Wave motion of an ideal fluid in a narrow open channel

This paper considers a nonlinear integrodifferential model constructed for the motion of an ideal incompressible fluid in an open channel of variable section using the long-wave approximation. A characteristic equation for describing the perturbation propagation velocity in the fluid is derived. Necessary and sufficient conditions of generalized hyperbolicity for the equations of motion are formulated, and the characteristic form of the system is calculated. In the case of a channel of constant width, the model reduces to the Riemann integral invariants which are conserved along the characteristics. It is found that, during the evolution of the flow, the type of the equations of motion can change, which corresponds to long-wave instability for a certain velocity distribution along the channel width. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 61–71, March–April, 2009.     

Incremental deformation model for a rod

A nonlinear deformation model for a rod with rigid cross sections is proposed. A complete system of local incremental equations, a variational equation equivalent to this system, and an equation of virtual work are formulated. Numerical analysis of the deformation of a ring transmission is performed. Institute of Computer Modeling, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 229–235, July–August, 1999.     

Nonstationary interaction of a short blunt body with a cavity in a compressible liquid

The shock-interaction problem for a rigid spherical body and a spherical cavity in a compressible liquid is formulated and solved. Three typical cases of typical dimensions of the body and cavity are examined. An asymptotic solution valid at the earliest stage of interaction is obtained. In the general case, the problem is reduced to an infinite system of integral equations of the second kind. It is numerically solved for the case of a nonsmall air gap __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 40–56, November 2006.     

Henry’ Law and Gas Phase Disappearance

In a two-phase (liquid–gas) two-component (water–hydrogen) system we discuss the formulation of the possible dissolution of hydrogen in the liquid phase. We show how Henry’s law fits in a phase diagram and the problem is formulated as a set of nonlinear partial differential equations with complementarity constraints.     

Incremental deformation model for a shell

A nonlinear deformation model for a shell with rigid transverse fibers is proposed. A complete system of incremental equations, a variational equation equivalent to this system, and a particular equation of virtual work are formulated. Numerical analysis of the nonlinear deformation of a spherical dome is performed using the complete equation. Institute of Computer Modeling, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 202–207, September–October, 1999.     

Discrete-continuum model of symmetric separation of a material

A problem of the beginning of motion of a finite-width cut in a linearly elastic plane under the action of symmetric external loading is formulated. The material on the way of cut propagation forms a layer (interaction layer). The stress-strain state of the material is postulated to be homogeneous across this layer. A system of integral boundary equations is obtained for determining the stress-strain state. Based on this system of equations, a discrete model of separation of the layer material is constructed under the assumption of a constant stress-strain state in an element of the interaction layer. The stress distribution in the pre-fracture zone is determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 134–140, January–February, 2009.     

Rheological characterization and constitutive modeling of bread dough

Bread dough (a flour–water system) has been rheologically characterized using a parallel-plate, an extensional, and a capillary rheometer at room temperature. Based on the linear and nonlinear viscoelastic and viscoplastic data, two constitutive equations have been applied, namely a viscoplastic Herschel–Bulkley model and a viscoelastoplastic K–BKZ model with a yield stress. For cases where time effects are unimportant, the viscoplastic Herschel–Bulkley model can be used. For cases where transient effects are important, it is more appropriate to use the K-BKZ model with the addition of a yield stress. Finally, the wall slip behavior of dough was studied in capillary flow, and an appropriate slip law was formulated. These models characterize the rheological behavior of bread dough and constitute the basic ingredients for flow simulation of dough processing, such as extrusion, calendering, and rolling.     

Equilibrium of an elastic body pierced by horizontal thin elastic bars

A simple mathematical model of a structure consisting of a three-dimensional body and rigid carrying bars is proposed. The estimated characteristics are the deflections of the bars, their reactions averaged over the sections, and the subsidence parameters of the body. The problem formulated on the basis of asymptotic analysis comprises the bending equations of the bars, the equations of equilibrium of the body, and a relation between the reactions and the deflections of the bars. In this problem, in addition to the moment of inertia, another cross-sectional characteristic, namely, the outer conformal radius, is involved. The method of solving the problem and the ways of its generalization are discussed. Makarov State Marine Academy, St. Petersburg 199026. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 236–242, July–August, 1999.     

Positive Solutions of Linear Impulsive Differential Equations

The paper deals with the existence of positive (nonnegative) solutions of linear homogeneous impulsive differential equations. The main result is also applied to the investigation of a similar problem for higher-order linear homogeneous impulsive differential equations. All results are formulated in terms of coefficients of the equations. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 291–297, July–September, 2005.     

Control of the spatial motions of a gantry manipulator with elastic links

A mathematical dynamic model is proposed for a controlled gantry robot with elastic compliance and inertia distributed along a two-link arm. The model includes a nonlinear system of hybrid differential equations. Kinematic and dynamic control problems for the robot are formulated. The dynamic characteristics of the robot are analyzed in comparison with an equivalent model of a robot manipulator with rigid links based on the Lagrangian formalism __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 121–128, February 2006.     

Shallow waves in a two-layer vortex fluid under a lid

A mathematical model of the vortex motion of an ideal two-layer fluid in a narrow straight channel is considered. The fluid motion in the Eulerian-Lagrangian coordinate system is described by quasilinear integrodifferential equations. Transformations of a set of the equations of motion which make it possible to apply the general method of studying integrodifferential equations of shallow-water theory, which is based on the generalization of the concepts of characteristics and the hyperbolicity for systems with operator functionals, are found. A characteristic equation is derived and analyzed. The necessary hyperbolicity conditions for a set of equations of motion of flows with a monotone-in-depth velocity profile are formulated. It is shown that the problem of sufficient hyperbolicity conditions is equivalent to the solution of a certain singular integral equation. In addition, the case of a strong jump in density (a heavy fluid in the lower layer and a quite lightweight fluid in the upper layer) is considered. A modeling that results in simplification of the system of equations of motion with its physical meaning preserved is carried out. For this system, the necessary and sufficient hyperbolicity conditions are given. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 68–80, May–June, 1999.     

Surface instability of orthotropic laminated materials under compression

The piecewise-homogeneous material model and the three-dimensional linearized theory of stability with the assumption of small subcritical strains are used to study the surface buckling of orthotropic and transtropic laminates. A plane problem is formulated, and characteristic equations are derived. A solution is found for a specific transtropic material with different orientations of the isotropy axis __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 64–72, July 2006.     

Three-dimensional flows of thin layers of viscoplastic media in deformable channels and cavities

The problem of the three-dimensional flow of a viscoplastic medium in a thin layer bounded by two deformable material surfaces whose points are displaced transversely across the layer is formulated and solved. Expressions for the velocity components and the flow-rate and equations for calculating the pressure and the boundary of the shear flow region are obtained. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 201–205, July–August, 1998.     

On the instability of steady motion

This paper deals with the instability of steady motions of conservative mechanical systems with cyclic coordinates. The following are applied: Kozlov’s generalization of the first Lyapunov’s method, as well as Rout’s method of ignoration of cyclic coordinates. Having obtained through analysis the Maclaurin’s series for the coefficients of the metric tensor, a theorem on instability is formulated which, together with the theorem formulated in Furta (J. Appl. Math. Mech. 50(6):938–944, 1986), contributes to solving the problem of inversion of the Lagrange-Dirichlet theorem for steady motions. The cases in which truncated equations involve the gyroscopic forces are solved, too. The algebraic equations resulting from Kozlov’s generalizations of the first Lyapunov’s method are formulated in a form including one variable less than was the case in existing literature.     

On the determination of loads upon the ground support in underground mine workings

The piecewise-homogeneous model and the three-dimensional linearized theory of stability for small subcritical strains are used to study the surface instability of a regularly layered rock mass under biaxial loading. A plane problem is formulated. Basic characteristic equations are derived. A specific problem is solved as an example to demonstrate the selection of loads and the interaction of support elements with the wall rock __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 38–46, November 2005.     

Variational approach to constructing hyperbolic models of two-velocity media

A generalized Hamilton variational principle of the mechanics of two-velocity media is proposed, and equations of motion for homogeneous and heterogeneous two-velocity continua are formulated. It is proved that the convexity of internal energy ensures the hyperbolicity of the one-dimensional equations of motion of such media linearized for the state of rest. In this case, the internal energy is a function of both the phase densities and the modulus of the difference in velocity between the phases. For heterogeneous media with incompressible components, it is shown that, in the case of low volumetric concentrations, the dependence of the internal energy on the modulus of relative velocity ensures the hyperbolicity of the equations of motion for any relative velocity of motion of the phases. the present location of work: Universite of Aix-Marseille III, Marseille 13397 Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 39–54, September–October, 1998.     

On a Generalized Nonlinear K–ε Model and the Use of Extended Thermodynamics in Turbulence

A resent extension of the nonlinear K–ε model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–ε model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–ε type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–ε model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed. Received 19 February 1998 and accepted 23 October 1998     

Wave processes in a perfect liquid layer impacted on by a blunted solid

The problem of impact of a smooth blunt solid upon a compressible-liquid layer of finite depth is addressed. A mixed initial-boundary-value problem with an unknown moving boundary is formulated. In the general case, the problem is reduced to an infinite system of integral Volterra equations of the second kind. It is solved numerically, using quadrature formulas and truncation method. An exact analytic solution to the problem is obtained in the special case where the body moves with a constant velocity at the initial stage of submergence. This solution makes it possible to examine the effect of successive wave reflections on the pressure at the frontal point and inside the layer __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 37–47, March 2007.     

Large Eddy Simulations Using the Subgrid-Scale Estimation Model and Truncated Navier–Stokes Dynamics

We describe a procedure for large eddy simulations of turbulence which uses the subgrid-scale estimation model and truncated Navier–Stokes dynamics. In the procedure the large eddy simulation equations are advanced in time with the subgrid-scale stress tensor calculated from the parallel solution of the truncated Navier–Stokes equations on a mesh two times smaller in each Cartesian direction than the mesh employed for a discretization of the resolved quantities. The truncated Navier–Stokes equations are solved through a sequence of runs, each initialized using the subgrid-scale estimation model. The modeling procedure is evaluated by comparing results of large eddy simulations for isotropic turbulence and turbulent channel flow with the corresponding results of experiments, theory, direct numerical simulations, and other large eddy simulations. Subsequently, simplifications of the general procedure are discussed and evaluated. In particular, it is possible to formulate the procedure entirely in terms of the truncated Navier–Stokes equation and a periodic processing of the small-scale component of its solution. Received 27 April 2001 and accepted 16 December 2001