Linearized equations of nonlinear elastic deformation of thin plates

A linearized system of equations governing elastic deformation of a thin plate with arbitrary boundary conditions at its faces in an arbitrary curvilinear coordinate system is proposed. This system of equations is the first approximation of a oneparameter sequence of equations of twodimensional problems obtained from the initial threedimensional problem by approximating unknown functions by truncated series in Legendre polynomials. The stability problem of an infinite plate compressed uniaxially is solved. The results obtained are compared with the existing solutions.     

Exact solutions and mathematical properties of boundary‐value problems for dynamic‐diffusion boundary layers

The paper studies boundaryvalue problems for dynamicdiffusion boundary layers occurring near a vertical wall at high Schmidt numbers and for dynamic boundary layers whose inner edge is adjacent to the dynamicdiffusion layers. Exact solutions for boundary layers at small and large times are derived. The wellposedness of the boundaryvalue problem for a steady dynamicdiffusion layer is studied.     

Approximate Solution of an Axisymmetric Contact Problem with Allowance for Tangential Displacements on the Contact Surface

A structurally nonlinear contact problem of a punch shaped like a paraboloid of revolution is studied. An equation for the contactpressure density is derived with allowance for the radial tangential displacements of the boundary points of an elastic halfspace. A method for constructing a closedform approximate solution is proposed. The effect of the tangential displacements on the main contact parameters is discussed.     

Analytical Modeling of Nonaqueous Phase Liquid Dissolution with Green's Functions

Equilibrium and bicontinuum nonequilibrium formulations of the advection–dispersion equation (ADE) have been widely used to describe subsurface solute transport. The Green's Function Method (GFM) is particularly attractive to solve the ADE because of its flexibility to deal with arbitrary initial and boundary conditions, and its relative simplicity to formulate solutions for multidimensional problems. The Green's functions that are presented can be used for a wide range of problems involving equilibrium and nonequilibrium transport in semiinfinite and infinite media. The GFM is applied to analytically model multidimensional transport from persistent solute sources typical of nonaqueous phase liquids (NAPLs). Specific solutions are derived for transport from a rectangular source (parallel to the flow direction) of persistent contamination using first, second, or thirdtype boundary or source input conditions. Away from the source, the first and thirdtype condition cannot be expected to represent the exact surface condition. The secondtype condition has the disadvantage that the diffusive flux from the source needs to be specified a priori. Near the source, the thirdtype condition appears most suitable to model NAPL dissolution into the medium. The solute flux from the pool, and hence the concentration in the medium, depends strongly on the mass transfer coefficient. For all conditions, the concentration profiles indicate that nonequilibrium conditions tend to reduce the maximum solute concentration and the total amount of solute that enters the porous medium from the source. On the other hand, during nonequilibrium transport the solute may spread over a larger area of the medium compared to equilibrium transport.     

Nonsimilar Solutions for Mixed Convection in Non‐Newtonian Fluids Along a Vertical Plate in a Porous Medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in powerlaw type nonNewtonian fluids along a vertical plate with powerlaw wall temperature distribution. The mixed convection regime is divided into two regions, namely,the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

Stress–strain state of a composite anisotropic plate with curvilinear cracks and thin rigid inclusions

A complexpotential solution of a mixed problem of the linear theory of elasticity is given for an infinite plate composed of two anisotropic halfplanes. The plate contains cuts and thin undeformable inclusions shaped like arbitrary open smooth curves that do not intersect each other and the interface between the halfplanes.     

Effect of the Chemical Reaction and Injection on Flow Characteristics in an Unsteady Upward Motion of an Isothermal Plate

The transient incompressible viscous fluid flow regime past a semiinfinite isothermal plate under conditions of natural convection is studied numerically. The solution obtained takes into account the firstorder homogeneous chemical reaction and the mass flux through the plate. The calculated velocity profile is in good agreement with the known exact solution. Velocity, temperature, and concentration profiles are presented. It is shown that the fluid velocity decreases with increasing chemical reaction parameter. The distributions of local and averaged values of skin friction and Nusselt and Sherwood numbers are analyzed.     

Effects of magnetic field and variable viscosity on forced non‐darcy flow about a flat plate with variable wall temperature in porous media in the presence of suction and blowing

This paper presents a study of the effect of a magnetic field and variable viscosity on steady twodimensional laminar nonDarcy forced convection flow over a flat plate with variable wall temperature in a porous medium in the presence of blowing (suction). The fluid viscosity is assumed to vary as an inverse linear function of temperature. The derived fundamental equations on the assumption of small magnetic Reynolds number are solved numerically by using the finite difference method. The effects of variable viscosity, magnetic and suction (or injection) parameters on the velocity and temperature profiles as well as on the skinfriction and heattransfer coefficients were studied. It is shown that the magnetic field increases the wall skin friction while the heattransfer rate decreases.     

On the Displacement of NAPL Lenses and Plumes in a Phreatic Aquifer

The movement of an LNAPL lens above a sloping or horizontal water table, and a DNAPL lens above an impermeable surface is discussed. The governing equations are derived, using the vertical equilibrium approach and assuming the water mobility to be much greater than that of the NAPL. Analytical solutions are obtained for onedimensional movement of a lens along a sloping water table. They describe the lens movement with the formation of a jump at the leading front (large-scale approximation), and the distribution of NAPL in the transition zone near the jump (smallscale approximation). A model,describing the movement of a lens, taking into account NAPL retention,is proposed. Approximate onedimensional solutions for the movement of a NAPL lens along sloping or horizontal surfaces under such conditions are presented for this model. Some approximate analytical solutions for twodimensional lens (plume) formation and movement are obtained for the case of a point source at a sloping surface.     

Generation of perturbations by a localized vibrator in the boundary layer of a nonswept wing

Origination and development of perturbations generated by a threedimensional vibrating surface in the boundary layer on an airfoil with a zero slip angle is experimentally studied. Surface vibrations were generated by a Mylar membrane. It is shown that highamplitude vibrations of a threedimensional surface lead to simultaneous formation of two types of perturbations in the boundary layer: quasistationary streamwise structures and wave packets accompanying them. The presence of regions with favorable and adverse pressure gradients does not exert a significant effect on evolution of streamwise structures but leads first to attenuation and then to amplification of wave packets.     

Origination of In‐Phase Oscillations of Thin Plates with Aeroelastic Interaction

Synchronization of oscillations of thin elastic plates that are walls of a gasfilled channel is considered. The gas motion is described by a system of Navier–Stokes equations, which is solved using the secondorder MacCormack method with time splitting. The motion of the channel walls is described by a system of geometrically nonlinear dynamic equations of the theory of this plates, which is solved by the finitedifference method. Kinematic and dynamic contact conditions are imposed at the interface between the media. A numerical experiment is performed to determine typical dynamic regimes and study the transition of the aeroelastic system to inphase oscillations.     

Mixed Formulations of Bending Problems for Homogeneous Elastic Plates and Beams

Mixed formulations of bending problems for homogeneous plates (beams) are proposed, whose essence is that the deformation of a plate (beam) near its fixed boundary is described by the threedimensional elasticity equations, and the remaining part by the conventional equations of plate (beam) bending. At the interface between these regions, the solutions of these equations are joined. The mixed formulation allows one to describe the threedimensional stress state in the neighborhood of the fixed boundaries of plates (beams) and take into account the complex nature of the fixing conditions. Finiteelement implementation is more efficient for the mixed formulations of plate (beam) bending problems than for the wellknown threedimensional formulations.     

Stresses in an elastic body under nonlinear antiplane deformation

The stress field in a cylindrical elastic body under antiplane deformation and certain constraints imposed on volume and surface forces is studied in a nonlinear formulation in actualstate variables. A boundaryvalue problem for independent stress components is formulated in Cartesian and complex variables, sufficient ellipticity conditions for this problem are indicated, and constraints on surface loading are imposed. Analytical solutions are given for linear and weak nonlinear elastic potentials. Similarity to a plane subsonic idealgas flow is established. An approximate method for the solution of the problem is developed.     

First Fundamental Axisymmetric Problem of Thermoelasticity for a Compressed Spheroid with a Concentric Spherical Cavity

An axisymmetric boundaryvalue problem of thermoelasticity for a compressed spheroid with a concentric spherical cavity is studied by the generalized Fourier method. The problem is reduced to an infinite system of linear algebraic equations with the Fredholm operator under the condition that the boundary surfaces are not crossed. Results of a numerical analysis of stresses in the case of loadfree boundary surfaces in the presence of a temperature field caused by a constant temperature distribution on the boundary surfaces are presented.     

Unsteady Radiative–Convective Heat Transfer in a Flow Emitting‐Absorbing and Scattering Medium around an Ablating Plate

A conjugate problem of radiative–convective heat transfer in a turbulent hightemperature gasdisperse flow around a thermally thin ablating plate is considered. The plate experiences intense radiative heating by an external source, which is a blackbody. The temperature fields and the distributions of heat fluxes along the plate under unsteady conditions are calculated. The data gained make it possible to examine the effect of the Stark number and phasetransition heat in the plate material on the time evolution of the thermal state of the boundarylayer medium and the plate itself being heated by a hightemperature radiation source.     

Hydrodynamic calculations of layered seams on the basis of modified relative permeabilities

The possibility of reducing the dimensionality of the problem of twophase filtration in layered seams by introducing modified phase permeabilities instead of initial relative permeabilities, which are coefficients of the initial system of equations within the framework of the Barkley–Leverett model, is studied. Modified permeabilities are proposed for the case where the relative permeabilities of each phase are represented by analytical dependences for individual interlayers. Onedimensional numerical calculations with these permeabilities are in good agreement with the solution of the twodimensional problem.     

Self‐Equilibrated Stress Fields in a Continuous Medium

It is proved that the solutions of the static equations of a continuous medium constructed in terms of a stress function are selfequilibrated. From a mathematical point of view, these functions can be treated as the connectivity coefficients of the intrinsic geometry of the medium. It is shown that from a physical point of view, the existence of selfequilibrated stress fields is due to a nonuniform entropy distribution in the medium. As an example, for a circle in polar coordinates and a cylindrical sample, a selfequilibrated stress field and an elastic field compensating for its surface component are constructed and it is shown how to write the equation for the intrinsic geometrical characteristics.     

Thermal stability of a disturbed fluid flow

Stability of periodic solutions of a nonselfsimilar nonlinear problem is studied. The problem describes the thermal state of an axial fluid flow with continuously distributed sources of heat. The flow experiences the action of external lowamplitude perturbations changing in time in accordance with known periodic laws. The spectral problem is solved by the method of parametrix, and the critical conditions of the thermal explosion are determined in the linear approximation. Stability of the periodic solution at the critical point is evaluated using the known theorem of factorization, which takes into account the effect of nonlinear terms of the heatbalance equation. The calculation results show that the periodic solution is stable if the total action of external periodic perturbations at the critical point is directed to reduction of the fluidflow temperature.     

Velocity Measurements in a One‐Dimensional Steady Flow by Methods of Emission Tomography

A new method for velocityfield measurements in a onedimensional steady flow is proposed. The method is based on principles of laserinduced fluorescence combined with emission tomography. Results of a numerical experiment are presented.     

On the Propagation of Long‐Wave Perturbations in a Two‐Layer Free‐Boundary Rotational Fluid

A mathematical model for the propagation of longwave perturbations in a freeboundary shear flow of an ideal stratified twolayer fluid is considered. The characteristic equation defining the velocity of perturbation propagation in the fluid is obtained and studied. The necessary hyperbolicity conditions for the equations of motion are formulated for flows with a monotonic velocity profile over depth, and the characteristic form of the system is calculated. It is shown that the problem of deriving the sufficient hyperbolicity conditions is equivalent to solving a system of singular integral equations. The limiting cases of weak and strong stratification are studied. For these models, the necessary and sufficient hyperbolicity conditions are formulated, and the equations of motion are reduced to the Riemann integral invariants conserved along the characteristics.