Motion of a solidcontaininga spherical damper

For the purpose of modeling the motion of a solid with a cavity filled with a viscous fluid, M. A. Lavrent'ev [1] has proposed a model in the form of a solid with a spherical cavity in which another solid, spherical in shape, is enclosed. The sphere is separated from the cavity walls by a small, clearance in which viscous forces act (a lubricating film). This simple model with a finite number of degrees of freedom possesses certain mechanical properties of a solid with a cavity containing a viscous fluid. Study of this model is therefore of interest.The present paper examines certain properties of the model, which will be termed a solid with a damper. It is shown that for a highviscosity lubricant the motion of a solid with a damper can be described by the same equations which pertain to the motion of a solid with a spherical cavity filled with a high-viscosity fluid. Expressions relating the parameters of the systems are obtained. If these relations are fulfilled, the systems become mechanically equivalent.The steady motions of a free solid with a damper and their stability conditions are determined.These motions and stability conditions hold for a body with a cavity filled with a viscous fluid [2].     

New solutions of classical problems in rigid body dynamics

The equations of motion of a rigid body acted upon by general conservative potential and gyroscopic forces were reduced by Yehia to a single second-order differential equation. The reduced equation was used successfully in the study of stability of certain simple motions of the body. In the present work we use the reduced equation to construct a new particular solution of the dynamics of a rigid body about a fixed point in the approximate field of a far Newtonian centre of attraction. Using a transformation to a rotating frame we also construct a new solution of the problem of motion of a multiconnected rigid body in an ideal incompressible fluid. It turns out that the solutions obtained generalize a known solution of the simplest problem of motion of a heavy rigid body about a fixed point due to Dokshevich.     

Ideal gas flow behind a finite-amplitude shock wave

The equations of one-dimensional (with a plane of symmetry) adiabatic motion of an ideal gas are transformed to a form convenient for studying flows between a moving piston and a shock wave of variable intensity. The solution is found for the equations of a motion containing a shock wave which propagates through a quiescent gas with variable initial density and constant pressure. This solution contains four arbitrary constants and, in a particular case, gives an example of adiabatic shockless compression by a piston of a gas initially at rest.     

Draft of a nozzle with the circulating outflow of a vortical flow of gas

The presence of circulation in an outflowing gas leads to a change in the working parameters of a nozzle. The question of the mass flow rate and the draft of a nozzle without a diffusor (a point) for twisted flows has been studied theoretically and experimentally [1–6]. The use of nozzles with a supersonic part introduces a considerable degree of complication into the method for the analytical calculation of the draft characteristics and the program for their experimental investigation. In [2, 7], a theory of a nozzle is formulated for a model of a potential circulating flow of gas; in [5, 8], an electronic computer was used to solve the complete system of the equations of gasdynamics for the motion of a rotating flow along a nozzle; in [7, 9], an investigation was made of a variational problem of the shaping of a diffusor for a circulation flow. The calculation of the draft, carried out in the above-mentioned communications (with the exception of [2], in which a study was made of a partial model of an eddyless rotational motion), is bound up with labor-consuming computer calculations. In the present article, in a development of [3, 6], a quasi-one-dimensional theory of a supersonic nozzle for a vortical flow of gas is formulated and verified experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–149, September–October, 1975.     

Three-dimensional isothermal lava flows over a non-axisymmetric conical surface

Within the thin-layer approximation for a highly-viscous heavy incompressible fluid, a hydrodynamicmodel of a 3D isothermal lava flow over a non-axisymmetric conical surface is constructed. Using analytical methods, a self-similar solution for the law of leading-edge propagation is obtained in the case of a flow from a non-axisymmetric source located at the apex of a conical surface with smoothly varying properties. In the case of a flow over a substantially non-axisymmetric surface, it is shown that there exists a self-similar solution for the free-surface shape and the law of leading-edge motion. This solution is studied numerically for particular examples of the substrate surface and the source. In the general case of a non-self-similar flow over a substantially non-axisymmetric conical surface, a local analytical solution is obtained for the free-surface shape and the velocity field near the leading flow front.     

Numerical Simulation of Coherent Structures over Plant Canopy

This paper reports large eddy simulations of the interaction between an atmospheric boundary layer and a canopy (representing a forest cover). The problem is studied for a homogeneous configuration representing the situation encountered above a continuous forest cover, as well as for a heterogeneous configuration representing the situation similar to an edge or a clearing in a forest. The numerical results reproduces correctly all the main characteristics of this flow as reported in the literature: the formation of a first generation of coherent structures aligned transversally with the wind flow direction, the reorganization and the deformation of these vortex tubes into horse-shoe structures. The results obtained when introducing a discontinuity in the canopy (reproducing a clearing or a fuel break in a forest), are compared with the experimental data collected in a wind tunnel; here, the results confirm the existence of a strong turbulence activity inside the canopy at a distance equal to 8 times the height of the canopy, referenced in the literature as the Enhance Gust Zone (EGZ) characterized by a local peak of the skewness factor.     

Dynamical regimes of fiber spinning

An analytical solution is obtained that describes fiber spinning with a given force on the receiving bobbin. As an example, a calculation is made of the response of the final fiber section to a periodically varying draw force; a solution is constructed that describes the propagation along the fiber of a finite perturbation associated with a change in the conditions at the spinneret for a fixed draw force. The problem of the small perturbations of a fiber spun at a given rate onto a bobbin is reduced to a linear integrodifferential equation with retardation whose characteristic equation determines the region of the “draw resonance” instability. The reasons for the occurrence for the instability are elucidated.     

Transient responses of beams and plates subject to travelling load. Miscellaneous results

The steady-state response of a free and infinite Timoshenko beam is specified analytically in terms of non-dimensional displacements and stresses. The beam is supposed loaded by a travelling concentrated force or a moving step load. By a validated explicit numerical calculation, it is shown how a load travelling on a beam at constant velocity, from defined time and abscissa, generates a response which evolves towards the steady-state solution for a part, and towards a quantified transient solution for another part. Asymptotic values are given for the transient displacements and stresses according to the time and the speed of the loading. The solution is also found for a plate subject to a pressure, which spreads respecting the cylindrical symmetry. It is possible to identify in the response a part which follows the pressure front, and which is comparable with the steady-state response of a beam, and another transient part, which generates displacements and stresses with a much less oscillating character. An asymptotic solution is also presented for the plate.The whole series of the results makes it possible to better understand qualitatively the beginning of the transient response of a beam or of a plate to a moving load, and also makes it possible to estimate the stresses and displacements without needing specialised numerical codes.     

Penetration of a cumulative jet into a barrier

A solution is given to the problem of the penetration of a cumulative jet with an arbitrary distribution of the velocity along it, taking account of the strength properties of the barrier. Using the example of a jet with a linear distribution of the velocity, the article demonstrates the possibility of obtaining a large puncturing capacity due to a change in the gradient along the jet as a function of the physicomechanical properties of the barrier and the jet. In addition, a distribution of the velocity along the jet is obtained which assures a maximal penetration in a barrier, arranged at a distance where a limiting elongation is not attained either partially or completely over the whole jet.     

A Eulerian/Lagrangian model to calculate the evolution of a water droplet spray

We introduce a Eulerian/Lagrangian model to compute the evolution of a spray of water droplets inside a complex geometry. To take into account the complex geometry we define a rectangular mesh and we relate each mesh node to a node function which depends on the location of the node. The time-dependent incompressible and turbulent Navier-Stokes equations are solved using a projection method. The droplets are regarded as individual entities and we use a Lagrangian approach to compute the evolution of the spray. We establish the exchange laws related to mass and heat transfer for a droplet by introducing a mass transfer coefficient and a heat transfer coefficient. The numerical results from our model are compared with those from the literature in the case of a falling droplet in the atmosphere and from experimental investigation in a wind tunnel in the case of a polydisperse spray. The comparison is fairly good. We present the computation of a water droplet spray inside a complex and realistic geometry and determine the characteristics of the spray in the vicinity of obstacles.     

A Lagrangian finite element method for the simulation of flow of non-newtonian liquids

A Lagrangian method for the simulation of flow of non-Newtonian liquids is implemented. The fluid mechanical equations are formulated in the form of a variational principle, and a discretization is performed by finite elements.The method is applied to the slow of a contravariant convected Maxwell liquid around a sphere moving axially in a cylinder. The simulations show that the friction factor for a sphere in a narrow cylinder is a rapidly decreasing function of the Deborah number, while the friction factor for a sphere in a very wide cylinder is not significantly affected by fluid elasticity. It is demostrated that the simulated wall effect on the motion of the sphere may be utilized in an experimental identification of a time constant for a given liquid.     

Effect of boundaries of various shapes on vortex dynamics

The motion of a vortex near a boundary of arbitrary shape is considered within the framework of a two-dimensional problem. Integrable differential equations of motion are obtained. Two forms of the algebraic equation of the vortex trajectories are derived. Examples of vortex motion near a straight-line boundary, in a channel, in an angular domain, in the neighborhood of a flat edge, in a round basin, and near a parabolic boundary.     

Configurations of gas-liquid two-phase bubbles in immiscible liquid media

The configuration of a “two-phase bubble” constituted of a gas phase and a liquid phase in an immiscible liquid medium is classified into three types: complete engulfing of a gas bubble inside a liquid shell, partial coalescence of a gas bubble and a liquid drop forming a three-phase contact line, and non-coalescence whereby a gas bubble and a liquid drop remain separated. Simple criteria have been presented by which the favorable type of configuration in a given system is predicted from the values of the spreading coefficients characterizing the system. Experiments using some combinations of liquids as well as air suggest the general validity of the criteria.     

The extension and oscillation of a non-Hooke's law spring

We derive a wave equation for small-amplitude, undamped, extensional oscillation of a spring-mass system consisting of a mass suspended on a spring governed by a quadratic force-extension relationship. We justify this quadratic model using a Taylor series expansion of the general elasticity equations for a helical spring. Transformation of the equation of motion of the spring leads to a separable wave equation with the spacial component being a transformation of Bessel's equation. The model is successful in predicting static extension and period of oscillation of a helical wire spring for which the wave equation based on Hooke's law is inadequate.     

A geometric study of shocks in equations that change type

In this paper we validate the generalized geometric entropy criterion for admissibility of shocks in systems which change type. This condition states that a shock between a state in a hyperbolic region and one in a nonhyperbolic region is admissible if the Lax geometric entropy criterion, based on the number of characteristics entering the shock, holds, where now the real part of a complex characteristic replaces the characteristic speed itself. We test this criterion by a nonlinear inviscid perturbation. We prove that the perturbed Cauchy problem in the elliptic region has a solution for a uniform time if the data lie in a suitable class of analytic functions and show that under small perturbations of the data a perturbed shock and a perturbed solution in the hyperbolic region exist, also for a uniform time.     

Fundamentals of three-dimensional vortex motion around solid bodies

An Action Principle describing the evolution of a 3-D inviscid vorticity field around a solid body is presented. This variational principle is used for the construction of a numerical algorithm. A formal solution of the inviscid vorticity transport equation is given by means of a Lie transform using a Dirac bracket and Hamiltonian functional. This solution is used in a stochastic algorithm for the simulation of the evolution of a viscous vorticity field around a body.     

Levitation of Magnets and Paramagnetic Bodies in Vessels Filled with Magnetic Fluid

Formulas are obtained for the forces and moments acting on a spherical body made of a paramagnetic material in an uniform applied magnetic field and a magnet in a spherical vessel filled with magnetic fluid. An approximate formula is found for the force acting on bodies in ellipsoidal and cylindrical vessels or in a plane channel with a magnetic fluid in an uniform magnetic field. An analogy between the forces acting on a magnet and a paramagnetic body is demonstrated. The possibility of levitation of magnets and paramagnetic bodies in a vessel with a magnetic fluid is investigated.     

Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.     

Behavior of perturbations of solitary and periodic waves on the surface of a heavy liquid

In [1] a system of equations was obtained for the case of a potential motion of an ideal incompressible homogeneous fluid; the system described the propagation of a train of waves in a medium with slowly varying properties, the motion in the train being characterized by a wave vector and a frequency. A solitary wave is a particular case of a wave train in which the length of the waves in the train is large. In [2, 3] a quasilinear system of partial differential equations was obtained which described two-dimensional and three-dimensional motion of a solitary wave in a layer of liquid of variable depth. It follows from this system that if the unperturbed state of the liquid is the quiescent state, then some integral quantity (the average wave energy [2–4]), referred to an element of the front, is preserved during the course of the motion. This fact is also valid for a train of waves, and can be demonstrated to be so upon applying the formalism of [1] to a Lagrangian similar to that used in [2]. In the present paper we obtain, for the case of a layer of liquid of constant depth, a solution in the form of simple waves for a system, equivalent to the system obtained in [3], describing the motion of a solitary wave and also the motion of a train of waves. We show that it is possible to have tilting of simple waves, leading in the case considered here to the formation of corner points on the wave front. We consider several examples of initial perturbations, and we obtain their asymptotics as t→∞. We make our presentation for the solitary wave case; however, in view of our statement above, the results automatically carry over to the case of a train of waves.