Self-similar solution of the Navier-Stokes equations for a completely ionized hydrogen plasma (the plane piston problem)

The self-similar motion of a completely ionized hydrogen plasma is considered in the two-temperature hydrodynamic approximation, i.e., we consider the plane piston problem and the problem on energy release at a fixed wall. Results obtained by numerical integration of the relevant system of ordinary differential equations are quoted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 34–39, May–June, 1969.     

Shock-wave focusing in a three-temperature plasm

Improvements in numerical methods have now enabled one to calculate the convergence and focusing of a spherical shock-wave SW in a three-temperature plasma. There are differences from the one-temperature case [1, 2] and the two-temperature case (T_e and T_i) [3, 4] in that the calculations reveal new regularities at short distances from the center.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 52–55, July–August, 1984.I am indebted to V. A. Lykov and A. I. Zuev for useful discussions and advice.     

Comparison of different models for non-equilibrium CO2 flows in a shock layer near a blunt body

The paper presents results of a numerical simulation of a supersonic two-dimensional (2D) viscous flow containing CO₂ molecules near a spacecraft entering the Mars atmosphere. The gas–dynamic equations in the shock layer are coupled to the equations of non-equilibrium vibrational and chemical kinetics in the five-component mixture CO₂/CO/O₂/C/O. Transport and relaxation processes in the flow are studied on the basis of the rigorous kinetic theory methods; the developed transport algorithms are incorporated in the numerical scheme. The influence of the vibrational excitation of CO₂ and chemical reactions on the gas flow parameters and heat transfer is analyzed. The obtained results are compared with those found using two simplified models based on the two-temperature and one-temperature vibrational distributions in CO₂. The accuracy of the simplified models and the limits of their validity within the shock layer are evaluated. The effect of bulk viscosity in a flow near a re-entry body is discussed. The role of different diffusion processes, chemical reactions, and surface catalytic properties in a flow of the considered mixture in the shock layer is estimated.     

Energy integrals for mechanical systems with nonlinear nonholonomic constraints

The work analyzes energy relations for nonholonomic systems, whose motion is restricted by nonlinear nonholonomic constraints. For the mechanical systems with linear constraints, the analysis of energy relations was carried out in [1], [2], [3], [4], [5], [6] …. On the basis of corresponding Lagrange’s equations, a general law of the change in energy dε/dt is formulated for mentioned systems by the help of which it is shown that there are two types of the laws of conservation of energy, depending on the structure of elementary work of the forces of constraint reactions. Also, the condition for existing the second type of the law of conservation of energy is formulated in the form of the system of partial differential equations. The obtained results are illustrated by a model of nonholonomic mechanical system.     

Equations of hydromechanics and compression shock in two-velocity and two-temperature continuum with phase transformations

The equations of motion of multiphase mixtures have been considered in [1–10] and several other studies. In [1] it is proposed that the mixture motion be considered as an interpenetrating motion of several continua when velocity, pressure, mean density, concentration, etc., fields for each phase are introduced in the flowfield. The equations of motion are written separately for each phase, and the force effect of the other components is considered by introducing the interaction forces, which for the entire system are internal. The assumption of component barotropy is used to close the system.The energy equations are used in [2, 3] in place of the component barotropy assumption. Moreover, mixtures without phase transformations are considered. In [4] an analysis is made of the equations of turbulent motion with account for viscous forces for a two-velocity, but single-temperature medium in which equilibrium phase transformations are assumed, i. e., a two-phase medium is considered in which the phase temperatures are the same, the composition is equilibrium, but the phase velocities are different. In [5] the equations are written on the interface in a multicomponent medium consisting of barotropic fluids. A discontinuity classification is also presented here. In the aforementioned work [3] the equations on the shock are written for a continuum with particles without the use of the property of barotropy of the carrier fluid. Various different aspects of the motion of multiphase mixtures are considered in [6–11], for example, the effect of particle collisions with one another, the effect of the volume occupied by the particles on the parameters stream, shock waves, etc. In [7] a study is made of the force effect of an agitated medium on a particle on the basis of the Basset-Boussinesq-Oseen equation.In the following we derive the equations of motion of a two-velocity and two-temperature continuum with drops or particles with nonequilibrium phase transformations, i. e., a medium in which the phase velocities and temperatures are different and the composition may be nonequilibrium. In addition, we study the effect of the presence of particles or drops on the gas parameters behind a shock. Further, the equations obtained here are used to study compression waves, and in particular shock waves.The author wishes to thank Kh. A. Rakhmatulin, S. S. Grigoryan, and Yu. A. Buevich for helpful discussions and valuable comments.     

Some self-similar solutions of two-temperature hydrodynamic equations with allowance for nonlinear heat conduction and exothermic reactions

Processes occurring in plasma produced as a result of the interaction of powerful radiation fluxes with matter can be divided into three stages: absorption of radiation on the matter boundary, subsequent heating and compression of the central part of the target for the purpose of creating the conditions necessary for the initiation of an exothermic reaction and, finally, propagation of an exothermic reaction wave through the ambient matter. The present paper is devoted to an investigation of the last stage, a reaction wave igniting initially cold matter. The main method for the theoretical investigation of the processes described is a numerical solution of the equations of motion of a two-temperature gas with allowance for the physical processes occurring in a completely ionized medium: electron heat conduction, radiative losses, energy transfer between electrons and ions, and others. In view of the complicated nonlinear nature of the system of partial differential equations describing the process, searches for possible self-similar solutions are of interest. These solutions can be used as tests in calculating a complete system of equations; by means of them it is also possible to investigate asymptotic laws of exothermic reaction wave propagation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 145–151, March–April, 1986.     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.     

Instability of a layer op lightly ionized plasma in crossed electric and magnetic fields

In a lightly ionized plasma, charged-particle drift due to collisions with neutral atoms occurs at different velocities: $$\begin{array}{*{20}c} {v_{Ea} = \mp \frac{{b_a E}}{{1 + (\omega _a \tau _a )^2 }},v_{ \bot a} = \frac{{b_a E(\omega _a \tau _a )}}{{1 + (\omega _a \tau _a )^2 }}} \\ {\left( {b_a = \frac{{|e|\tau _a }}{{m_a }},\omega _a = \frac{{|e|\tau _a }}{{m_a }}} \right),} \\ \end{array} $$ where b_a is the mobility of particles of the type a;ω_a is the Larmor frequency; the upper sign refers to electrons and the lower sign to ions. A difference in the charged-particle drift velocities can cause instability of an inhomogeneous lightly ionized plasma. Let us consider the following example. Assume that in the initial state of the plasma there is a concentration gradient along the x-axis, that the external electric field is directed along the x-axis, and that the magnetic field coincides with the z-axis. In this system, under the influence of a Lorentz force the charged particles will move in a direction opposite to the y-axis. Since electrons have a higher velocity than ions, an electric field is induced in this direction. This electric field, together with the magnetic field, causes particle drift in the negative direction of the x-axis. Consequently, if the concentration gradient in the initial state is directed opposite to the x-axis this state cannot be stable. Instability of this kind has been examined by Simon [1]. On the basis of studies by Kadomtsev and Nedospasov [2], as well as by Rosenbluth and Longmire [3], Simon developed a theory of instability of a lightly ionized plasma in crossed fields with an inhomogeneous density distribution in the direction of the external electric field. Somewhat later, Simon's theory was developed [4]. In devices with inhomogeneous plasma flow in which the plasma (conducting) layers alternate with nonconducting layers, the external electric field and concentration are normal to one another. We shall bear this case in mind below and shall examine the instability of a lightly ionized plasma in crossed fields when the concentration inhomogeneity is in a direction perpendicular to the external electric field.     

A solar system model with dissipation

A problem of motion for an arbitrary number of planets is discussed with consideration of the forces of gravitational interaction according to the law of universal gravitation. The planets are assumed to be homogeneous viscoelastic spheres. In the process of motion, the planets are deformed and the dissipation of energy takes place due to internal viscous forces. On the basis of the motion separation method, an approximate system of equations is obtained to describe the motion of planet centers of mass and the variation of planet angular momenta with respect to the centers of mass. The equations of motion contain small conservative corrections to the law of universal gravitation and small dissipative forces whose influence causes a decrease of the total mechanical energy. The motion under consideration admits the following first integral: the law of angular momentum conservation for the system with respect to the centers of mass. When the system executes the steady motion corresponding to its rotation with a constant angular velocity as a rigid body, the dissipative forces do not perform work, since the deformed planets have no time-dependent deformations.     

Articular cartilage with intra- and extrafibrillar waters – mass transfer and generalized diffusion

The present analysis complements the chemo-mechanical model of articular cartilage developed in Loret and Simões [Loret, B., Simões, F.M.F., 2004. Articular cartilage with intra- and extrafibrillar waters. A chemo-mechanical model. Mech. Mater. 36 (5–6), 511–541; Loret, B., Simões, F.M.F., 2005a. Mechanical effects of ionic replacements in articular cartilage. Part I – The constitutive model. Biomech. Model. Mechanobiol. 4 (2–3), 63–80. Part II – Simulations of successive substitutions of NaCl and CaCl₂, 81–99], where only equilibria were considered, and therefore time was absent. The focus here is, first, to present how transport phenomena are aggregated to the porous media framework, and, second, to detail the constitutive equations of these transports. Indeed, these equations are developed in the context of a three-phase multi-species electro-chemo-mechanical model that accounts for the effects of two water compartments, namely intrafibrillar water stored between collagen fibrils and extrafibrillar water covering the negatively charged proteoglycans. The electrolyte circulating the two fluid phases contains ions sodium Na⁺, calcium Ca²⁺ and chloride Cl⁻.Species diffuse within their phase. They transfer from one fluid phase to the other. The various sources of dissipation are built in a thermodynamic framework, segregated and decoupled via the Clausius–Duhem inequality.Linear and non-linear equations of mass transfer are proposed along an onsagerist approach.The generalized diffusion in the extrafibrillar compartment accounts for Darcy's law of seepage through the porous solid skeleton, Fick's law of ionic diffusion, and Ohm's law of electric flow. An original derivation of the constitutive equations of generalized diffusion is proposed. Indeed, the dissipation inequality is written in two forms, which are required to be equivalent. This approach has the advantage of delivering the general structure of the diffusion matrix. It also displays in explicit form the degrees of freedom for possible refinements. Simple assumptions, phrased in terms of entities that are standard in transport of porous media, allow to recover arrowhead diffusion matrices. Comparison with an earlier proposal is detailed.An osmotic coefficient is found to be hidden in the equations, and anomalous negative osmosis is observed to take place for both sodium chloride and calcium chloride electrolytes.Finally, an experimental setup to measure transport properties is analyzed. The model describes correctly the increase and leveling of the experimental diffusion coefficient, and no additional ad hoc constitutive assumptions are needed in contrast to some suggestions in the literature.The results are presented for sodium chloride NaCl and calcium chloride CaCl₂.     

The DRBEM solution of incompressible MHD flow equations

This paper presents a dual reciprocity boundary element method (DRBEM) formulation coupled with an implicit backward difference time integration scheme for the solution of the incompressible magnetohydrodynamic (MHD) flow equations. The governing equations are the coupled system of Navier‐Stokes equations and Maxwell's equations of electromagnetics through Ohm's law. We are concerned with a stream function‐vorticity‐magnetic induction‐current density formulation of the full MHD equations in 2D. The stream function and magnetic induction equations which are poisson‐type, are solved by using DRBEM with the fundamental solution of Laplace equation. In the DRBEM solution of the time‐dependent vorticity and current density equations all the terms apart from the Laplace term are treated as nonhomogeneities. The time derivatives are approximated by an implicit backward difference whereas the convective terms are approximated by radial basis functions. The applications are given for the MHD flow, in a square cavity and in a backward‐facing step. The numerical results for the square cavity problem in the presence of a magnetic field are visualized for several values of Reynolds, Hartmann and magnetic Reynolds numbers. The effect of each parameter is analyzed with the graphs presented in terms of stream function, vorticity, current density and magnetic induction contours. Then, we provide the solution of the step flow problem in terms of velocity field, vorticity, current density and magnetic field for increasing values of Hartmann number. Copyright © 2010 John Wiley & Sons, Ltd.     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

LINEAR AND NONLINEAR DYNAMICS OF CANTILEVERED CYLINDERS IN AXIAL FLOW. PART 2: THE EQUATIONS OF MOTION

In this paper a nonlinear equation of motion is derived for the dynamics of a slender cantilevered cylinder in axial flow, generally terminated by an ogival free end. Inviscid forces are modelled by an extension of Lighthill's slender-body work to third-order accuracy. The viscous, hydrostatic and gravity-related terms are derived separately, to the same accuracy. The equation of motion is obtained via Hamilton's principle. The boundary conditions related to the ogival free end are also derived separately. The paper is concluded by a discussion of the methods used to obtain the solutions presented in Part 3 of this study.     

Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source

In this work, a problem of thermoelastic interactions in an elastic infinite medium with cylindrical cavity thermally shocked at its bounding surface and subjected to moving heat source with constant velocity has been solved. The governing equations are taken in the context of two-temperature generalized thermoelasticity theory (Youssef model). The analytical solution with direct approach in the Laplace transforms domain has been obtained. The derived analytical expressions have been computed for specific situations. Numerical results for the dynamical and conductive temperatures, stress, strain, and displacement are represented graphically with comparisons by one-temperature generalized thermoelasticity (Lord–Shulman model).     

Equations of electrohydrodynamics of weakly ionized aerosols with diffusion charging of dispersed phase particles

A closed model describing the motion of weakly ionized aerosols with allowance for dispersed phase particle charging processes due to ion deposition is constructed within the framework of continuum mechanics [1]. Both the general process of particle charging in a weakly ionized gas and its limiting cases, where the limiting stage of the process is the diffusion of the ions towards the particles or the reactions leading to their deposition on the particles, are investigated. Expressions are obtained for the positive and negative ion flows to a particle in a weakly ionized gas. The basic equations of electrohydro-dynamics of weakly ionized aerosols, in which the dispersed phase particle charging mechanism in question leads to the interphase transfer of elctrical charge, are formulated. Cases where the system of equations of electrohydrodynamics obtained can be simplified by replacing the differential equations of motion and charging of the dispersed phase and, moreover, the positive and negative ion balance equations by algebraic relations such as Ohm's law and Saha's equation are investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 54–60, January–February, 1986.     

A streamfunction–velocity approach for 2D transient incompressible viscous flows

Electrodeposition is a widely used technique for the fabrication of high aspect ratio microstructures. In recent years, much research has been focused within this area aiming to understand the physics behind the filling of high aspect ratio vias and trenches on substrates and in particular how they can be made without the formation of voids in the deposited material. This paper reports on the fundamental work towards the advancement of numerical algorithms that can predict the electrodeposition process in micron scaled features. Two different numerical approaches have been developed, which capture the motion of the deposition interface and 2‐D simulations are presented for both methods under two deposition regimes: those where surface kinetics is governed by Ohm's law and the Butler–Volmer equation, respectively. In the last part of this paper the modelling of acoustic forces and their subsequent impact on the deposition profile through convection is examined. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-equilibrium relativistic M.H.D. with finite electrical conductivity

A system of balance laws for relativistic m.h.d, with finite eIectrical conductivity, heat flux and viscosity is proposed, starting from the properties of the systems of conservation laws compatible with a supplementary balance law (entropy balance). Adopting a two-fluid scheme the plasma is treated as a mixture of a neutral fluid and a charged fluid. Following the approach ofextended thermodynamics heat flux, viscous stress and electric current density are considered as new field variables contributing to non equilibrium entropy density and flux.     

Shock wave structure for a fully ionized plasma

We study planar shock wave structure in a two-temperature model of a fully ionized plasma that includes electron heat conduction and energy exchange between electrons and ions. For steady flow in a reference frame moving with the shock, the model reduces to an autonomous system of ordinary differential equations which can be numerically integrated. A phase space analysis of the differential equations provides an additional insight into the structure of the solutions. For example, below a threshold Mach number, the model produces continuous solutions, while above another threshold Mach number, the solutions contain embedded hydrodynamic shocks. Between the threshold values, the appearance of embedded shocks depends on the electron diffusivity and the electron–ion coupling term. We also find that the ion temperature may achieve a maximum value between the upstream and downstream states and away from the embedded shock. We summarize the methodology for solving for two-temperature shocks and show results for several values of shock strength and plasma parameters in order to quantify the shock structure and explore the range of possible solutions. Such solutions may be used to verify hydrodynamic codes that use similar plasma physics models.