Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.     

On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one “wall” is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional “wall.” With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a two-dimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.     

On a convective flow of a binary mixture in a vertical layer

Under study is an invariant solution of the equations of thermal diffusive convection which describes a stationary process of a binary mixture flow in a vertical layer under the action of the pressure gradient and the buoyancy force that depends nonlinearly on temperature and concentration. Some general properties of this solution are established and an existence theorem is proved. Analysis of the numerical solution of the problem is carried out in the cases of a power-law and exponential dependence of the buoyancy force on its argument.     

Motion of a container as a nonholonomous system in a curved section of a horizontal pipeline

The problem of the motion of a container in a curved section of a horizontal pipeline is solved using second-order Lagrange equations in the presence of nonholonous couplings. The special case of the motion of a container in a circular curve is examined.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 90–95, 1987.     

Hydrodynamics of a particle impact on a wall

The problem of a particle impacting on a wall, a common phenomenon in particle-laden flows in the minerals and process industries, is investigated computationally using a spectral-element method with the grid adjusting to the movement of the particle towards the wall. Remeshing is required at regular intervals to avoid problems associated with mesh distortion and the constantly reducing maximum time-step associated with integration of the non-linear convective terms of the Navier–Stokes equations. Accurate interpolation between meshes is achieved using the same high-order interpolation employed by the spectral-element flow solver. This approach allows the full flow evolution to be followed from the initial approach, through impact and afterwards as the flow relaxes. The method is applied to the generic two-dimensional and three-dimensional bluff body geometries, the circular cylinder and the sphere. The principal case reported here is that of a particle colliding normally with a wall and sticking. For the circular cylinder, non-normal collisions are also considered. The impacts are studied for moderate Reynolds numbers up to approximately 1200. A cylindrical body impacting on a wall produces two vortices from its wake that convect away from the cylinder along the wall before stalling while lifting induced wall vorticity into the main flow. The situation for a sphere impact is similar, except in this case a vortex ring is formed from the wake vorticity. Again, secondary vorticity from the wall and particle plays a role. At higher Reynolds number, the secondary vorticity tends to form a semi-annular structure encircling the primary vortex core. At even higher Reynolds numbers, the secondary annular structure fragments into semi-discrete structures, which again encircle and orbit the primary core. Vorticity fields and passive tracer particles are used to characterize the interaction of the vortical structures. The evolution of the pressure and viscous drag coefficients during a collision are provided for a typical sphere impact. For a Reynolds number greater than approximately 1000 for a sphere and 400 for a cylinder, the primary vortex core produced by the impacting body undergoes a short-wavelength instability in the azimuthal/spanwise direction. Experimental visualisation using dye carried out in water is presented to validate the predictions.     

On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of a fractional order

This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.     

The transmission of a flexural-gravitational wave through a rigid end-stop in a floating plate

An elastic plate is located on the surface of a liquid, in continuous contact with it and rigidly clamped in a support along a certain straight line. The orthogonal incidence of a small amplitude flexural-gravitational wave on the support is considered. Exact expressions are obtained for the wave field in the fluid and the flexural field in the plate. The transmission coefficient of the incident flexural-gravitational wave through the support and its reflection coefficient from it are determined. The forces which arise in the support are found. The investigation is carried out for liquids of finite and infinite depth. The effect of the depth of the liquid on the wave processes is indicated. The liquid is assumed to be inviscid and its friction on the bottom and the lower surface of the plate in the neighbourhood of the support is therefore ignored.     

Torsional Oscillations of a Punch on a Layer Bonded to a Half-Space Containing a Cylindrical Cavity

A previously published algorithm [9] is implemented in application to the Reissner–Sagoci problem for a layer bonded to a half-space containing a cylindrical cavity. The influence of the mechanical and geometrical parameters of the layer and the half-space on the amplitude-frequency response curves of the punch oscillations is analyzed. Practical applications of the results are proposed for ensuring the seismic isolation of buildings on the investigated foundation in the presence of dynamic torsional excitations.     

The rolling of a wheel with a pneumatic tyre on a planet

A model of a pneumatic tyre as a system with an infinite number of degrees of freedom is proposed, when its surface is represented by the deformed surface of a torus. Using a number of hypotheses a functional of the potential energy of the deformations of the tyre is obtained as a function of the deformations of its tread. A complete system of equations of motion is obtained, assuming that the wheel rolls without slipping in the area of contact of the tread with the plane, with respect to the previously unknown part of the tread. In two special cases of the rolling of a wheel with breakaway and on a banking, all the characteristics of the motion (the contact area, the tyre deformation, and the forces and moments applied to the disc of the wheel) are obtained.     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

On a Reliable Solution of a Volterra Integral Equation in a Hilbert Space

We consider a class of Volterra-type integral equations in a Hilbert space. The operators of the equation considered appear as time-dependent functions with values in the space of linear continuous operators mapping the Hilbert space into its dual. We are looking for maximal values of cost functionals with respect to the admissible set of operators. The existence of a solution in the continuous and the discretized form is verified. The convergence analysis is performed. The results are applied to a quasistationary problem for an anisotropic viscoelastic body made of a long memory material.     

Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

On moving a sofa around a corner

A necessary condition is given for a region of the plane to have the greatest possible area of any region able to move around a right-angled corner in a hallway of unit width. A region is constructed, with area 2.2195... and bounded by 18 analytic pieces, which satisfies this condition. It is conjectured that this is the unique region of maximum area.     

Fast computation of a rational point of a variety over a finite field

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time-space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system. This invariant, called the degree, is bounded by the Bézout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.

     

Point force in a stratified fluid in a cylinder under a magnetic field

The flow induced by a body moving in an inviscid incompressible density stratified fluid in an infinite circular cylinder under the influence of a uniform axial magnetic field is studied using the method of replacing the body by an isolated point force. This method was adopted by Childress and others in discussing the body effects in a viscous fluid. The solution is obtained using the Fourier transformation and the Lighthill’s radiation condition. The cases of weak and strong magnetic fields are discussed.     

Stressed state of a rock mass around a working with a relief cavity

The stressed state of a rock mass in the vicinity of a preparatory working protected by a relief cavity is studied. A numerical study is made of the stress field and the relief zone, and the effect of length and slope of a slot on formation of the relief zone is clarified.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 31–36, 1989.     

Factoring a Lie Group into a Compactly Embeddedand a Solvable Subgroup

 Let G be a real connected Lie group. A subgroup K is called compactly embedded if the closure of Ad(K) is compact in Aut(). If K is, in addition, maximal with respect to this property, then there exists a solvable subgroup S containing the nilradical such that and is the one-component of the center of G. (Received 1 June 1999; in revised form 28 December 1999)     

The rolling of a wheel with a reinforced tyre along a plane without slip

A model of a wheel with a reinforced tyre, whose surface is simulated by a flexible strip (tread) attached to parts of two tori (the sidewalls of the tyre) is considered. The disk of the wheel (a rigid body) has six degrees of freedom and is in contact with the plane along part of the tread. Based on several assumptions, the potential energy functional of the deformed wheel is found as a function of the deformations of the centre line of the tread. On the assumption that the wheel is rolling without slip in the region of contact of the tread with the plane along a previously unknown section of the tread, the complete system of equations of motion is obtained. The equilibrium of the wheel and the steady state of rolling in a straight line with given swivel and tilt are investigated, and all characteristics of the motion are found (the contact region, the tyre deformation, and the forces and torques applied to the wheel disk).     

Unsteady draining of a fluid from a circular tank

Three-dimensional draining flow of a two-fluid system from a circular tank is considered. The two fluids are inviscid and incompressible, and are separated by a sharp interface. There is a circular hole positioned centrally in the bottom of the tank, so that the flow is axially symmetric. The mean position of the interface moves downwards as time progresses, and eventually a portion of the interface is withdrawn into the drain. For narrow drain holes of small radius, the interface above the centre of the drain is pulled down towards the hole. However, for drains of larger radius the portion of the interface above the drain edge is drawn down first, rather than the central section. Non-linear results are obtained with a novel spectral technique, and are also compared against the predictions of linearized theory. Unstable Rayleigh–Taylor type flows, in which the upper fluid is heavier than the lower one, are also discussed.     

Embedding a graph-like continuum in a surface

Graph-like continua provide a very natural setting for generalizing finite graphs to infinite, compact structures. For example, the Freudenthal compactification of a locally finite graph, exploited by Diestel and his students in their study of the cycle space of an infinite graph, is an example of a graph-like continuum. Generalizing earlier works in special cases, the authors, along with Christian, have proved MacLane’s and Whitney’s characterizations of planarity for graph-like continua (Electron. J. Combin. 17 (2010)). In this work, we consider embeddings of graph-like continua in compact surfaces and show that: (i) every edge is in an open disc that meets the graph-like continuum precisely in that edge; (ii) there are natural analogues of face boundary walks; (iii) there is a graph-like continuum triangulating the same surface and containing as a sub-graphlike continuum the original embedded graph-like continuum; (iv) the face boundaries generate a subspace of the cycle space; and (v) the quotient of the cycle space by the boundary cycles is the homology of the surface. These all generalize results known for embeddings of finite graphs.