A finite-difference method for solving initial-boundary value problems for loaded differential and integro-differential equations

Бондарев, Э. А.; Voevodin, A. F.

doi:10.1007/bf02757374

Cited by 12 publications

(6 citation statements)

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“…8 Difference methods of solving initial-boundary-value problems for loaded differential and integrodifferential equations have been investigated. 9 The special case of a problem with a loaded equation was considered in Voronin's thesis. 1 The initial-boundary-value problems that arise during the search for the invariant solution (2.1) in a vertical strip will be considered in dimensionless form.…”

Section: Statement Of the Problemmentioning

confidence: 99%

The modelling of microconvection in an infinite strip

Goncharova¹

2009

Journal of Applied Mathematics and Mechanics

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Section: Statement Of the Problemmentioning

confidence: 99%

The modelling of microconvection in an infinite strip

Goncharova¹

2009

Journal of Applied Mathematics and Mechanics

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“…If we additionally consider equations obtained by equating terms at higher power of ζ in Eqs. (11), (12) or Eqs. (13), (14), then each such equation adds one unknown.…”

Section: And the Seriesmentioning

confidence: 99%

“…Following the terminology of [11], this system is called loaded because its right side contains unknown functions and their derivatives at the origin ζ = 0. For instance, u 0 = U (1) (0, t) in Eq.…”

Section: Introductionmentioning

confidence: 99%

Fourier method in the problem of small perturbations of colliding jets

Karabut

2010

J Appl Mech Tech Phy

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Small unsteady perturbations of the solution of the known problem of a steady collision of two jets of an ideal incompressible fluid having identical velocities are studied. A discrete spectrum of the linear problem is found through direct calculations. It is demonstrated that the Fourier method is inapplicable because the eigenfunctions are not bounded.Key words: ideal incompressible fluid, free surface, stability in the linear approximation, method of separation of variables, numerical determination of the spectrum.

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“…Nonlinear equations were obtained which describe the evolution of jets for the case where the perturbations are not assumed small and which can be used for the numerical solution of the problem. Linearization of these equations leads to nonstandard equations which in [17] were called loaded equations because their right sides contain the solution and its derivatives taken at the coordinate origin.…”

Section: The Function Iumentioning

confidence: 99%

Small perturbation in the problem of jet impingement: Constitutive equations

Karabut

2009

J Appl Mech Tech Phy

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The well-known steady-state problem of impingement of two plane jets of an ideal incompressible fluid moving with the same velocity is refined and extended to the case of unsteady interaction. Equations describing perturbation propagation on the free surface of the impinging jets are obtained and linearized on the steady-state solution.Introduction. The problem of impingement of two jets moving at an angle toward each other with the same velocity which can be considered equal to unity without loss of generality is one of the simplest problems solved by the Kirchhoff method [1]. It arises from the problem of fluid flow from two hoses (more precisely, slots) of different thicknesses (Fig. 1a) in the limit where these hoses move to infinity in directions opposite to the directions of the jet velocities. This results in a steady-state jet flow which is shown schematically in Fig. 1b.Investigation of unsteady flow is necessary, for example, for impingement of jets moving at different velocities. In this case, a continuous steady-state solution cannot be constructed because different Bernoulli constants cannot provide pressure continuity at the critical point. A moving coordinate system in which the flow is steadystate can be found only in the particular case of head-on impingement where the jet velocities are different and directed toward each. In the case where the jets impinge at an angle, such a system does not exist. Unlike in the case of head-on impingement, in this case there is a distinguished point, for example, the point of intersection of the midline of the impingement jets (point A in Fig. 1a). In a coordinate system rigidly attached to this point, the flow is apparently periodic in time, which is confirmed by experimental studies [2] which showed the existence of aperiodic waves on the free surface during interaction of two oil flows moving at different velocities. This periodic interaction of jets can be regarded as the simplest model of wave formation during explosion welding [3]. More complex models taking into account acoustic perturbations, viscosity, thermal conductivity, surface tension, instability of the von Kármán vortex street, etc., are presented in [4].The equations obtained below can be used to describe the interaction of jets moving at different velocities. However, first of all, it is necessary to derive equations to study the stability of steady-state impingement of jets moving at the same velocity. In this case, it is assumed that perturbations decrease at infinity, i.e., do not influence the steady-state flow velocity at points at infinity.Slightly perturbed jet flows are commonly studied by using the equations given in [5] with the boundary conditions extended to the unperturbed free surface. The question of the validity of this procedure for the case of unsteady interaction of jets where there are four points at infinity remains open. In this case, in fact, a conformal mapping of the perturbed flow region onto the unperturbed region is performed. It is not clear whether it is possible ...

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A finite-difference method for solving initial-boundary value problems for loaded differential and integro-differential equations

Cited by 12 publications

References 1 publication

The modelling of microconvection in an infinite strip

The modelling of microconvection in an infinite strip

Fourier method in the problem of small perturbations of colliding jets

Small perturbation in the problem of jet impingement: Constitutive equations

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