Asymptotic analysis of the equations of a three-dimensional boundary layer

The asymptotic solutions of the self-similar equations of two- and three-dimensional boundary layers have been investigated by many authors (see, for example, [1–3]). In [4, 5], asymptotic solutions were found for non-self-similar equations for two-dimensional flow, and the propagation of perturbations near the external edge of the boundary layer was analyzed. In the present paper, asymptotic solutions are obtained for the non-self-similar equations of a three-dimensional laminar boundary layer of an incompressible fluid. It is shown that the conclusion drawn in [5] — that the boundary conditions can be transferred from infinity to a finite distance from the wall — is also true for three-dimensional flow. The obtained solutions explain the experimentally well-known phenomenon of the conservativeness of the secondary currents. The essence of this phenomenon is that a change in the sign of the transverse (along the normal to a streamline of the external flow) pressure gradient is accompanied by a very rapid change in the direction of the secondary flow near the wall, whereas in the upper layers of the boundary layer the direction remains unchanged for a substantial time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 155–157, September–October, 1979.     

A continuum model for size-dependent deformation of elastic films of nano-scale thickness

Ultra-thin elastic films of nano-scale thickness with an arbitrary geometry and edge boundary conditions are analyzed. An analytical model is proposed to study the size-dependent mechanical response of the film based on continuum surface elasticity. By using the transfer-matrix method along with an asymptotic expansion technique of small parameter, closed-form solutions for the mechanical field in the film is presented in terms of the displacements on the mid-plane. The asymptotic expansion terminates after a few terms and exact solutions are obtained. The mid-plane displacements are governed by three two-dimensional equations, and the associated edge boundary conditions can be prescribed on average. Solving the two-dimensional boundary value problem yields the three-dimensional response of the film. The solution is exact throughout the interior of the film with the exception of a thin boundary layer having an order of thickness as the film in accordance with the Saint-Venant’s principle.     

The distributions of the interlaminar stress in a laminated shell with free end boundary

This paper studies the free edge effect yielded by interlaminar stress in a laminated cylindrical shell made up of fiber reinforced layer [0°], [90°] and the isotropic material layer under axisymmetric thermal load or radial pressure. Both ends of the shell are in free boundary condition. The exact solution of the problem can be obtained by using the three-dimensional theory of elasticity. For illustration, the numerical laminar stresses in a double-layer laminated shell under thermal load or radial pressure are calculated.     

Propagation of acoustic waves and shock waves radiated by a sinusoidal pulsation of a sphere during a single period

A spherical sound wave is emitted by a sphere which executes a small sinusoidal pulsation of a single period at high frequency in an inviscid fluid. Nonlinear propagation of the waves is formulated as an initial boundary value problem and is analysed in detail. The governing equation is linear near the sphere, while it is a nonlinear hyperbolic equation in a far field. The nonlinearity has a significant effect there, leading to the formation of two shocks. The exact solution to match the near field solution can easily be obtained for the far field equation. The nonlinear distortion of waveform and the shock formation distance are evaluated from the representation of the solution with strained coordinates. The evolution and nonlinear attenuation of the two shock discontinuities are also examined by making use of the equal-areas rule. In its asymptotic form the entire profile is an N wave with a long tail.     

Contact strength of a two-layer covering under friction forces in the contact region

Friction and antifriction composite materials of multilayer structure [1] have recently become very popular in the engineering industry. Antifriction materials are widely used in sliding bearings, and friction materials are widely used in brakes. In the first case, the friction forces between the contacting surfaces are negligible, but in the second case, they are rather large. We use two examples of two plane problems from the theory of elasticity concerning the interaction between a die and a base formed by two elastic layers with different mechanical properties, which are rigidly connected with each other and with an undeformable support, to study how the geometric and mechanical parameters of the problem affect the stress-strain state of such a base, both on its surface and at its internal points, and to find the optimal parameters ensuring the required operation resources of the friction units thus modeled. We assume that the die foot is parabola-shaped or plane, the normal and tangential stresses in the contact region are related to each other by the Coulomb law, and the die is subjected to normal and tangential forces. In this case, the die-two-layer base is in the limit equilibtrium, and the die does not rotate in the process of deformation of the layer. In this setting, the problems were studied in [2] by solving the integral equations (IE) by the asymptotic method of large λ (see [3–7], etc.), which permits finding the effective solution only for relatively large thicknesses of layers compared with the dimensions of the contact region. But in real friction units mentioned above, the layers can have rather small relative thicknesses, and the large λ method cannot be used. We note that the other asymptotic methods (e.g., see [3]) efficient in the case of relatively small thicknesses of layers cannot yet be adapted to the case of friction forces in the contact region. In the present paper, we propose to use the collocation method following the scheme given in [8] to solve the corresponding integral equation of the first kind with logarithmic kernel. This method allows one to obtain sufficiently exact solutions practically for all values of the parameters of the problem with relatively small expenditure of the computer time for modern computers. The contact problem for a two-layer base was used in [9] for a close statement of the problem without friction forces in the contact region.     

To the theory of analysis of a two-layer cylindrical bearing

We consider the plane contact problem of elasticity concerning the interaction between an absolutely rigid cylinder and the internal cylindrical surface of the cylindrical base, which consists of two circular cylindrical layers with different elastic constants. The base external surface is fixed, the layers are rigidly connected with each other, and the friction forces are absent in the contact region. Such problems sufficiently well model the operation of a composite cylindrical slider bearing, especially in the case of loads for which the angular dimension of the contact site is commensurable with the bearing width and the moduli of the insert liner and of the support are different and significantly less than the modulus of elasticity of the other details of the bearing.For the above-stated problem of elasticity, we first construct integral equations, which are solved by the direct collocation method [1, 2] and by the asymptotic method [3, 4].In contrast to the similar problems considered earlier (e.g., see [3, 4]) for a single-layer cylinder, the collocation method used here permits studying the problem practically for any parameter values. The asymptotic approach gives an efficient solution in the case of relatively thin layers in simple analytic form. We also compare the two solutions numerically and determine the scope of the asymptotic method.     

Multi-directional and multi-time step absorbing layer for unbounded domain

Variant techniques are proposed for reproducing the elastic wave propagation in an unbounded medium such as the infinite elements, the absorbing boundary conditions or the perfect matched layers. Here, a simplified approach is adopted by considering absorbing layers characterized by the viscous Rayleigh matrix as studied by Semblat et al. [16] and Rajagopal et al. [14]. Here, further improvements to this procedure are provided. First, we start by establishing the strong form for the elastic wave propagation in a medium characterized by the Rayleigh matrix. This strong form will be used for deriving optimal conditions for damping out in the most efficient way the incident waves while minimizing the spurious reflected waves at the interface between the domain of interest and the Rayleigh damping layer. A procedure for designing the absorbing layer is proposed by targeting a performance criterion expressed in terms of logarithmic decrement of the wave amplitude in the layer thickness. Second, the GC subdomain coupling method, proposed by Combescure and Gravouil [9], is introduced for enabling the choice of any Newmark time integration schemes associated with different time steps depending on subdomains. When wave propagation is predicted by an explicit time integrator, the subdomain strategy is of great interest because it enables a different time integrator for the absorbing layer to be adopted. An external coupling software, based on the GC method, is used to carry out multi=time step explicit/implicit co-computations, making interact in time an explicit FE code (Europlexus) for the domain of interest, with an implicit FE code (Cast3m) handling the absorbing boundary layers. The efficiency of the approach is shown in 1D and 2D elastic wave propagation problems.     

An Existence Theorem for Nonlinearly Elastic ‘Flexural’ Shells

The two-dimensional equations of a nonlinearly elastic ‘flexural’ shell have been recently identified and justified by V. Lods and B. Miara, by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. These equations can be recast as a minimization problem for a ‘two-dimensional energy’ over a manifold of ‘admissible deformations’. The stored energy function is a quadratic expression in terms of the exact difference between the curvature tensor of the deformed middle surface and that of the undeformed one; the admissible deformations are those that preserve the metric of the undeformed middle surface and satisfy boundary conditions of clamping or of simple support. We establish here that this minimization problem has at least one solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Propagation of quasishear waves in prestressed materials with unbonded layers

The propagation of quasi-shear waves along the layers of a prestressed composite material is described using the three-dimensional linearized theory of elasticity. The composite consists of two alternating layers that are free to slip relative each other. The dispersion equation is analyzed numerically. The effect of the prestresses on the wave velocity is studied     

Interaction between the boundary layer and a supersonic flow when part of the wall is rapidly heated

There have been many theoretical studies of aspects of the unsteady interaction of an exterior inviscid flow with a boundary layer [1–9]. The mathematical flow models obtained in these studies by the method of matched asymptotic expansions describe a wide range of phenomena observed experimentally. These include boundary layer separation near the hinge of a flap, the flow in the neighborhood of the trailing edge of an oscillating airfoil [1–2], and the development and propagation of perturbations in a boundary layer excited by an oscillating wall or some other way [3–5]. The present paper studies the interaction of an unsteady boundary layer with a supersonic flow when a small part of the surface of a body in the flow is rapidly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–70, January–February, 1984.     

3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid

The free vibration of an arbitrarily thick orthotropic piezoelectric hollow cylinder with a functionally graded property along the thickness direction and filled with a non-viscous compressible fluid medium is investigated. The analysis is directly based on the three-dimensional exact equations of piezoelasticity using the so-called state space formulations. The original functionally graded shell is approximated by a laminate model, of which the solution will gradually approach the exact one when the number of layers increases. The effect of internal fluid can be taken into consideration by imposing a relation between the fluid pressure and the radial displacement at the interface. Analytical frequency equations are derived for different electrical boundary conditions at two cylindrical surfaces. As particular cases, free vibration of multi-layered piezoelectric hollow cylinder and wave propagation in infinite homogeneous cylinder are studied. Numerical comparison with available results is made and dispersion curves predicted from the present three-dimensional analysis are given. Numerical examples are further performed to investigate the effects of various parameters on the natural frequencies.     

A general vibration model of angle-ply laminated plates that accounts for the continuity of interlaminar stresses

This paper presents the generalisation of a well documented two-dimensional shear deformable laminated shell theory [Compos. Struct. 25 (1993) 165] that, based on a fixed number of unknown variables, was initially proposed for laminates made of specially orthotropic layers only. The theory is here specialised for laminated plates but is able to encompass monoclinic layers in a general multilayered configuration. Moreover, it is able to account for the interlaminar continuity of both displacements and transverse shear stresses. Higher-order effects, as shear deformation and rotary inertia, are naturally included into the formulation. In order to obtain the relevant governing differential equations, both Hamilton's variational principle and a recently proposed vectorial approach [Compos. Engng. 3 (1993) 3] have been independently used. The effectiveness of the present model is tested numerically by comparing its results with exact three-dimensional elasticity results obtained under the particular condition that the plates vibrate in cylindrical bending.     

Contact problems for a two-layer elastic wedge

We obtain integral equations for plane contact problems for a two-layer wedge (composite) under three types of boundary conditions on one of its sides (absence of stresses, sliding, or rigid fixation). The composite consists of two wedges completely linked with each other, which have different opening angles and elasticity parameters. Using the symbols (Mellin transforms) of the kernels of integral equations for the two-layer wedge, one can derive the symbols of the kernels of integral equations for symmetric problems about a crack in a three-layer wedge or a three-layer strip and for contact problems for a two-layer strip (by passing to the limit in a special way). The complex zeros of the Mellin transform determine the asymptotics of the normal contact pressure at the corner point of the composite as the contact region approaches this point. It is important that this asymptotics is also preserved in three-dimensional contact problems as the die enters the edge of a two-layer wedge (outside the corner points of the die itself). Taking into account this asymptotics, we obtain solutions of the contact problems as the die enters the vertex of the composite. We show that by appropriately choosing the materials and the internal angle of the two-layer wedge one can avoid contact pressure oscillations at the vertex, which occur in the case of a homogeneous wedge and result in loss of contact. The contact pressure at the wedge vertex can be made zero for a composite, while for a homogeneous wedge with the same opening angle it increases unboundedly. We construct asymptotic solutions of the contact problems for a plane die located relatively close or to the vertex of a two-layer wedge or relatively far from the vertex. The asymptotic and other methods were earlier used to solve similar plane contact problems for a homogeneous wedge [1, 2]. In the case of sliding fixation of one of the sides of a plane homogeneous wedge, the closed solution of the contact problem is known for a die entering the corner point [3, p. 131]. Two-dimensional contact problems were studied for a truncated wedge [4] and for a wedge supported by a rod of equal resistance [5]. The out-of-plane shear vibrations were studied for wedge-shaped composites [6, 7]. The spatial contact problems were considered for a homogeneous wedge [8]. The plane contact problem was analyzed for a continuously inhomogeneous wedge one of whose sides was rigidly fixed (the shear modulus continuously depends on the angular coordinate and the Poisson coefficient is constant). For a two-layer composite, which is studied in the present paper, the kernel symbol has different asymptotic properties, which are used in asymptotic methods for solving the problem. A similar distinction of the symbol properties takes place in contact problems for a continuously inhomogeneous layer and a layered packet.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case.Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].     

On the stability of a plate under longitudinal compression on a two-layer half-space with lower layer prestressed by gravity forces

In the plane (plane strain) and axially symmetric statements, we study the problem of stability, under the action of longitudinal compressing forces, of an infinite elastic plate in two-sided contact with an elastic half-space. The upper layer of finite depth is described by the usual equations of linear theory of elasticity; the lower layer, which is geometrically nonlinear, incompressible, and infinite in depth, is prestressed by gravity forces. The total adhesion between the layer of finite depth and the lower half-space is realized. It is also assumed that the same adhesion takes place between the upper layer of the half-space and the plate with the contact tangential stresses taken into account.The results can be used to calculate the working capacity of coated bodies and layered composites and in problems of geophysics.The problem of stability of an infinite elastic plate under longitudinal compression under conditions of two-sided contact with an elastic base was studied earlier in the monograph [1] (Fuss-Winkler base) and in [2–4].     

Experimental investigation of three-dimensional supersonic flow of a gas in the interference region of intersecting surfaces

A contemporary high-speed aircraft represents a complex three-dimensional configuration, where supersonic gas flow is accompanied by numerous local flow interaction zones, in particular, near the intersection of different surfaces. Such a flow is characterized by three-dimensional systems of shock and expansion waves, and close to the surfaces one finds interaction of boundary layers and, above all, interaction of shock waves with the boundary layer. In general, the angular configurations are formed by intersection or contact of nonplanar surfaces with swept-back or blunted leading edges. This makes it practically impossible to obtain a rigorous theoretiical solution to the problem of gas flow over these surfaces, and presents considerable difficulty in an experimental investigation. It is therefore of interest to study the physical features of gas flow in corner configurations of very simple form [1–3]. The present paper examines the results of an experimental investigation of typical features of symmetric and asymmetric interaction of compressive, expansive, and mixed flows in the interference region of planar surfaces intersecting at an angle of less than 180?.     

Theory of short waves in gas dynamics

An examination is made of the two-dimensional, almost stationary flow of an ideal gas with small but clear variations in its parameters. Such gas motion is described by a system of two quasilinear equations of mixed type for the radial and tangential velocity components [1, 2]. Partial solutions [3, 4], characterizing the variation in the gas parameters in the vicinity of the shock wave front (in the short-wave region), are known for this system of equations. The motion of the initial discontinuity of the short waves derived from the velocity components with respect to polar angle and their damping are studied in the report. A solution of the equations characterizing the arrangement of the initial discontinuity derived from the velocities is presented for one particular case of the class of exact solutions of the two parameter type [4]. Functions are obtained which express the nature of the variation in velocity of the front of the damped wave and its curvature.Translation from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–58, May–June, 1973.     

Development of three-dimensional disturbances in a boundary layer

In the region of transition from a two-dimensional laminar boundary layer to a turbulent one, three-dimensional flow occurs [1–3]. It has been proposed that this flow is formed as the result of nonlinear interaction of two-dimensional and three-dimensional disturbances predicted by linear hydrodynamic stability theory. Using many simplifications, [4, 5] performed a calculation of this interaction for a free boundary layer and a boundary layer on a wall with a very coarse approximation of the velocity profile. The results showed some argreement with experiment. On the other hand, it is known that disturbances of the Tollmin—Schlichting wave type can be observed at sufficiently high amplitude. This present study will use the method of successive linearization to calculate the primary two- and three-dimensional disturbances, and also the average secondary flow occurring because of nonlinear interaction of the primary disturbances. The method of calculation used is close to that of [4, 5], the disturbance parameters being calculated on the basis of a Blazius velocity profile. A detailed comparison of results with experimental data [1] is made. It developed that at large disturbance amplitude the amplitude growth rate differs from that of linear theory, while the spatial distribution of disturbances agree s well with the distribution given by the natural functions and their nonlinear interaction. In calculating the secondary flow an experimental correction was made to the amplitude growth rate.     

Exact Solutions for a Thick Elastic Plate with a Thin Elastic Surface Layer

A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate. The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material. In this paper we consider a thick plate of isotropic elastic material with a thin surface layer of different isotropic elastic material. It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory. Two illustrative examples are described.