One-dimensional Unsteady Motion Research Articles

Non-isothermal filtration of a viscous compressible fluid in a viscoelastic porous medium

The system of equations of one-dimensional unsteady fluid motion in a viscous heat-conducting medium is considered. The mathematical model is based on the equations of conservation of mass for liquid and solid phases, Darcy’s law, rheological relation, the law of conservation of balance of forces and the equation for the temperature of the medium. The transition to Lagrange variables in the case of an incompressible fluid allows us to reduce the initial system of governing equations to a third-order equation for porosity and a second-order equation for temperature, respectively. A calculation algorithm is proposed and a numerical study of the obtained initial - boundary value problem is carried out.

Filtration of Liquid in a Non-isothermal Viscous Porous Medium

The solvability of the initial-boundary value problem is proved for the system of equations of one-dimensional unsteady fluid motion in a heat-conducting viscous porous medium

Galerkin approximations in the problem of one-dimensional unsteady motion of a viscous compressible two-component fluid

Galerkin approximations in the problem of one-dimensional unsteady motion of a viscous compressible two-component fluid

An Efficient Analytical Solution of Blast Wave Problem in Real Gas Flow under the Influence of Dust Laden Particles

In the present paper, we investigated the problem of the propagation of blast waves governed by a nonhomogeneous quasilinear hyperbolic system of partial differential equations (PDEs), describing one-dimensional unsteady motion with generalized geometries in a real gas flow (van der Waals gas) in which the influence of the dust particles is significant. An efficient analytical approach has been used to the governing hyperbolic system with respect to the Rankine-Hugoniot (RH) conditions to obtain an exact solution in terms of flow parameters density, velocity and the pressure, which exhibits space-time dependence. Further, an analytical expression for the total energy influenced by real gas effects (consisting of non-ideal gas and small solid dust-laden particles) is derived. The results obtained significantly explore the effect of dust-laden particles on the propagation of blast waves in a van der Waals gas.

Investigation of Shock Waves in Non-Ideal Gas Under the Action of Magnetic Field

In present paper, certain aspect of shock wave in non-ideal gas, when magnetic field is orthogonal to the trajectories of the gas particles and electrical conductivity is taken to be infinite, is investigated. Considering one-dimensional unsteady non-planer motion, basic equations, its general solution and formation of shock-wave, conservation laws and jumps conditions, variation of area of non-uniform cross section and analytical solution of strong non planer shock is obtained.

Similarity Solutions of Cylindrical Shock Waves in Magnetogasdynamics With Thermal Radiation

Using Lie group of transformations, here we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of an ideal gas in the presence of radiative cooling and idealized azimuthal magnetic field. The similarity solutions are investigated behind a cylindrical shock wave which is produced as a result of a sudden explosion or driven out by an expanding piston. The shock is assumed to be strong and propagates into a medium which is at rest, with nonuniform density. The total energy of the wave is assumed to be time dependent obeying a power law. Indeed, with the use of the similarity solution, the problem is transformed into a system of ordinary differential equations (ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.

One-dimensional unsteady fluid motion between two infinite walls

We considered an incompressible fluid motion driven by space-dependent body force. For a one-dimensional case, the problem was solved analytically, with the arbitrary choice of body force coordinate dependence. It was shown that unsteady fluid flow can be represented as a series of separate modes, each with its own characteristic response time.

Dynamics of Instabilities in a Viscous Fluid Rotating with an Acceleration

The flow of a viscous incompressible fluid between coaxial cylinders rotating with a constant rigid-body acceleration about the axis is studied numerically. A one-dimensional time-dependent solution of the Navier-Stokes equations is constructed analytically for the motion starting from the state of rest. On the initial time interval the one-dimensional unsteady fluid motion is unstable. Small perturbations introduced into the flow initiate the formation of secondary vortex flows with a velocity component along the axis. The dynamics of the developing instabilities and their dissipation are studied numerically. A condition determining the dimensions of the unsteady secondary flow zone is formulated. The unsteady regime is transient and starting from a certain instant of time the flow becomes stable.

Diagnosis of the properties of a structurized medium by long nonlinear waves

It was traditionally believed that long-wave processes in inhomogeneous media can be simulated within the framework of a homogeneous medium. It is known [1-3] that, at the acoustic level, the structure of a medium for long waves can be taken into account by means of disperse-dissipative properties of a homogeneous medium. At the same time, the evolution of finite-amplitude long waves is directly affected by the medium structure, as is shown by a rigorous mathematical analysis using asymptotic averaging [4-6l. This paper proves that the effect of the structure on nonlinear long-wave perturbations is so significant that one can predict the properties of the medium from the features of evolution of the wave field. 1. Asymptotic Averaged Model. Media with regular structures are elementary inhomogeneous media for which the effect of the structure can be analyzed. The mechanism of propagation of long-wave perturbations is studied for a periodic medium with equalization of stresses and mass velocities on the boundaries of adjacent components. It is assumed that the microstructural element of the medium is sufficiently large that the laws of classical continuum mechanics can be applied to it. The medium is barotropic. We consider media in a hydrodynamic approximation ignoring shear stresses. The specific volume V = p-1 and the sound velocity c are considered periodically varying properties of undisturbed media. One method of the averaged description of media of regular structure is that of asymptotic averaging [7, 8]. It is used to simulate long waves in compressible media [4]. Taking into account the well-known results for the case of plane symmetry [4-6], we derive an averaged system of equations for one-dimensional motion of arbitrary symmetry. The assumed equations of one-dimensional unsteady motion are the equations of motion for each individual component in Lagrangian variables:

On the periodic solutions of the problem of the one-dimensional unsteady motion of a fluid in a tube

The periodic motion of a fluid in a tube of finite length and circular cross-section is considered. The tube bas a unit at one end which changes the fluid flow rate. This unit is separated from the tube by a chamber which serves to control the flow rate and to dampen pressure oscillations. The problem of under what conditions such periodic motions are possible and what is their form and dependence on the period and durations of the first and second stages of the oscillation of the parameters determining the motion is considered.

On the evolution of plane detonations

Numerical solutions of the Navier–Stokes equations for the plane one-dimensional unsteady motion of a compressible, combustible gas mixture are used to follow the history of events that are initiated by addition of large heat power through a solid surface bounding an effectively semi­-infinite domain occupied by the gas. Plane Zel’dovich–von Neumann–Doring detonations eventually appear either at the precursor shock (which exists in every set of circumstances) or in the regions, occupied by an unsteady induction-domain and an initially quasi-steady fast-flame, that lie behind the precursor shock.

A new theory of the instability of a uniform fluidized bed

The form of the momentum equation for one-dimensional (vertical) unsteady mean motion of solid particles in a fluidized bed or a sedimenting dispersion is established from physical arguments. In the case of a fluidized bed that is slightly non-uniform this equation contains two dependent variables, the local mean particle velocity V and the local concentration ϕ, and several statistical parameters of the particle motion in a uniform bed. All these parameters are functions of ϕ with clear physical meanings, and the important ones are measurable. It is a novel feature of the equation that it contains two explicit contributions to the bulk modulus of elasticity of the particle configuration, one arising from the transfer of particle momentum by velocity fluctuations and one arising from the effective repulsive force exerted between particles in random motion. This latter contribution, which proves to be the more important of the two, is related to the gradient diffusivity of the particles, a key quantity in the new theory.The equation of mean motion of the particles and the equation of particle conservation are sufficient to determine the behaviour of a small disturbance with sinusoidal variation of V and ϕ in the vertical direction. Particle inertia forces in such a propagating wavy disturbance may promote amplitude growth, whereas particle diffusion tends to suppress it, and instability occurs when the particle Froude number exceeds a critical value. Rough estimates of the relevant parameters allow the criterion for instability to be put in approximate numerical from for both gas-fluidized beds (for which the flow Reynolds number at marginal stability is small) and liquid-fluidized beds of solid spherical particles (for which the Reynolds number is well above unity), although more information about the particle diffusivity in particular is needed. The predictions of the theory appear to be in qualitative accord with the available observational data on instability of gas- and liquid-fluidized beds.

Stability of certain one-dimensional unsteady motions of a viscous heat-conducting gas

Stability of certain one-dimensional unsteady motions of a viscous heat-conducting gas

One-dimensional unsteady motions of a viscous heat-conducting gas with a linear velocity distribution with respect to the coordinate

The one-dimensional motions of a perfect gas are considered in cases of spherical, cylindrical and plane symmetry, when the velocity is proportional to the distance to the center of symmetry. The solutions obtained are an extension of the known solutions of Sedov [1, 2] to the case of a viscous heat-conducting gas with a power-law temperature dependence of the coefficient of viscosity and thermal conductivity.

Stability of certain one-dimensional unsteady gas motions

This paper studies the stability of the adiabatic unsteady motions of a perfect gas for which the velocity is proportional to the distance to the center of symmetry. The solutions of the gas dynamic equations of such a form and their physical interpretation are considered in [1]. The stability of one of the solutions, which corresponds to the pulsations of uniform gravitating spheres, was studied in paper [2]. The stability of the other solutions (gas motions in the absence of Newtonian gravitation, nonperiodic motions of gravitating gas) has not previously been investigated. The stability of such solutions is investigated in this paper and it is shown that the motions in which gas compression occurs are, as a rule, unstable.

Semiempirical theory of the generation of discrete tones by a supersonic underexpanded jet flowing over an obstacle

i. G. G. Chernyi, Gas Flow with a High Supersonic Velocity [in Russian], Fizmatgiz, Moscow (1959). 2. Modern Problems of Gasdynamics [Russian translation], Mir, Moscow (1971). 3. O. S. Ryzhov and E. D. Terent'ev, "On the entropy layer in hypersonic flows containing compression shocks whose form is given by a power-law function," Prikl. Mat. Mekh., 34, No. 3 (1970). 4. O. S. Ryzhov and E. D. Terent'ev, "Theory of the high-entropy layer in hypersonic flows~' Zh. Vychisl. Mat. Mat. Fiz., Ii, No. 2 (1971). 5. V. G. Dulov, "Equations of three-dimensional steady gas flows in special dynamic variables," in: Proceedings of Second International Colloquium of the Gasdynamics of Detonation and Reactive Systems, Novosibirsk, 1969. Section of Numerical Methods in Gasdynamics [in Russian], Izd. Vychisl. Tsentr. Akad. Nauk SSSR, Moscow (1969). 6. J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, Massachusetts (1968). 7. S. S. Grigoryan, "The Cauchy problem and the piston problem for one-dimensional unsteady gas motions (self-similar motions)," Prikl. Mat. Mekh., 22, No. 2 (1958). 8. V. P. Korobeinikov, "Problems of the theory of a point explosion in gases," in: Proceedings of the V. A. Steklov Moscow Mathematics institute [in Russian] (1973). 9. G.M. Fikhtengol'ts, A Course in Differential and Integral Calculation [in Russian], Vol. 2, Nauka, Moscow (1969). I0. V. V. Sychev, "Method of small perturbations in problems of flow over thin blunt bodies by a hypersonic gas stream," Zh. Prikl. Mekh. Tekh. Fiz., No. 6 (1962).

Singular perturbations in one-dimensional unsteady motion of a real gas: PMM vol. 39, n≗ 6, 1975, pp. 1051–1059

Navier-Stokes equations for one-dimensional motion of gas are reduced to a special dimensionless form convenient for investigations involving a perturbation front. In new variables the transition from limit conditions of motion of an inviscid non-heat-conducting gas to the case of small but finite coefficients of viscosity and thermal conductivity, which is simulated by a perfect gas with singular perturbations induced by the indicated dissipative factors. We establish the inevitability of existence of two regions of singular perturbations, the neighborhood of the perturbation front and that of the point (line, surface) where the investigated motion is generated. The derivation of equations for both boundary layers, which is valid for a fairly general statement of problems of this kind, is presented and conditions of merging with the external (adiabatic) flow are formulated. Examples of computation of motion in boundary layers in problems of piston and point explosion are presented.

The effect of heat transfer on unsteady periodic motion of a gas in tubes

The one-dimensional unsteady motion of a gas in a straight tube is described with the aid of the equations of continuity, of motion, of state, and of energy; in addition, friction and thermal flow are assumed to be quasistationary. Solution of the linearized equations shows that when heat transfer is taken into account, “entropie” waves of pressure and velocity appear in the gas. As the result of heat transfer changes also occur in the propagation constants for ordinary waves; there is, compared with an adiabatic process, an increase in the distribution of friction.

A cavitation model in one-dimensional unsteady motion

A cavitation model in one-dimensional unsteady motion