Nonlinear Effects in Osmotic Volume Flows of Electrolyte Solutions through Double-Membrane System

The results of experimental study of volume osmotic flows in a double-membrane system are presented in this article. The double-membrane system consists of two membranes (M _u, M _d) oriented in horizontal planes and three identical compartments (u, m, d), containing unstirred binary or ternary ionic solutions. In this system concentrations of the solutions fulfil the following conditions C _us  = C _ds  < C _ms (s = 1 or 2). Solutions of aqueous potassium chloride or ammonia were used as binary solutions, whereas potassium chloride dissolved in aqueous ammonia solution or ammonia dissolved in aqueous potassium chloride solution were used as ternary solutions. For binary solutions, the dependencies of a volume flux (J _v) on potassium chloride or ammonia concentration (C _ms ) are linear, whereas for ternary solutions these dependencies are nonlinear. The volume flux amplification and the osmotic conductivity coefficients were calculated on the basis of experimental data. The coefficient of the volume flux amplification for ternary solutions in comparison to binary ones depends on solutes concentrations and has maximum values dependent on solutes concentrations. Similarly, the osmotic conductivity coefficient has maximal values dependent on solutes concentrations. Moreover, the thermodynamic model of the osmotic volume flux was developed and the results were interpreted within the gravitational instability category.     

Certain features of similarity solutions for flows in viscous vortex cores

A class of steady similarity solutions of the equations for viscous vortex cores which correspond to external inviscid similarity solutions with a power-law variation of the circumferential velocityv-r ^−m near the rotation axis is considered. It is found that if the Bernoulli function in external flow is constant, then these solutions will exist only on a certain range of the indexm of the exponential. For eachm on this range there are two solutions. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–43, January–February, 1998. The work was financially supported by the Russian Foundation for Fundamental Research (project No. 95-01-00483).     

General high-order rational solutions and their dynamics in the (3+1)-dimensional Jimbo–Miwa equation

General high-order rational solutions are derived for the (3+1)-dimensional Jimbo–Miwa equation based on the Hirota bilinear form. The solutions are presented in terms of Gram determinants; the elements of determinants are connected to Schur polynomials and have simple algebraic expressions. Their dynamic behaviors are researched using three-dimensional imagery and contour plots. It is revealed that different kinds of solutions appear in (x, y) plane and (y, z) plane. When one of these internal parameters in the rational solutions is sufficiently large, in (x, y) plane Lump solutions appear with obvious geometric structures, which are deconstructed by a first-order Lump such as triangle, pentagon, and nonagon, among others; in (y, z) plane rational line soliton solutions with maximum background amplitude changing over time appear. These findings might help us comprehend the nonlinear wave propagation processes in the many nonlinear physical systems.

     

Numerical and experimental study of driven flow in a polar cavity

A numerical and an experimental study of the flow of an incompressible fluid in a polar cavity is presented. The experiments included flow visualization, in two perpendicular planes, and quantitative measurements of the velocity field by a laser Doppler anemometer. Measurements were done for two ranges of Reynolds numbers; about 60 and about 350. The stream function-vorticity form of the governing equations was approximated by upwind or central finite-differences. Both types of finite-difference approximations were solved by a multi-grid method. Numerical solutions were computed on a sequence of grids and the relative accuracy of the solutions was studied. Our most accurate numerical solutions had an estimated error of 0.1 per cent and 1 per cent for Re = 60 and Re = 350, respectively. It was also noted that the solution to the second order finite difference equations was more accurate, compared to the solution to the first order equations, only if fine enough meshes were used. The possibility of using extrapolations to improve accuracy was also considered. Extrapolated solutions were found to be valid only if solutions computed on fine enough meshes were used. The numerical and the experimental results were found to be in very good agreement.     

Fracture dynamics problems of mode I semi-infinite crack for anisotropic orthotropic body

By means of the theory of complex functions, fracture dynamics problems of mode I semi- infinite crack for anisotropic orthotropic body were researched. Analytical solutions of stress, displacement, and dynamic stress intensity factor under the action of moving increasing loads Px ³/t ³, Pt ⁴/x ³, respectively, are very easily obtained utilizing the approaches of self-similar functions. In the light of relevant material’s coefficients, the alterable rule of dynamic stress intensity factor was depicted very well. The correlative closed solutions are attained based on the Riemann–Hilbert problems. After those analytical solutions were applied by the superposition principle, the solutions of discretional complex problems could be attained.     

UNSTEADY CONFINED VISCOUS FLOWS WITH OSCILLATING WALLS AND MULTIPLE SEPARATION REGIONS OVER A DOWNSTREAM-FACING STEP

This paper presents computational solutions for unsteady viscous flows in channels with a downstream-facing step, followed by an oscillating floor. These solutions of the unsteady Navier–Stokes equations are obtained with a time-integration method using artificial compressibility in a fixed computational domain, which is obtained via a time-dependent coordinate transformation from the fluid domain with moving boundaries. The computational method is first validated for steady viscous flows past a downstream-facing step by comparison with previous numerical solutions and experimental results. This method is then used to obtain solutions for unsteady viscous flows with multiple separation regions over a downstream-facing step with oscillating walls, for which there are no previously known solutions. Thus, the present results may be used as benchmark solutions for the unsteady viscous flows with multiple separation regions between fixed and oscillating walls.     

Wronskian,Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

Application of the shallow water waves in environmental engineering and hydraulic engineering is seen. In this paper, a (3+1)-dimensional generalized nonlinear evolution equation (gNLEE) for the shallow water waves is investigated. The Nth-order Wronskian, Gramian and Pfaffian solutions are proved, where N is a positive integer. Soliton solutions are constructed from the Nth-order Wronskian, Gramian and Pfaffian solutions. Moreover, we analyze the second-order solitons with the influence of the coefficients in the equation and illustrate them with graphs. Through the Hirota-Riemann method, one-periodic-wave solutions are derived. Relationship between the one-periodic-wave solutions and one-soliton solutions is investigated, which shows that the one-periodic-wave solutions can approach to the one-soliton solutions under certain conditions. We reduce the (3+1)-dimensional gNLEE to a two-dimensional planar dynamic system. Based on the qualitative analysis, we give the phase portraits of the dynamic system.

     

Resonant solutions and breathers to the BKP equation

In this paper, we derive resonant and breather solutions from multi-soliton solutions of the B-type Kadomtsev–Petviashvili (BKP) equation of fourth order via the Hirota bilinear method. We first discuss N-soliton solutions of the BKP equation and use the linear superposition principle to generate N-resonant solutions. Subsequently, we construct complexiton and breather solutions and finally, study the dynamics of some selected solutions with the aid of 3D plots, contour plots and density plots.

     

Motion of a Graph by <Emphasis Type="Italic">R</Emphasis>-Curvature

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R ^d , by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.     

Removing Collision Singularities from Action Minimizers for the N-Body Problem with Free Boundaries

For the planar and spatial N-body problems, it has been proved by Marchal and Chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions. This important result has been extended by Ferrario & Terracini to Newtonian-type problems and equivariant problems. It has also been used to construct many symmetric solutions for the N-body problem. In this paper we are interested in action minimizing solutions in function spaces with free boundaries. The function spaces are imposed with boundary conditions, such that every mass point starts and ends on two transversal proper subspaces of ℝ^d, d≥2. We will prove that solutions for this minimizing problem with free boundaries are always free from collisions, including boundary collisions. This result can be used to construct certain classes of relative periodic solutions of the N-body problem.     

Long-time Behavior of Solutions of Hamilton–Jacobi Equations with Convex and Coercive Hamiltonians

We investigate the long-time behavior of viscosity solutions of Hamilton–Jacobi equations in \mathbbRⁿ{\mathbb{R}^n} with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.     

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u _t+uu_x)_x=1/2u _x ² . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L ^p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.     

The creep deformation of vibrating structures

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.     

NGPG-Stability of Linear Multistep Methods for Systems of Generalized Neutral Delay Differential Equations

The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations. After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP _G -stable if and only if it is A-stable.     

Auto-Bäcklund transformations and soliton solutions on the nonzero background for a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schif equation in a fluid

In this paper, a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schif equation in a fluid is investigated. By the virtue of the truncated Painlevé expansion, a set of the auto-Bäcklund transformations of that equation is worked out. Based on the auto-Bäcklund transformations with certain non-trivial seed solutions, one-, two-, three- and N-soliton solutions on the nonzero background of that equation are derived with N as a positive integer. According to those two-soliton solutions, X- and inelastic-type soliton solutions are obtained. Via the asymptotic analysis, influence of the coefficients for the above equation is discussed and the interactions between the solitons are also studied. Then, those solitons and interactions are shown graphically.

     

Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ^∞ as t→0₊ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation. Accepted: October 26, 1999     

Harmonic Maps and Ideal Fluid Flows

Using harmonic maps we provide an approach towards obtaining explicit solutions to the incompressible two-dimensional Euler equations. More precisely, the problem of finding all solutions which in Lagrangian variables (describing the particle paths of the flow) present a labelling by harmonic functions is reduced to solving an explicit nonlinear differential system in \mathbb Cⁿ{\mathbb {C^n}} with n = 3 or n = 4. While the general solution is not available in explicit form, structural properties of the system permit us to identify several families of explicit solutions.     

The Generalized Proudman�CJohnson Equation Revisited

We demonstrate the existence of solutions to the inviscid generalized Proudman–Johnson equation for parameters a lying in the open interval (−5,−1) which develop singularities in finite time; moreover, we show that there are solutions which exist for all times if a = −1. Finally, a simple blow-up criterion for solutions arising from a special class of initial data is given.     

A ship motion forecasting approach based on empirical mode decomposition method hybrid deep learning network and quantum butterfly optimization algorithm

Studies of the shallow water waves are active, possessing the applications in ocean engineering, marine environment, atmospheric science, etc. In this paper, we investigate a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Hirota method and symbolic computation help us work out (1) a bilinear form, (2) N-soliton solutions with N being a positive integer, (3) the higher-order breather solutions, (4) periodic-wave solutions and (5) hybrid solutions composed of one first-order breather and one soliton/two solitons. Moreover, we provide some nonlinear phenomena described by the associated solutions. All of the obtained results are determined via the time-dependent coefficients of that equation.

     

Novel evolutionary behaviors of localized wave solutions and bilinear auto-Bäcklund transformations for the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation

Water waves are common phenomena in nature, which have attracted extensive attention of researchers. In the present paper, we first deduce five kinds of bilinear auto-Bäcklund transformations of the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation starting from the specially exchange identities of the Hirota bilinear operators; then, we construct the N-soliton solutions and several new structures of the localized wave solutions which are studied by using the long wave limit method and the complex conjugate condition technique. In addition, the propagation orbit, velocity and extremum of the first-order lump solution on (x, y)-plane are studied in detail, and seven mixed solutions are summarized. Finally, the dynamical behaviors and physical properties of different localized wave solutions are illustrated and analyzed. It is hoped that the obtained results can provide a feasibility analysis for water wave dynamics.