic. The “Makar-Limanov invariant” of locally nilpotent derivations played a pivotal role in solving the linearization conjecture in the 1990s, while invariants of exponential maps were central to N. Gupta’s resolution of the Zariski cancellation problem in positive characteristic. In the study of locally nilpotent derivations on commutative algebras containing $$\mathbb {Q}$$, Freudenburg and Moser-Jauslin (Mich Math J 62:227–258, (2013), Theorem 6.1) have introduced a new invariant called “rigid core” and used it to formulate an alternative version of Mason’s theorem and to prove a well-known analogue of Fermat’s last theorem for rational functions (Freudenburg and Moser-Jauslin (2013), Corollary 6.1). In this note, we consider the concept of the rigid core in the framework of exponential maps on commutative algebras over an algebraically closed field k of arbitrary characteristic. We observe that for any factorial k-domain B with $${\text {tr.deg}}_k(B)=2$$, the concept of rigid core coincides with the Makar-Limanov invariant. We also show that over any affine two-dimensional normal k-domain B, its rigid core is a stable invariant.     

Frictional contact of elastomer materials on rough rigid surfaces

The frictional behavior of elastomer materials is still unexplored, but very important for many industrial applications. Special attention is turned to rubber friction on rough road tracks. Due to the non‐rigid material characteristics of the rubber it is not sufficient to use a constant friction coefficient like Coulombs law. The frictional qualities depend on many different influences like sliding velocity, applied normal stress, surface roughness, material properties and the temperature in the contact zone. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damageable contact between an elastic body and a rigid foundation

In this work, the contact problem between an elastic body and a rigid obstacle is studied, including the development of material damage which results from internal compression or tension. The variational problem is formulated as a first-kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field. The existence of a unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, three two-dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme.