Exact Partial Solution Research Articles

Energy Transfer Optimization in the Case of Single-Stage Acceleration of a Piston by Compressed Gas

The limiting possibilities and energy efficiency of single-stage acceleration of a piston by compressed gas in a barrel (ballistic Lagrange problem) is under study. An exact partial solution of the ballistic Lagrange problem in a homogeneous deformation approximation is obtained. The calculation results are compared by different methods with the experimental data and the calculation results from other papers. It is shown that the use of an efficiency coefficient as a criterion for choosing an optimal solution prevents from determining the effective configurations of ballistic launchers.

Solution for the space-charge-limited current in coaxial vacuum diodes

This paper presents a new relativistic solution for the space-charge-limited current for coaxial vacuum diodes. The solution is deduced using a methodology that combines exact partial solutions and numerical fitting. The expression obtained presents a maximum error of 2% in the verified voltage range from 0 V to 846 MV and the radius ratio from 0.05 to 0.95.

MULTI-COMPONENT BIANCHI I COSMOLOGIES WITH TORSION

The Bianchi I cosmological models containing radiation, stiff fluid and a nonminimally coupled material scalar field with polynomial potentials of the fourth degree are considered in the framework of the Einstein–Cartan theory (ECT). Exact partial solutions are obtained for arbitrary positive values of the coupling constant ξ. The restrictions on ξ are found from the integrability conditions. It is demonstrated that singular models with the asymptotical isotropization by the de Sitter type and by the power-law (t4/3) type at late-times are possible. It is shown that the bouncing models with the isotropization by the de Sitter law of expansion at late-times are admissible. The role of sources in the evolution of models is elucidated.

Analytical model of a corona discharge from a conical electrode under saturation

Exact partial solutions are found for the electric field distribution in the outer region of a stationary unipolar corona discharge from an ideal conical needle in the space-charge-limited current mode with allowance for the electric field dependence of the ion mobility. It is assumed that only the very tip of the cone is responsible for the discharge, i.e., that the ionization zone is a point. The solutions are obtained by joining the spherically symmetric potential distribution in the drift space and the self-similar potential distribution in the space-charge-free region. Such solutions are outside the framework of the conventional Deutsch approximation, according to which the space charge insignificantly influences the shape of equipotential surfaces and electric lines of force. The dependence is derived of the corona discharge saturation current on the apex angle of the conical electrode and applied potential difference. A simple analytical model is suggested that describes drift in the point-plane electrode geometry under saturation as a superposition of two exact solutions for the field potential. In terms of this model, the angular distribution of the current density over the massive plane electrode is derived, which agrees well with Warburg’s empirical law.

Exact partial solutions for the surface dynamics of a dielectric liquid with a charged surface in the gravitational field

We study the evolution of instability in the boundary of a perfect dielectric liquid with a free surface charge in an external electric field. Conformal variables are used to find exact partial solutions to the equations of motion for the case when the charge completely shields the field above the liquid, the electrostatic and gravitational forces being taken into account. The solutions describe the development of instability of the initially planar boundary until sharp dimples are formed on it.

Equilibrium surface configurations of a conducting liquid in external spatially periodic electric field

The problem of possible equilibrium configurations of the free surface of a conducting liquid in an external spatially periodic electric field is considered. It is shown that, under identical conditions, the problem admits solutions with different wavelengths, including the wavelength corresponding to the field period and the half shorter wavelength corresponding to the period of the electrostatic pressure oscillations. In the former case, an exact partial solution is found that describes a surface profile with allowance for the capillary forces.

Construction of exact partial solutions of nonintegrable systems by means of formal Laurent and Puiseux series

A package facilitating search for analytic solutions of nonintegrable systems of autonomous ordinary differential equations is suggested. The algorithm discussed, which is a generalization of the Conte-Musette method, allows one to obtain single-valued and multivalued analytical solutions that were known to exist only in the form of formal Laurent and Puiseux series (obtained, for example, by means of the Painleve analysis), respectively. Solutions in the form of the formal Laurent series and procedures from the package discussed are used not only for constructing elliptic solutions but also for proving impossibility of such construction. The suggested procedures have been implemented as packages in Maple and REDUCE.

EVOLUTION GALERKIN SCHEMES APPLIED TO TWO-DIMENSIONAL RIEMANN PROBLEMS FOR THE WAVE EQUATION SYSTEM

The subject of this paper is a demonstration of the accuracy and robust- ness of evolution Galerkin schemes applied to two-dimensional Riemann problems with finitely many constant states. In order to have a test case with known exact solution we consider a linear first order system for the wave equation and test evo- lution Galerkin methods as well as other commonly used schemes with respect to their accuracy in capturing important structural phenomena of the solution. For the two-dimensional Riemann problems with finitely many constant states some parts of the exact solution are constructed in the following three steps. Using a self-similar transformation we solve the Riemann problem outside a neighborhood of the origin and then work inwards. Next a Goursant-type problem has to be solved to describe the interaction of waves up to the sonic circle. Inside it a system of composite elliptic- hyperbolic type is obtained, which may not always be solvable exactly. There an interesting local maximum principle can be shown. Finally, an exact partial solution is used for numerical comparisons.

Evolution Galerkin schemes applied to two-dimensional Riemann problems for the wave equation system

The subject of this paper is ademonstration of the accuracy and robustness of evolutionGalerkin schemes applied to two-dimensional Riemann problems withfinitely many constant states.In order to have a test case with known exact solution weconsider a linear first ordersystem for the wave equation and test evolution Galerkin methodsas well as other commonly used schemes with respect to theiraccuracyin capturing important structural phenomenaof the solution. For the two-dimensional Riemann problems withfinitely many constant states some parts of the exact solution areconstructed in thefollowing three steps.Using a self-similar transformationwe solve the Riemann problem outside a neighborhoodof the origin and then work inwards.Next a Goursant-typeproblem has to be solved to describe the interaction of waves upto the sonic circle. Inside it a systemof composite elliptic-hyperbolic type is obtained, whichmay not always be solvable exactly.There an interesting local maximum principle can be shown.Finally, an exact partial solution is used for numericalcomparisons.

Analytic examples of force-free toroidal MHD equilibria

Analytically described toroidal (axisymmetric and three-dimensional) equilibrium magnetic field configurations with a “flat” current density, j=λB (λ = const), are proposed. Such configurations are superpositions of several force-free two-dimensional configurations with plane, axial, or helical coordinate symmetry. Each of them is generated by an exact partial solution to the corresponding Grad-Shafranov equation. A variety of toroidal configurations thus obtained allows one to model topological changes of magnetic surfaces, such as magnetic axis splitting (doublets) in axisymmetric equilibrium configurations and the appearance and interaction of magnetic islands and ergodic lines in three-dimensional configurations.

Nonlinear wave structures in a thermally relaxing gas-liquid mixture

An evolutionary equation describing a nonlinear wave process in a gas-bubble-liquid mixture is derived. In the mixture interphase heat transfer takes place due to deviation of the gas behavior from adiabatic. Exact partial solutions describing the structures of both shock waves and solitons are given. The mechanism of maximum compression in a shock wave structure propagating in a mixture containing bubbles of a dissolving gas is elucidated. The interval of variation of the input bubble radius on which, as a result of compression, the steady-state wave profile is nonmonotonic is found. A wave profile with an oscillatory structure is shown to exist. Numerical calculations based on the formulas obtained are found to be in fairly satisfactory agreement, at least qualitatively, with the experimental data known to the author.

Equilibrium Nonlinear Distributions of One-Component Order Parameter in Systems with Competitive Interaction

The approach which enables to obtain the solutions of the variational equation as the series [Formula: see text] is considered. It is shown that when N=1 it gives well-known cos x solutions, while for N>1 it leads to essentially nonlinear functions. The φ2φ′2-term in the thermodynamical potential provides the proper nonlinear behavior of the order parameter and the possibility of existence of kink-like distributions, particularly. In special case N=2 there exist the exact partial solutions which, however, do not give the absolute minima to the thermodynamical potential.

An exact partial solution to the compressible flow problems of jet formation and penetration in plane, steady flow

The particle solution of the problem of the symmetric impact of two compressible fluid stream is derived. The plane two-dimensional flow is assumed to be steady, and the inviscid compressible fluid is of the Chaplygin (tangent gas) type. The equations governing this flow are transformed to the hodograph plane where an exact, closed-form solution for the stream function is obtained. The distribution of fluid properties along the plane of symmetry and the shape of free surface streamlines are determined by transformation back to the physical plane. The problem of a compressible fluid jet penetrating an infinite target of similar material is also solved by considering a limiting case of this solution. Differences between compressible and incompressible flows of the type considered are illustrated.