Forced capillary-gravity wave motion of two-layer fluid in three-dimensions

A class of problems associated with forced capillary-gravity wave motion in a channel are analyzed in the presence of surface and interfacial tensions in a two-layer fluid in both the cases of finite and infinite water depths. The two and three-dimensional Green functions associated with the capillary-gravity wave problems in the presence of surface and interfacial tensions are derived using the fundamental source potentials. Using the two-dimensional Green function along with Green’s second identity, the expansion formulae for the velocity potentials associated with the capillary-gravity wavemaker problems in two-dimensions are obtained. The two-dimensional results are generalized to derive the expansion formulae for the velocity potentials associated with the forced capillary-gravity wave motion in the presence of surface and interfacial tensions in three-dimensions. Certain characteristics of the eigen-system associated with the expansion formulae are derived. The velocity potentials associated with the free oscillation of capillary-gravity waves in a closed basin and semi-infinite open channel in the presence of surface and interfacial tensions are obtained. The utility of the forced motion in a channel is demonstrated by analyzing the capillary-gravity wave reflection by a wall in a channel in the presence of surface and interfacial tensions. Long wave equations associated with capillary-gravity wave motion in the presence of surface and interfacial tensions are derived under shallow water approximation and the associated dispersion relation are obtained. Various expansion formulae and Green functions derived in the present study will be useful for analyzing a large class of physical problems in ocean engineering and mathematical physics.     

A reduction method to quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction

A reduction method is worked out for determining a class of exact solutions with inherent wave features to quasilinear hyperbolic homogeneous systems of N>2 first-order autonomous PDEs. A crucial point of the present approach is that in the process the original set of field equations induces the hyperbolicity of an auxiliary 2×2 subsystem and connection between the respective characteristic velocities can be established. The integration of this auxiliary subsystem via the hodograph method and through the use of the Riemann invariants provides the searched solutions to the full governing system. These solutions also represent invariant solutions associated with groups of translation of space/time coordinates and involving arbitrary functions that can be used for studying non-linear wave interaction. Within such a theoretical framework the two-dimensional motion of an adiabatic fluid is considered. For appropriate model pressure-entropy-density laws, we determine a solution to the governing system of equations which describes in the 2+1 space two non-linear waves which were initiated as plane waves, interact strongly on colliding but emerge with unaffected profile from the interaction region. These model material laws include the classical pressure-entropy-density law which is usually adopted for a polytropic fluid.     

Existence and algorithm of solutions for general multivalued mixed implicit quasi-variational inequalities

A new class of general multivalued mixed implicit quasi-variational inequalities in a real Hilbert space was introduced, which includes the known class of generalized mixed implicit quasi-variational inequalities as a special case , introduced and studied by Ding Xie-ping . The auxiliary variational principle technique was applied to solve this class of general multivalued mixed implicit quasi-variational inequalities. Firstly, a new auxiliary variational inequality with a proper convex , lower semicontinuous , binary functional was defined and a suitable functional was chosen so that its unique minimum point is equivalent to the solution of such an auxiliary variational inequality . Secondly , this auxiliary variational inequality was utilized to construct a new iterative algorithm for computing approximate solutions to general multivalued mixed implicit quasi-variational inequalities . Here , the equivalence guarantees that the algorithm can generate a sequence of approximate solutions. Finally, the e     

Free surface waves in equilibrium with a vortex

Finite amplitude solitary waves of uniform depth which interact with a stationary point vortex are considered. Waves both with and without a submerged obstacle are computed. The method of solution is collocation of Bernoulli's equation at a finite number of points on the free surface coupled with equations for equilibrium of a point vortex. The stream function and vortex location are found by computing a conformal map of the flow domain to an infinite strip. For a given obstacle the solutions are parametrized with respect to Froude number and vortex circulation. When no obstacle is present there are two families of solutions, in one of which the amplitude of the wave increases by increasing the circulation, while in the other amplitude increases by decreasing the circulation. Beyond a certain critical Froude number the maximum amplitude wave has a sharp crest with an angle of 120 degrees. Similar behavior is observed for the flow past a submerged obstacle except that there is a critical Froude number below which there is no solution at all.     

Pseudolocalized three-dimensional solitary waves as quasi-particles

A higher-order dispersive equation is introduced as a candidate for the governing equation of a field theory. A new class of solutions of the three-dimensional field equation are considered, which are not localized functions in the sense of the integrability of the square of the profile over an infinite domain. For this new class of solutions, the gradient and/or the Hessian/Laplacian are square integrable. In the linear limiting case, an analytical expression for the pseudolocalized solution is found and the method of variational approximation is applied to find the dynamics of the centers of the quasi-particles (QPs) corresponding to these solutions. A discrete Lagrangian can be derived due to the localization of the gradient and the Laplacian of the profile. The equations of motion of the QPs are derived from the discrete Lagrangian. The pseudomass (“wave mass”) of a QP is defined as well as the potential of interaction. The most important trait of the new QPs is that, at large distances, the force of attraction is proportional to the inverse square of the distance between the QPs. This can be considered analogous to the gravitational force in classical mechanics.     

Response of a non-linear system with strong damping to multifrequency excitations

Summary In this paper, a new asymptotic calculation method is presented for a class of nonlinear nonautonomous vibration systems with multiple external periodic interferences. Simple calculation formulae of an asymptotic solution for resonance and off-resonance vibration are derived. The paper is concerned with a vibration system representing a class of nonlinear oscillators. Consequently, the calculation of a class of nonlinear oscillators is routinized. Two different Duffing equations are verified which shows that the results are completely in accordance with the solutions of references [3, 4, 6]. The derivation of solutions of Duffing equations becomes easier and simpler. In addition, some errors in reference [6] are pointed out.     

Theory of nonstationary interaction between pressure waves in a fluid and unclosed surfaces

Problems of nonstationary scattering of incident waves by unclosed surfaces have been solved in the general formulation under the usual assumptions of the linear mechanics of ideal compressible fluids. Such problems are encountered in a number of important hydrodynanic applications. In this case the nonstationary wave field must be known at any distance from the scatterer, and in particular in its immediate vicinity. The known methods of stationary diffraction cannot be used in nonstationary problems, when it is not possible to obtain exact expressions for the Fourier transforms of the unknown solutions and hence guarantee the unique recovery of the inverse transforms. However, the direct solution of the nonstationary problem is possible only in very simple situations: scattering of pressure waves by a plate, diffraction at the edge of a half-plane or at a slit in a flat screen, etc. These circumstances make it necessary to develop special approaches to the solution of the problem of the nonstationary scattering of pressure waves in a fluid by arbitrary unclosed surfaces. This paper outlines a method which leads to the construction of the Laplace transforms of the unknown solutions and is based on a unique means of satisfying the boundary conditions with the subsequent obtaining of exact expressions for the coefficients (densities) of the expansions employed. The class of problems solvable by this method is confined to those for which it is possible to obtain corresponding solutions by expansion in series or integrals over the complete orthogonal system of eigenfunctions on the assumption that the surface of the obstacle is closed. The Laplace transforms of the solutions can be inverted by any approximate method. The solutions constructed in accordance with the formalism developed are in satisfactory agreement with the experimental data and coincide with the classical results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 84–91, November–December, 1985.     

The boundary integral equation method for the solution of wave-obstacle interaction

The boundary integral equation method constitutes the basis of a number of computer programs used for the solution of wave-obstacle interaction problems. For the case of obstacles in a constant depth fluid, the method assumes that the velocity potential at any point in the fluid may be represented by a distribution of Green's function sources over the immersed surface of the obstacle. Application of the obstacle kinematic boundary condition gives rise to an integral equation which may be solved, using numerical discretization, for the unknown source strength distribution function. Subsequent evaluation of the discretized velocity potential permits evaluation of the hydrodynamic interaction parameters. A series of numerical solutions have been carried out for a range of substantially rectangular obstacles, in a two-dimensional domain, using varying levels of immersed profile discretization. The results, presented in the form of fixed and floating mode wave reflection and transmission, together with the motion response of the floating obstacle, demonstrate the significant sensitivity of the evaluated parameters to variations in the level of discretization.     

Class of solutions for flow near a stagnation point

This paper considers a class of solutions for flow of a perfect gas near the stagnation point on a two-dimensional obstacle, where the flow is rotational far upstream from the obstacle. It is shown that the potential flow near the stagnation point is a special case of this class of solutions. Solutions accounting for the rotationality of the outer flow are obtained for flow in the mixing layer with an obstacle, and these solutions differ appreciably from the analogous Jimenez solution for potential flow near the stagnation point on a two-dimensional obstacle.     

Explicit solutions for shearing and radial stresses in curved beams

In this paper the formulae for the shearing and radial stresses in curved beams are derived analytically based on the solution for a Volterra integral equation of the second kind. These formulae satisfy both the equilibrium equations and the static boundary conditions on the surfaces of the beams. As some applications, the resulting solutions are used to calculate the shearing and radial stresses in a cantilevered curved beam subjected to a concentrated force at its free end. The numerical results are compared with other existing approximate solutions as well as the corresponding solutions based on the theory of elasticity. The calculations show a better agreement between the present solution and the one based on the theory of elasticity. The resulting formulae can be applied to more general cases of curved beams with arbitrary shapes of cross-sections.     

Supersonic flow of a combustible gas mixture past a sphere

In recent years considerable interest has developed in the problems of steady-state supersonic flow of a mixture of gases about bodies with the formation of detonation waves and slow combustion fronts. This is due in particular to the problem of fuel combustion in a supersonic air stream.In [1] the problem of supersonic flow past a wedge with a detonation wave attached to the wedge apex is solved. This solution is based on using the equation of the detonation polar obtained in [2]-the analog of the shock polar for the case of an exothermic discontinuity. In [3] a solution is given of the problem of cone flow with an attached detonation wave, and [4] presents solutions of the problems of supersonic flow past the wedge and cone with the formation of attached adiabatic shocks with subsequent combustion of the mixture in slow combustion fronts. In the two latter studies two different solutions were also found for the problem of flow past a point ignition source, one solution with gas combustion in the detonation wave, the other with gas combustion in the slow combustion front following the adiabatic shock. These solutions describe two different asymptotic pictures of flow of a combustible gas mixture past bodies.In an experimental study of the motion of a sphere in a combustible gas mixture [5] it was found that the detonation wave formed ahead of the sphere splits at some distance from the body into an ordinary (adiabatic) shock and a slow combustion front. Arguments are presented in [6] which make it possible to explain this phenomenon and in certain cases to predict its occurrence.The present paper presents examples of the calculation of flow of a combustible gas mixture past a sphere with a detonation wave in the case when the wave does not split. In addition, the flow near the point at which the detonation wave splits is analyzed for the case when splitting occurs where the gas velocity behind the wave is greater than the speed of sound. This analysis shows that in the given case the flow calculation may be carried out without any particular difficulties. On the other hand, the calculation of the flow for the case when the point of splitting is located in the subsonic portion of the flow behind the wave (or in the region of influence of the subsonic portion of the flow) presents difficulties. This flow case is similar to the problem of the supersonic jet of finite width impacting on an obstacle.     

Bifurcation of the solution to the problem of steady traveling waves in a layer of viscous liquid on an inclined plane

Traveling waves in a viscous liquid flowing down an inclined plane can be described at small and moderate Reynolds numbers by an ordinary differential equation in the thickness of the layer [1, 2]. As the Reynolds number tends to zero, this equation goes over into an equation of third order with quadratic nonlinearity [3]. Periodic solutions of this last equation bifurcating from the plane-parallel solution have been investigated by Nepomnyashchii and Tsvelodub [3–6]. In the present paper, a study is made of the bifurcation of periodic solutions from periodic solutions, namely, an investigation is made of the values of the wave number for which a periodic solution is not unique; a bifurcation equation is derived, the number of bifurcating solutions is found, and their behavior near a bifurcation point is considered; and the bifurcating solutions are continued numerically with respect to a parameter (the wave number) from the neighborhoods of the bifurcation points.     

复杂形状及开孔功能梯度板的三维分析

In this paper, the basic formulae for the semi-analytical graded FEM on FGM members are derived. Since FGM parameters vary along three space coordinates, the parameters can be integrated in mechanical equations. Therefore with the parameters of a given FGM plate, problems of FGM plate under various conditions can be solved. The approach uses 1D discretization to obtain 3D solutions, which is proven to be an effective numerical method for the mechanical analyses of FGM structures. Examples of FGM plates with complex shapes and various holes are presented.     

任意外型薄板非线性后屈曲问题的边界元解

本文就薄板后屈曲问题建立一组新型的边界元计算公式，用这组公式求解能方便处理各种边界问题，另外文中将面内应力分解成基本部份和附加部份，并利用微分算子分解理论导得了挠度的一个不同形式的基本解，由于计算公式中，实现了面内位移和挠度的解耦，从而使迭代过程得到简化，文末还对圆板后屈曲路径进行了计算，得到了满意的结果。     

New numerical method for volterra integral equation of the second kind in piezoelastic dynamic problems

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.     

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.     

STABILITY OF SYSTEM OF TWO-DIMENSIONAL NON-HYDROSTATIC REVOLVING FLUIDS

Applying the theory of stratification, it is proved that the system of the two-dimensional non-hydrostatic revolving fluids is unstable in the two-order continuous function class. The construction of solution space is given and the solution approach is offered. The sufficient and necessary conditions of the existence of formal solutions are expressed for some typical initial and boundary value problems and the calculating formulae to formal solutions are presented in detail.     

A general nonlocal nonlinear Schrödinger equation with shifted parity,charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a \(\beta \)-plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.     

Embedding formulae for Laplace-Beltrami problems on the sphere with a cut

We consider the problem of diffraction of a plane wave by a quarter-plane (a flat cone) with Dirichlet boundary conditions. The most efficient approach to this problem is the technique of Smyshlyaev’s formulae or a modified Smyshlyaev’s formula, both of which are representations of the diffraction coefficient as contour integrals over a complex parameter. These representations have been proven independently. Here we are demonstrating a link between these classes of formulae. The link is established by developing an embedding procedure (in the very special sense of diffraction theory) on the unit sphere.     

Bilinear forms and soliton interactions for two generalized KdV equations for nonlinear waves

Korteweg–de Vries (KdV)-type equations can describe the nonlinear waves in fluids, plasmas, etc. In this paper, two generalized KdV equations are under investigation. Bilinear forms of which are constructed with the Bell polynomials and an auxiliary variable. \(N\) -soliton solutions are given through the Hirota direct method. Via the asymptotic analysis, the soliton interactions of the first generalized KdV equation are analyzed, which turn out to be elastic. Singular breather solutions have been derived from the two-soliton solutions. The collision between soliton and singular breather appears to be elastic, and the bound states of soliton and singular breather are exhibited. Unlike the first one, the other generalized KdV equation can only support the bound states of solitons, for the regular and singular solitons alike.