One-dimensional Nonstationary Problem Research Articles

Исследование процессов передачи тепла в системе «трубопровод – датчик давления»

The one-dimensional non-stationary problem of temperature determination in the mechanical system «pipeline‒ pressure sensor» is studied. The solution of the problem is constructed using the method of separation of variables. Temperature distributions in the working medium in the pipeline and in the material of the sensor sensing element are obtained

SIMULATION OF TEMPERATURE EFFECTS IN MASSIVE BODIES USING THE MODIFIED METHOD OF LINES

This article presents the main ideas and possibilities of the modified straight line method for modeling temperature effects in massive bodies. The calculation model of a rectangular beam with a defined thermal state is adopted. The procedure for reducing the dimensionality of the original equations using the modified method of straight lines and the Bubnov-Galyorkin-Petrov method for boundary conditions is applied. The reduced equation of thermal conductivity and the reduced equation of heat balance are given. Conclusions are made regarding the addition of reduced second-order equations with components from the boundary conditions on the lateral surfaces, and finally reduced second-order equations in spatial variables are given.
 We are considering a beam of rectangular cross-section, the three dimensions of which are of the same order. The thermal state of such an object is three-dimensional and is therefore described systematically using three-dimensional heat conduction equations. When applying the modified method of straight lines to reduce the dimensionality of the original equations, it is convenient to use them in the form of a system of partial differential equations of the first order in spatial coordinates.
 According to the modified method of straight lines, two systems of straight lines parallel to the coordinate axes Oy and Oz are applied to the beam (a decrease in dimensionality in terms of spatial variables y and z is assumed). The choice of such straight lines is due to the fact that the reduced equations will depend on the variables x and t, that is, they will resemble the one-dimensional equations of the one-dimensional non-stationary problem of thermal conductivity, for which efficient and convenient numerical methods have been developed in the literature of mathematical physics at the moment, to which it is easy to adapt the reduced equation.

Accuracy and stability of a finite difference scheme for two-dimensional problems of soil dynamics

Problems of numerical methods for calculating hyperbolic systems of equations with several variables are considered topical. At present, for most technical issues, the solution to one-dimensional non-stationary problems can be conducted with sufficient accuracy. It is too early to assert the mass solution of three-dimensional problems. Therefore, the present study is focused on the consideration of the accuracy and stability of the two-dimensional difference method for solving non-stationary dynamic problems. Unlike conventional difference schemes, the M. Wilkins difference scheme, based on the integral definition of partial derivatives, is considered here. The order of approximation error for arbitrary quadrangular cells is shown. Analytical methods are used to determine the accuracy and stability of the difference relations for the linear equations of the dynamics of a deformable rigid body, as applied to the problems of the interaction of rigid bodies with soil.

Perturbation Method in the Problem of Compression-Shear Shock Load for a Nonlinear Elastic Half-Space

On the example of a one-dimensional nonstationary problem of oblique impact on the boundary of a nonlinear elastic isotropic half-space, the question of the manifestation of nonlinear deformation effects via basic evolution equations is studied. Much attention is given to the behavior of the solution behind the leading edge of a quasi-transverse shock wave. For particular cases of boundary conditions, it is shown that the onset region of the evolution equation of a quasi-transverse wave is preceded by a series of preliminary transitions to the intermediate internal problems of the small parameter method determined by the type of preliminary bulk deformation. This deformation consistently affects the distortion of the characteristic coordinates and the leading edge of the quasitransverse process. As a consequence, the transition to the evolution equation of quasi-transverse waves occurs with simultaneous change of all independent variables of the boundary value problem.

Unsteady expansion of thick-wall spherical and cylindrical viscoplastic shells

One-dimensional nonstationary problems of adiabatic expansion for thick-wall spherical and cylindrical viscoplastic shells are solved exactly under the assumption that, at the initial instant of time, the distributions of radial velocities satisfy the condition of incompressibility of the shell material. The resulting solutions can easily be modified for the case of compression of such shells.

Analytical solutions of problems of the adiabatic compression of thick-walled spherical and cylindrical shells made of incompressible viscoplastic material

Exact solutions of the one-dimensional non-stationary problems of the adiabatic compression of thin-walled spherical and cylindrical shells made of incompressible viscoplastic material are obtained, assuming that, at the initial instant of time, the radial velocity distributions satisfy the conditions of incompressibility of the shell material. From these solutions it is also easy to obtain the solutions for cases of expansion of the shells.

Investigation of the Influence of Properties of a "Third Body" on Heat Generation Due to Friction

We describe a method of determining the thermal conductance of a contact in simulation of contact thermoelasticity with regard for friction and heat generation. Using the results of numerical analysis, we construct the dependence of the thermal conductance of a contact on input parameters and show that some of them substantially affect this phenomenon. By the example of a one-dimensional nonstationary problem, we study the effect of the parameters of contacting surfaces on heat generation.

Finite-difference method for solving a one-dimensional nonstationary problem of radiative-conductive heat transfer

Finite-difference method for solving a one-dimensional nonstationary problem of radiative-conductive heat transfer

Nonstationary problems of the combustion of aerosuspensions in fuel that contains the oxidant

A study is made of one-dimensional nonstationary problems of the combustion and detonation of aerosuspensions of unitary fuels or propellants, which contain the oxidant as well as the combustible material (gunpowder, high explosives). A numerical analysis is made of the damping of the convective combustion which occurs at relatively low mass concentrations of the fuel; the critical concentration dividing the damped and the detonation regimes is determined. It is shown that the realization of the damped or detonation regime of convective combustion at a given concentration is completely determined by the gas dynamics of the relative motion of the gas and the particles (two-velocity effect), this being governed by the coefficient of friction.

The computation of one-dimensional non-stationary problems of gravitational gas dynamics

SOME features of the numerical solution of one-dimensional non-stationary problems of gas dynamics when gravity forces are present are studied. The problem of the spread of shock waves in the sun's atmosphere is discussed. The effectiveness of the use of completely conservative difference schemes in problems of this class is demonstrated. The process of periodic formation of shock waves in the atmosphere for a monotonic dynamic action on its lower boundary is explained.

Electromagnetic field generated in air by a nonstationary gamma-ray source in an ideally conducting plane

We consider a model problem on the generation of a radio signal by a nonstationary gamma-ray source. The problem is essentially two-dimensional in space but is reduced to a number of one-dimensional nonstationary problems. The results of a numerical solution of the problem are discussed.

Model of the motion and shock waves in two-phase solids with phase transitions

The one-velocity and one-temperature model of the motion of a two-phase solid, in which each phase occupies a certain part of the volume, is considered. The investigation is carried out in Lagrangian variables, which offers certain advantages in solving one-dimensional nonstationary problems. The stress tensor for the mixture is decomposed into two parts -hydrostatic pressure, common to the two phases, associated with the three-term equation of state, and the deviator, which varies elastically up to a certain value and then remains constant. A certain relation, determined by the characteristic reaction time, is proposed for the phase transition kinetics. Then a solution is obtained for the problem of the nonstationary one-dimensional motion of a metal (iron) resulting from the impact of a plate against a target. The phase transitions (Feα⇌Feɛ) behind the wave and their characteristic time have an important effect on the damping of the disturbance and on the zone in which these transitions go to completion. A method is proposed for determining the coefficient in the relation for the phase transition rate from the residual effect (hardening) after impact.

A finite-difference method for the solution of one-dimensional non-stationary problems in magneto-hydrodynamics

A finite-difference method for the solution of one-dimensional non-stationary problems in magneto-hydrodynamics