Some characteristic features of diffusion of a magnetic field into a moving conductor

The study of the diffusion of a magnetic field into a moving conductor is of interest in connection with the production of ultra-high-strength magnetic fields by rapid compression of conducting shells [1,2]. In [3,4] it is shown that when a magnetic field in a plane slit is compressed at constant velocity, the entire flux enters the conductor. In the present paper we formulate a general result concerning the conservation of the sum current in the cavity and conductor for arbitrary motion of the latter. We also consider a special case of conductor motion when the flux in the cavity remains constant despite the finite conductivity of the material bounding the magnetic field.Notation ₁, * flux which has diffused into the conductor - ₂ flux in the cavity - ₀ sum flux - r radius - r* cavity boundary - thickness of the skin layer - (r) delta function of r - t time - q intensity of the fluid sink - v velocity - flux which has diffused to a depth larger than r - x self-similar variable - dimensionless fraction of the flux which has diffused to a depth larger than r - * fraction of the flux which has diffused into the conductor - a conductivity - c electrodynamic constant - R_m magnetic Reynolds number - dimensionless parameter     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

Laminar assisting mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties

The steady laminar boundary layer flow, with an external force, along a vertical isothermal plate is studied in this paper. The external force may be produced either by the motion of the plate or by a free stream. The fluid is water whose density-temperature relationship is non-linear at low temperatures and viscosity and thermal conductivity are functions of temperature. The results are obtained with the numerical solution of the boundary layer equations with , k and variable across the boundary layer. Both upward and downward flow is considered. It was found that the variation of , k and with temperature has a strong influence on mixed convection characteristics.Nomenclature c_p water specific heat - f dimensionless stream function - g gravitational acceleration - Gr_x local Grashof number - k thermal conductivity - Nu_x local Nusselt number - Pr Prandtl number - Pr_a ambient Prandtl number - Re_x local Reynolds number - s salinity - T water temperature - T_a ambient water temperature - T_o plate temperature - u vertical velocity - u_a free stream velocity - u_o plate velocity - v horizontal velocity - x vertical coordinate - y horizontal coordinate - pseudo-similarity variable - nondimensional temperature - dynamic viscosity - _f film dynamic viscosity - _o dynamic viscosity at plate surface - kinematic viscosity - buoyancy parameter - water density - _a ambient water density - _f film water density - _o water density at plate surface - physical stream function     

Screening of a surface vaporizing under laser bombardment,at temperature and ionization disequilibrium

The kinetics of the process through which vapor released from the surface of a solid is heated by laser radiation are calculated. The data indicate the existence of temperature and ionization disequilibrium (for Al at incident flux densities q over 50 MW/cm² in times of the order of 1sec or less). Treatment of this disequilibrium lowers the critical flux density q_* corresponding to the onset of screening within a specified time t, or within the time t_* required for screening to develop at a specified flux density q, appreciably, as compared to the values obtained in the equilibrium theory of the process. Some of the other physical processes that might have some effect on flare behavior are also discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 35–45, September–October, 1971.In conclusion, the authors express their thanks to A. I. Petrukhin and V. I. Bergel'son for valuable discussions and for kindly making available experimental and theoretical data, to V. A. Onishchuk for performing calculations of the absorption coefficients, and to V. V. Novikov for helpful assistance in the calculations.     

Two-dimensional flow of a perfect conducting gas in crossed electric and magnetic fields

Reference [1, 2] give a solution of the problem of the two-dimen-, sional flow of an inviscid thermally-nonconducting gas with constant conductivity in a channel of constant cross section for particular forms of the given applied magnetic field. The present paper obtains a solution of the problem of the two-dimensional flow of a gas with variable conductivity in crossed electric and arbitrary magnetic fields by means of the small parameter method. The magnetic Reynolds number R_m and the magnetohydrodynamic interaction parameter S are chosen as parameters. The international system of units is employed.Notation V flow velocity - j electric current density - p pressure in the flow - E electric field strength - gas density - electrical conductivity of the gas - T gas temperature - ratio of specific heats at constant pressure and volume - L channel half-height - ] permeability (magnetic) - B magnetic induction vector - B₀ applied magnetic field     

Dynamic Response of Non-Linear Systems to Random Trains of Non-Overlapping Pulses

The stochastic excitation considered is a random train of rectangular, non-overlapping pulses, with random durations completed at latest at the next pulse arrival. For Erlang distributed interarrival times and for the actual distributions of pulse durations determined from the primitive Erlang distribution, the formulation of the problem in terms of a Markov chain allows to evaluate the mean value, the autocorrelation function and the characteristic function of the excitation process. However, the state vector of the dynamical system is a non-Markov process. The train of non-overlapping pulses with parameters , 1 and , 1 is then demonstrated to be a process governed by a stochastic equation driven by two independent Poisson processes, with parameters and , respectively. Hence, the state vector of the dynamical system augmented by this additional variable becomes a Markov process. The generalized Itôs differential rule is then used to derive the equations for the characteristic function and for moments of the response of a non-linear oscillator.     

Reynolds number dependence of the drag coefficient for laminar flow through fine-scale photoetched screens

The laminar steady flow downstream of fine-mesh screens is studied. Instead of woven-wire screens, high-uniformity screens are fabricated by photoetching holes into 50.8 m thick Inconel sheets. The resulting screens have minimum wire widths of 50.8 m and inter-wire separations of 254 m and 318 m for the two screens examined. A flow facility has been constructed for experiments with these screens. Air is passed through the screens at upstream velocities yielding wire width Reynolds numbers from 2 to 35. To determine the drag coefficient, pressure drops across the screens are measured using pressure transducers and manometers. Threedimensional flow simulations are also performed. The computational drag coefficients consistently overpredict the experimental values. However, the computational results exhibit sensitivity to the assumed wire cross section, indicating that detailed knowledge of the wire cross section is essential for unambiguous interpretation of experiments using photoetched screens. Standard semi-empirical drag correlations for woven-wire screens do not predict the present experimental results with consistent accuracy.List of symbols A ₁, A ₂ screen aspect ratios - c _d screen drag coefficient - d woven-wire diameter - D photoetched minimum wire width (spanwise) - f woven-wire screen drag function - M distance between adjacent wires - N spectral-element order - o woven-wire open area fraction - O photoetched open area fraction - _p pressure drop across screen - Re _d woven-wire diameter Reynolds number - Re _D photoetched wire width Reynolds number - U fluid velocity upstream of screen - W photoetched sheet thickness (streamwise) - x, y, z spatial coordinates - fluid density - fluid viscosity     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Heat transfer in a laminar wall jet over a curved surface

The steady state heat transfer characteristics of the wall jet over a curved surface are obtained for constant wall temperature and constant wall heat flux boundary conditions. Both concave and convex curvatures have been considered. Numerical results for the temperature distribution are obtained and solutions for the wall values of the temperature functions have been tabulated for Prandtl number ranging from 0.01 to 100 while the curvature parameter was varied from –0.03 to 0.07.Nomenclature f velocity profile function - h heat transfer coefficient - K thermal conductivity - Nu Nusselt number - Pr Prandtl number - q _w heat flux at the wall - Re Reynolds number - R ₀ surface radius of curvature - T temperature - U characteristic velocity - u velocity component in x direction - v velocity component in y direction - x distance parallel to the surface - y distance normal to the surface - curvature parameter - dimensionless coordinate - dimensionless temperature - dynamic viscosity - kinematic viscosity - fluid density - shear stress - w conditions at the wall - conditions far away from the surface     

Heat transfer to a biased wall of an electric arc chamber

A DC arc is operated in a rectangular arc chamber with currents of 50 to 150 amperes and at pressures of 100 to 760 mm Hg in an axially superimposed flow of argon. A segment of the chamber wall between cathode and anode is biased both positively and negatively relative to the cathode and the corresponding current-voltage measurements at the wall segment reveal two different types of characteristics depending on the particular arc parameters. One of the characteristics shows electron current saturation accompanied by a breakdown forming a secondary arc. The other type of characteristic shows neither electron current saturation nor breakdown. The heat transferred to the wall segment reflects the behavior or the current-voltage characteristics and suggests that variations in the potential distribution of the plasma in the immediate vicinity of the surface of the biased segment and/or changes of the electron flux are responsible for the observed effects.

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Zusammenfassung Ein axial angeblasener Gleichstrombogen in Argon wird in einer Entladungskammer mit rechteckigem Querschnitt mit Strömen im Bereich von 50–150 amp und Drucken von 100 bis 760mm Hg betrieben. Ein Segment der unterteilten Entladungskammer wird mit sowohl negativen als auch positiven Spannungen bezüglich der Bogenkathode beaufschlagt und die entsprechenden, gemessenen Strom-Spannungskennlinien fallen in zwei verschiedene Kategorien, die durch die Bogenparameter bestimmt werden. Die erste Kategorie ist durch Elektronensättigungsströme mit nachfolgendem elektrischem Durchbruch und der Ausbildung eines Sekundärbogens gekennzeichnet, während die zweite Kategorie weder Elektronensättigung noch elektrischen Durch-bruch zeigt. Der Wärmeübergang zu diesem Segment spiegelt das Verhalten der Strom-Spannungskennlinien wieder, was darauf hinweist, daß Änderungen der Potentialverteilung in unmittelbarer Nähe der Oberfläche und/oder Anderungen des Elektronenstromes für die beobachteten Effekte verantwortlich sind.

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Nomenclature C mean thermal speed - e elementary charge - E_i ionization potential - I arc current, or total current to biased segment - j current density - K_B Boltzmann constant - n number density - P pressure - Q heat flow - q_b heat flux to wall segment carried by the current - q_conv heat flux to wall segment due to convection - q_w total heat flux to wall segment - t time - T temperature - U potential of surface - U₁ potential at edge of sheath - V superimposed gas velocity - _D Debye length - mean free path - work function of wall segment material - e electrons - i ions - f at floating potential     

Zum Impuls-, Wärme- und Stoffaustausch in turbulenten Strömungen reagierender Binärgemische

Zusammenfassung Für ein reagierendes Binärgemisch konstanter Dichte werden die Reynolds'schen Gleichungen angegeben. Die Transportkoeffizienten sowie die Wärmekapazitäten der beiden Komponenten werden als konstant angenommen. Für die unbekannten Reynolds'schen Terme werden Transportgleichungen angegeben. Weiter wird der Einfluß der Turbulenz auf die chemische Produktionsdichte diskutiert.

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About the transfer of momentum, heat and mass in turbulent flows of binary mixturesPart I: The reynolds equations and the transport equations

The Reynolds equations for a reacting binary mixture of constant density are given. The transport coefficients as well as the specific heats of the components are assumed to be constant. Transport equations for the unknown Reynolds-terms are given. The influence of turbulence on the chemical production of species in discussed.

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Formelzeichen c Massenkonzentration - c_p soezifische Wärme bei konstantern Druck - D binärer Diffussionskoeffizient - h spezifische Enthalpie - h_o=h + v _k ² /2 totale spezifische Enthalpie - h^o Reaktionsenthalpie - j_k Massendiffusionsstromvektor - k Reaktionsgeschwindigkeitskonstante - p Druck - Pr= c_p/ Prandtl-Zahl - q²/2 kinetische Energie der Schwankungsbewegung - q_k Energiestromvektor - R universelle Gaskonstante - Sc=/D Schmidt-Zahl - t Zeit - T absolute Temperatur - v_k Geschwindigkeitsvektor - x_k Ortsvektor Griechische Symbole Dissipationsfunktion - Wärmeleitfähigkeit - dynamische Viskosität - =/ kinematische Viskosität - Dichte - Produktionsdichte - _jk viskoser Spannungstensor Indizes auf die Komponente bezogen - 1 auf die Komponente 1 bezogen - 2 auf die Komponente 2 bezogen - mol molekularer Anteil - tur turbulenter Anteil - res resultierender Anteil     

Combined natural and forced convection in a laminar wall jet along a vertical plate with uniform surface heat flux

The steady state flow and heat transfer characteristics of the combined natural and forced convection in a two dimensional, laminar, incompressible wall jet over a vertical wall are obtained for constant wall heat flux boundary condition. The velocity and temperature distribution are assumed to be power series, where the zeroth term corresponds to that for a plane wall jet in the absence of buoyancy effects. Numerical results for the momentum and thermal series functions are presented for a Prandtl number of 0.73. Wall values of the momentum and thermal series functions are presented for Prandtl numbers ranging from 0.01 to 1000.Nomenclature Gr* modified Grashof number - k thermal conductivity - Nu Nusselt number - Pr Prandtl number - q _w heat flux at the wall - Re Reynolds number - T temperature - u velocity component in x-direction - v velocity component in y-direction - x co-ordinate along the plane wall - y co-ordinate normal to the wall - () gamma function - non-dimensional co-ordinate defined in (6) - non-dimensional temperature - dynamic viscosity - kinematic viscosity - non-dimensional co-ordinate defined in (6) - density - w values at the wall - values at large distances away from the wall     

Dependence of the stability criterion of nucleate boiling on the compressibility of the vapor

The article gives the results of a study of the motion of bubbles and their deformation near the heating surface at different pressures. It was observed that, during the time of their growth, the gaseous medium in the bubbles is in a compressed state.Nomenclature R) radius of bubble - R_h) maximul radius of a deformed bubble in the horizontal plane - R_v) maximal radius of a deformed bubble in the vertical plane - ) specific weight - B) universal gas constant - ) surface-tension coefficient - p) pressure - ) edge wetting angle - g) acceleration due to gravity - V) volume - ) molecular weight - C_T) isothermal velocity of sound Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 77–81, July–August, 1971.     

Geometric properties of two phase flow in geothermal reservoirs

The two phase flow equations frequently used in geothermal engineering ignore capillary pressure, which results in a singular system of equations. Analysis of these equations reveals three mechanisms for altering saturation: local boiling, the spatial dependence of flowing enthalpy due to the convective transport of fluid, and counterflow. A scalar function is associated with each of these three mechanims. At each point in space, flows are essentially two dimensional, with gravity establishing a vertical hierarchy, in that volumetric, energy and mass fluxes can never point below a lower member in this triple. With increasing liquid saturation, the characteristics associated with the saturation equation move up from below this grouping of directions, and eventually may even point above volumetric fluxes. Finally, weak shocks and the associated entropy condition are considered. The characteristics of the saturation equation coincide with the velocity of extremely weak shocks, and saturation increases with the passage of a weak shock, provided the magnitude of the characteristic speed increases with saturation.Notation C_l liquid heat capacity - C_m rock heat capacity - C_v vapour heat capacity - G counterflow energy flux - h flowing enthalpy - hl liquid enthalpy - h_v vapour enthalpy - k permeability - k downward vector - P pressure - S liquid saturation - T temperature - dT/dP derivative at saturation - z vertical coordinate - l liquid viscosity - v vapour viscosity - Pl liquid density - _m rock density - _v vapour density - porosity     

A simplified model of bubble-driven flow in an axisymmetric container

A simplified mathematical model simulating a gas bubble agitation system is here examined for the case when the orifice is located at the centre of the base of a cylindrical vessel. The two phase flow which is confined to a cone region is approximated by the drift flux model. The governing equations for the recirculating liquid flow are quai-linearized and the flow domain is transformed by a simple transformation into a cylindrical region. Using standard finite difference techniques numerical solutions are obtained for Reynolds numbers in the range 0–10⁴; Re=U ₀ R ₀/ _eff, where R ₀ is the radius of the vessel, U ₀ the velocity of injection of the gas, the density of the liquid phase & _eff the constant effective turbulent viscosity. For large Re it is shown that the primary recirculating flow is confined to a narrow region adjacent to the two phase/liquid interface.     

Untersuchungsergebnisse über die kapillare Leitfähigkeit von Baustoffen

Zusammenfassung Krischer hat die kapillare Flüssigkeitsbewegung als Potentialströmung beschrieben, deren Ursache ein Feuchtegefälle ist und führte als Stoffeigenschaft die Flüssigkeitsleitzahl als Funktion des Feuchtegehaltes ein. Trennt man durch einen modifizierten Ansatz Kapillar- und Reibungskräfte, so erhält man Kapillarfunktionen, die für den Fall der stationären Strömung bei horizontaler Flüssigkeitsbewegung oder bei lotrechter Flüssigkeitsbewegung unter Vernachlässigung der Schwerkraft in der Krischerschen Flüssigkeitsleitzahl (Kapillarleitkoeffizient) zusammengefaßt werden können.Diese Kapillarfunktionen für Wasser wurden von Quarzsand, Ziegel, Kalksandstein, Gasbeton und Bimsbeton ermittelt und der Kapillarleitkoeffizient als Funktion des Feuchtegehaltes für den Befeuchtungsvorgang angegeben. Zur experimentellen Bestimmung des Feuchtegehaltes war das Durchstrahlungsverfahren mit Gammastrahlen gewählt worden, um den volumenbezogenen Feuchtegehalt während eines quasistationären Vorganges der kapillaren Flüssigkeitsbewegung in Abhängigkeit von Zeit und Ort ohne Störung des Vorganges ermitteln zu können.

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Results of investigations on the capillary motion of moisture in building materials

Krischer described the capillary motion of moisture as a water transfer proportional to the gradient of water content by volume, and defined a coefficient of capillary conductivity as a function of moisture content. Equations of general validity, however, can be developed by separation in terms for capillary and gravity forces and capillary resistance. These capillary functions can be transferred in the coefficient for processes with horizontal motion and for those cases where gravity does not have any impact on the motion in small capillary pore spaces.The capillary functions and the coefficients of capillary conductivity for quasi-steady processes of humidification were determined of quartz sand, brick, sandlime brick, cellular concrete and pumice concrete. The temporally and locally changing moisture content during capillary rising tests was measured non-destructively by means of the attenuation effect of penetrating gamma rays.

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Formelzeichen F Stoffquerschnitt - H() feuchtigkeitsabhängige maximale kapillare Steighöhe - H_max maximale kapillare Steighöhe beim maximalen Feuchtegehalt - I₀ Intensität der auffallenden Gammastrahlung - I Intensität der durchfallenden Gammastrahlung - R() feuchtigkeitsabhängiger kapillarer Reibungskoeffizient - R_max kapillarer Reibungskoeffizient beim maximalen Feuchtegehalt - V Volumstrom - h kapillare Steighöhe - q_S Volumanteil des Feststoffes - q_W Volumanteil des Wassers - q_L Volumanteil der Luft - s Weglänge - t Zeit - x Schichtdicke - y Impulszahl - Neigungswinkel gegen die Lotrechte - statistischer Fehler bei der Impulsmessung - Kapillarleitkoeffizient bzw. Flüssigkeitsleitzahl na ch Krischer - Schwächungskoeffizient für Gammastrahlen - Dichte - / Massenschwächungskoeffizient - volumenbezogener Feuchtegehalt - _max maximaler volumenbezogener Feuchtgehalt - _S Schwächungskoeffizient des Feststoffes - _W Schwächungskoeffizient des Wassers - _L Schwächungskoeffizient der Luft Herrn Professor Dr.-Ing. H. Glaser, Stuttgart, zum 70. Geburtstag gewidmet.Die Untersuchungen erfolgten mit Mitteln der AIF (Arbeitsgemeinschaft industrieller Forschungsvereinigungen e.V., Köln). Der Aufbau der Versuchs-anordnung und die Gammastrahlungsmessungen mit Auswertung wurden von H. Perk durchgeföhrt, der zugleich der för den Strahlenschutz Verantwortliche des Instituts im Sinne des § 20 der I. Strahlenschutzverordnung ist.     

Screening of selective radiation in a boundary layer

An analysis of numerous calculations of the flow and radiative-convective heat exchange in a hypersonic shock layer near a blunt body, both at an impermeable surface [1] and in the presence of ablation [1–4], made it possible to establish some relationships connected with the screening in the boundary layer of radiation from the high-temperature part of the shock layer. It was established that the dependence of the screening coefficient of the boundary layer, which consists of the ratio of the integral flux q⁺ _RW of radiation incident on the body to the flux q⁺ _RW arriving at the outer limit of the boundary layer from the high-temperature region behind the shock wave, on the radius has a nonmonotonic character. In this case the absolute value of the minimum _min is determined mainly by the temperature T_S behind the shock wave and by the injection velocityf _W = (v)_W/(v), while its position (R_min) is determined by the pressure P_S in the shock layer. An analysis of the spectral characteristics of the screening of the boundary layer showed that the region of wavelengths > 0.5 m is very important from the point of view of the radiative heating and destruction of the surface. Components which efficiently absorb high-temperature radiation in this spectral interval are absent from the gaseous products of the disintegration of specific heat-protection coatings. An analysis is made permitting an evaluation of the possibility of introducing into the coatings additives possessing those optical and thermochemical properties which would provide for screening of radiation in this part of the spectrum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 71–77, September–October, 1978.The authors thank A. B. Karasev for advice and a discussion.     

On the properties of compressible gas flow in a porous media

The flow of an adiabatic gas through a porous media is treated analytically for steady one- and two-dimensional flows. The effect on a compressible Darcy flow by inertia and Forchheimer terms is studied. Finally, wave solutions are found which exhibit a cut-off frequency and a phase shift between pressure and velocity of the gas, with the velocity lagging behind the pressure.Nomenclature A area of tube for one-dimensional flow - B drag coefficient associated with Forchheimer term - c speed of sound - M Mach number - p ^* gas pressure - p dimensionless gas pressure - s coordinate along the axis of tube - t ^* time variable - t dimensionless time variable - V^* gas velocity in the porous media - V dimensionless gas velocity Greek Letters ratio of specific heat capacities - phase angle between gas pressure and velocity for linear waves - parameter indicating the importance of the inertia term - viscosity - _p natural frequency of the porous media - ^* gas density - dimensionless gas density - parameter indicating the importance of the Forchheimer term - porosity of porous media - velocity potential - stream function     

Codimension two bifurcations of symmetric cycles in Hamiltonian systems with an antisymplectic involution

Consider a Hamiltonian system with parameters, such that there exists an involution which reverses this Hamiltonian system. Let us assume the linear part L at =0 has only nonzero purely imaginary eigen-values ±ib₁,..., ±ib_n. In this paper, we classify the typical bifurcations of families of symmetric periodic solutions of this system at resonance if b_i/b_j=±1, ±2, or ±1/2 and the number of parameters needed is one or two. First, one puts the Hamiltonians into a convenient normal form. Next, applying a Lyapunov-Schmidt reduction and making further manipulations, one can geta reduced bifurcation equation which can possess certain symmetry. Finally, by using elementary methods from singularity theory or isotopy methods, one obtains the desired bifurcation diagrams.     

Self-similar solutions for displacement of non-Newtonian fluids through porous media

This paper presents a class of self-similar solutions describing piston-like displacement (single-phase flow is included as a special case) of one slightly compressible non-Newtonian, power-law, dilatant fluid by another through a homogeneous, isotropic porous medium. These solutions can be used to evaluate the validity and accuracy of existing approximate solutions, such as the assumption of constant flow rate at each radial distance that Ikoku and Ramey use to linearize the partial differential equation for the flow of non-Newtonian, power-law fluid through a porous medium.Nomenclature a parameter, defined by (A8) - A cross-section area of linear reservoir - B constant - c fluid compressibility - c _f formation compressibility - c _t system compressibility - c t dimensionless system compressibility, defined by (24) - C constant of integration - D _I dimensionless coefficient, directly proportional to injection rate, for linear displacement case, defined by (22). - D ₂ dimensionless coefficient, directly proportional to injection rate, for radial displacement case, defined by (55) - erf(x) error function - ercf(x) complementary error function - Ei(x) exponential integral - f dimensionless pressure, defined by (10) - h formation thickness - k permeability - l linear location of moving boundary between the displacing and displaced fluids - n flow behavior parameter - p pressure - p _i injection pressure - p ₀ initial pressure; reference pressure - p ₀ dimensionless initial pressure, defined by (19) - q injection rate - r radial distance - R radial location of moving boundary between the displacing and displaced fluids - t time - u superficial velocity - U substitution of variable - x linear distance - _e effective viscosity - _e dimensionless effective viscosity, defined by (24) - dimensionless variable, defined by (9) or (45) - _i0 value of corresponding to the location of the moving boundary between the displacing and displaced fluids - density - ₀ value of density at reference pressure - porosity - ₀ value of porosity at reference pressure - 1 displacing fluid - 2 displaced fluid