Nonlinear periodic waves in a gas

A system of equations describing the one-dimensional time-dependent polytropic motion of a gas is considered. In special cases the general solutions of this system of equations are obtained and exact solutions with the initial conditions which are periodic with respect to the spatial variable are found. For an arbitrary polytropic exponent an asymptotic solution, which is uniformly suitable till the onset of the gradient catastrophe, is constructed in the form of expansions in series in a small parameter, namely, the initial wave amplitude. Asymptotic dependences of the time of onset and the location of the gradient catastrophe are obtained. The complex correspondence between the initial system of equations and the system of equations describing the motion of quasi-gas media is given. An example of using this correspondence is considered.     

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.     

Edge effects in a sandwich plate: a plane problem

Edge effects in a rectangular sandwich plate with isotropic components are studied. The mathematical model is represented by the homogeneous equations of linear elasticity, which is indicative of an approximate approach in edge-effect theory. The initial equations are reduced to inhomogeneous ones and an exact problem is formulated. Approximate solutions are found by the mesh method. Discrete problems are based on the concept of base scheme. The mesh equations are written in an explicit form and then solved using a computation optimization procedure. As an example, edge-effect zones in a real composite are analyzed.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 124–133, December 2004.     

Free Convection in a Square Cavity Filled with a Tridisperse Porous Medium

This work is dealing with the natural convection heat transfer in a square filled with porous medium that has been extended according to the Nield and Kuznetsov model to tridisperse porous medium. Considering impermeable walls which the horizontal ones are insulated and vertical ones are assumed to be isothermal, the governing equations are set as the three equations for momentum and three equations for energy for three phases of porosity and are numerically solved utilizing finite element method. In this study isothermal contours, streamlines and Nusselt number values are foremost criteria which are presented for three levels of porosity. The influence of various governing parameters on the heat transfer is investigated.     

Stress intensity factors for a moving cylindrical crack in a nonhomogeneous cylindrical layer in composite materials

Stresses are determined for a finite cylindrical crack that is propagating with a constant velocity in a nonhomogeneous cylindrical elastic layer, sandwiched between an infinite elastic medium and a circular elastic cylinder made from another material. The Galilean transformation is employed to express the wave equations in terms of coordinates that are attached to the moving crack. An internal gas pressure is then applied to the crack surfaces. The solution is derived by dividing the nonhomogeneous interfacial layer into several homogeneous cylindrical layers with different material properties. The boundary conditions are reduced to two pairs of dual integral equations. These equations are solved by expanding the differences in the crack surface displacements into a series of functions that are equal to zero outside the crack. The Schmidt method is then used to solve for the unknown coefficients in the series. Numerical calculations for the stress intensity factors were performed for speeds and composite material combinations.     

Free Convection Heat Transfer over a Vertical Cylinder in a Saturated Porous Medium Using a Local Thermal Non-equilibrium Model

In this article, free convection heat transfer over a vertical cylinder with variable surface temperature distributions in a porous medium is analyzed. It is assumed that the fluid and solid phases are not in local thermal equilibrium and, therefore, a two-temperature model of heat transfer is applied. The coupled momentum and energy equations are presented and then they are transformed into ordinary differential equations. The similarity equations are solved numerically. The resulting velocity, streamlines, temperature distributions for fluid and solid phases are shown for different values of parameters entering into the problem. The calculated values of the local Nusselt numbers for both solid and fluid phases are also shown.     

The uniqueness of the Einstein field equations in a four-dimensional space

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of g _ij and its first two derivatives are investigated. In general these equations will be of fourth order in g _ij. Necessary and sufficient conditions for these Euler-Lagrange equations to be of second order are obtained and it is shown that in a four-dimensional space the Einstein field equations (with cosmological term) are the only permissible second order Euler-Lagrange equations. This result is false in a space of higher dimension. Furthermore, the only permissible third order equation in the four-dimensional case is exhibited.     

Vortex motions of liquid in a narrow channel

Quasi-linear integrodifferential equations that describe vortex flows of an ideal incomparessible liquid in a narrow curved channel in the Eulerian-Lagrangian coordinate system are considered. The necessary and sufficient conditions for hyperbolicity of the system of equations of motion are obtained for flows with a monotonic velocity depth profile. The propagation velocities of the characteristics and the characteristic form of the system are calculated. A particular solution is given in which the system of integrodifferential equations changes type with time. The solution of the Cauchy problem is given for linearized equations. An example of initial data for which the Cauchy problem is ill-posed is constructed. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 38–49, July–August, 1998.     

Film boiling on a hot body moving in a fluid

We investigate the transient film boiling in the vicinity of a stagnation point on the frontal surface of a very hot blunt body which moves with a constant velocity in an incompressible viscous fluid in the presence of a vapor layer near the body surface. Within the unsteady two-phase boundary layer approximation, the equations of motion of the liquid and vapor phases are formulatedwith account of the conservation of mass, momentum, and energy on the a priori unknown phase interface. In the vicinity of the stagnation point on the body surface, the solution of the boundary layer equations is sought in the form of series in the longitudinal coordinate. For the leading terms of the series, a parabolic system of partial differential equations is obtained, which is solved numerically. The similarity parameters controlling the film boiling process are determined. On the basis of parametric numerical calculations, the dynamics of the vapor layer are investigated for the case of a plane hot body moving in water with the room pressure and temperature. In the space of governing parameters, the limits of the existence of steady and unsteady film boiling regimes are found.     

Screw flow of a liquid in a cylindrical tube

A study is made of the flow of a viscous incompressible liquid with helical streamlines in an infinite cylindrical tube within which a screw rotates (auger). Generalized linearized Oseen equations are derived, and one class of the exact solutions of these is identical with the corresponding class of exact solutions of the complete Navier-Stokes equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 6, pp. 3–7, November–December, 1979.     

Surface Waves in a Stratified Periodic Medium with a Liquid Upper Layer

An approach is proposed to set up the dispersion equations for surface waves in a periodically stratified half-space contacting with a layer of a perfect compressible liquid. The approach is based on the formalism of periodic Hamiltonian systems. The dispersion equations derived are valid for an arbitrary law of variation in the properties with respect to the coordinate of periodicity. The effects of the liquid layer and the inhomogeneity of the elastic medium on the dispersion spectra of surface waves are studied     

SIMILARITY SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A SPECIAL NON-NEWTONIAN FLUID IN A SPECIAL COORDINATE SYSTME

Two dimensional equations of steady motion for third order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the inviscid flow around an arbitrary object, the streamlines are the phicoordinates and velocity potential lines are psi-coordinates which form an orthogonal curvilinear set of coordinates. The outcome, boundary layer equations, is then shown to be independent of the body shape immersed into the flow. As a first approximation, assumption that second grade terms are negligible compared to viscous and third grade terms. Second grade terms spoil scaling transformation which is only transformation leading to similarity solutions for third grade fluid. By ~sing Lie group methods, infinitesimal generators of boundary layer equations are calculated. The equations are transformed into an ordinary differential system. Numerical solutions of outcoming nonlinear differential equations are found by using combination of a Runge-Kutta algorithm and shooting technique.     

Chemically reactive hydromagnetic flow of a second-grade fluid in a semi-porous channel

An analysis of a second-grade fluid in a semi-porous channel in the presence of a chemical reaction is carried out to study the effects of mass transfer and magnetohydrodynamics. The upper wall of the channel is porous, while the lower wall is impermeable. The basic governing flow equations are transformed into a set of nonlinear ordinary differential equations by means of a similarity transformation. An approximate analytical solution of nonlinear differential equations is constructed by using the homotopy analysis method. The features of the flow and concentration fields are analyzed for various problem parameters. Numerical values of the skin friction coefficient and the rate of mass transfer at the wall are found.     

An embedded crack in a functionally graded orthotropic coating bonded to a homogeneous substrate under a frictional Hertzian contact

In this paper, we consider the elasto-static problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subject to statically applied normal and tangential surface loading. The crack direction is parallel to the free surface. The coating is graded in the thickness direction and is orthogonal to the crack direction. This coating is modelled as a non-homogeneous medium with an orthotropic stress–strain law. The equivalent crack surface stresses are first obtained and substituted in the plane elasticity equations. Using integral transforms, the governing equations are converted into singular integral equations which are solved numerically to yield the displacement field as well as the crack-tip stress intensity factors. This study presents a complete theoretical formulation for the problem in the static case. A numerical predictive capability for solving the singular integral equations and computing the crack-tip stress intensity factors is proposed. Since the loading is compressive, a previously developed crack-closure algorithm is applied to avoid interpenetration of the crack faces. The main objective of the paper is to investigate the effects of the material orthotropy and non-homogeneity of the graded coating on the crack-tip stress intensity factors, with and without using the crack-closure algorithm, for the purpose of gaining better understanding on the behavior and design of graded coatings.     

Incremental elastic motions superimposed on a finite deformation in the presence of an electromagnetic field

The equations describing the interaction of an electromagnetic sensitive elastic solid with electric and magnetic fields under finite deformations are summarized, both for time-independent deformations and, in the non-relativistic approximation, time-dependent motions. The equations are given in both Eulerian and Lagrangian form, and the latter are then used to derive the equations governing incremental motions and electromagnetic fields superimposed on a configuration with a known static finite deformation and time-independent electromagnetic field. As a first application the equations are specialized to the quasimagnetostatic approximation and in this context the general equations governing time-harmonic plane-wave disturbances of an initial static configuration are derived. For a prototype model of an incompressible isotropic magnetoelastic solid a specific formula for the acoustic shear wave speed is obtained, which allows results for different relative orientations of the underlying magnetic field and the direction of wave propagation to be compared. The general equations are then used to examine two-dimensional motions, and further expressions for the wave speed are obtained for a general incompressible isotropic magnetoelastic solid.     

On a supplementary conservation law for the energy equation in a rigid heat conductor

We consider a rigid heat conductor with specified constitutive equations and show that the internal energy equation may be written in the form of a symmetric and conservative hyperbolic system of first order quasi-linear equations for which the Cauchy problem is well-posed. Moreover, such a system is useful to study shocks. Several particular cases are examined.     

Free Convection Boundary Layer Flow Past a Vertical Surface in a Porous Medium with Temperature-Dependent Properties

Numerical solutions for the free convection heat transfer in a viscous fluid at a permeable surface embedded in a saturated porous medium, in the presence of viscous dissipation with temperature-dependent variable fluid properties, are obtained. The governing equations for the problem are derived using the Darcy model and the Boussinesq approximation (with nonlinear density temperature variation in the buoyancy force term). The coupled non-linearities arising from the temperature-dependent density, viscosity, thermal conductivity, and viscous dissipation are included. The partial differential equations of the model are reduced to ordinary differential equations by a similarity transformation and the resulting coupled, nonlinear ordinary differential equations are solved numerically by a second order finite difference scheme for several sets of values of the parameters. Also, asymptotic results are obtained for large values of | f _w|. Moreover, the numerical results for the velocity, the temperature, and the wall-temperature gradient are presented through graphs and tables, and are discussed. It is observed that by increasing the fluid variable viscosity parameter, one could reduce the velocity and thermal boundary layer thickness. However, quite the opposite is true with the non-linear density temperature variation parameter.     

Analytic Approximate Solution for a Flow of a Second-Grade Viscoelastic Fluid in a Converging Porous Channel

The problem of a two-dimensional steady flow of a second-grade fluid in a converging porous channel is considered. It is assumed that the fluid is injected into the channel through one wall and sucked from the channel through the other wall at the same velocity, which is inversely proportional to the distance along the wall from the channel origin. The equations governing the flow are reduced to ordinary differential equations. The boundary-value problem described by the latter equations is solved by the homotopy perturbation method. The effects of the Reynolds and crossflow Reynolds number on the flow characteristics are examined.     

Development of lattice turbulence in a stream with a constant velocity gradient

On the basis of the equations for the Reynolds stresses and the equation for the scale of the turbulence, an analysis is made of the development of lattice turbulence in a stream with a constant velocity gradient. The constants in the equations are determined under the assumption that, far from the lattice and with large Reynolds numbers, the structure of the turbulence tends toward a limiting state with constant values of the correlation coefficient, the degree of anisotropy, and the dimensionless velocity gradient. The constants in terms containing the viscosity are determined from a consideration of the flow beyond the lattice without a velocity gradient in the final stage of decay of the turbulence. The equations obtained were solved in an electronic computer. The calculation is in satisfactory agreement with the existing experimental data. For calculating flows with a variable velocity gradient, instead of the equation of the scale, it is proposed to use an equation for the frequency of the turbulent pulsations obtained in the present work. The computer calculations were made by S. I. Bekritskaya.     

Rolling of a Rigid Cylinder Over a Strip with Initial Stresses

An asymmetric quasistationary problem for a strip with initial stresses is formulated and solved. Resolving differential equations are derived and then reduced to five functional equations by the Hankel integral transform. Two of these equations are reduced to the Fredholm integral equations of the second kind, which are solved by the method of degenerated kernels. The normal and tangential stresses are plotted versus the elongation.