Analytical Modeling of Nonaqueous Phase Liquid Dissolution with Green's Functions

Equilibrium and bicontinuum nonequilibrium formulations of the advection–dispersion equation (ADE) have been widely used to describe subsurface solute transport. The Green's Function Method (GFM) is particularly attractive to solve the ADE because of its flexibility to deal with arbitrary initial and boundary conditions, and its relative simplicity to formulate solutions for multidimensional problems. The Green's functions that are presented can be used for a wide range of problems involving equilibrium and nonequilibrium transport in semiinfinite and infinite media. The GFM is applied to analytically model multidimensional transport from persistent solute sources typical of nonaqueous phase liquids (NAPLs). Specific solutions are derived for transport from a rectangular source (parallel to the flow direction) of persistent contamination using first, second, or thirdtype boundary or source input conditions. Away from the source, the first and thirdtype condition cannot be expected to represent the exact surface condition. The secondtype condition has the disadvantage that the diffusive flux from the source needs to be specified a priori. Near the source, the thirdtype condition appears most suitable to model NAPL dissolution into the medium. The solute flux from the pool, and hence the concentration in the medium, depends strongly on the mass transfer coefficient. For all conditions, the concentration profiles indicate that nonequilibrium conditions tend to reduce the maximum solute concentration and the total amount of solute that enters the porous medium from the source. On the other hand, during nonequilibrium transport the solute may spread over a larger area of the medium compared to equilibrium transport.     

Effect of Transitional Nonequilibrium on the Molecular–Dissociation Rate in a Hypersonic Shock Wave

The problem of the influence of a nonequilibrium (non–Maxwellian( distribution of translational energy over the degrees of freedom of molecules on the rate of their dissociation in a hypersonic shock wave is considered. An approximate beam—continuous medium model, which was previously applied to describe a hypersonic flow of a perfect gas, was used to study translational nonequilibrium. The degree of dissociation of diatomic molecules inside the shock–wave front, which is caused by the nonequilibrium distribution over the translational degrees of freedom, is evaluated. It is shown that the efficiency of the first inelastic collisions is determined by the dissociation rate exponentially depending on the difference in the kinetic energy of beam molecules and dissociation barrier.     

Exact solutions and mathematical properties of boundary‐value problems for dynamic‐diffusion boundary layers

The paper studies boundaryvalue problems for dynamicdiffusion boundary layers occurring near a vertical wall at high Schmidt numbers and for dynamic boundary layers whose inner edge is adjacent to the dynamicdiffusion layers. Exact solutions for boundary layers at small and large times are derived. The wellposedness of the boundaryvalue problem for a steady dynamicdiffusion layer is studied.     

Application of the method of integral diffusion to investigation of turbulent transfer

A number of authors have critically examined semiempirical mixing length theories [1]. A defect of these theories is connected with the fact that the magnitude of the mixing length, which is assumed to be small in constructing the theory, turns out in experiments to be comparable with the characteristic dimensions of the flow region. Thus, the concept of volume convection [2–4] or integral diffusion [5], which is understood to be a transfer mechanism in which the friction stress is not expressed in terms of the velocity gradient, is introduced along with the concept of gradient diffusion. In addition, there are a number of experimental papers [6] in which it is shown that the turbulent friction stress cannot be equal to zero at the place in the flow where the derivative of the velocity is equal to zero. Mixing length theory does not describe this effect.It is possible to generalize mixing length theory [7–9] in a way which eliminates these defects. Flow of an incompressible fluid is considered.     

Calculation of heat transfer accompanying hypersonic three-dimensional flow of air over slender blunt-nosed cones

The effective length method [1, 2] has been used to make systematic calculations of the heat transfer for laminar and turbulent boundary layers on slender blunt-nosed cones at small angles of attack ( + 5° in a separationless hypersonic air stream dissociating in equilibrium (half-angles of the cones 0 20°, angles of attack 0 15°, Mach numbers 5 M 25). The parameters of the gas at the outer edge of the boundary layer were taken equal to the inviscid parameters on the surface of the cones. Analysis of the results leads to simple approximate dependences for the heat transfer coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1981.     

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

Ionization and nonequilibrium radiation of air behind strong shock waves

It is shown that at high velocities of shock waves (V 9.5 km/sec) an important factor influencing the rate of ionization is the depletion of the number of excited states of the atoms through de-excitation. In the case of low pressures (p 1 torr) and for a bounded and optically transparent region of gas heated by the shock wave (for example, for the motion of gas in a shock tube or in a shock layer near a blunt body), the effective ionization rate k_f depends on the pressure [1], which leads to violation of the law of binary similarity which holds under these conditions without allowance for de-excitation. On leaving the relaxation zone, the gas arrives at a stationary state with constant parameters differing from those in thermodynamic equilibrium. The electron concentration and also the radiation intensity in the continuum and the lines are lower than the values for thermodynamic equilibrium. These considerations explain the results of known experiments and some new experiments on ionization and radiation of air behind a travelling shock wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 105–112, January–February, 1980.     

Instability of a plane axisymmetric flow with a shear discontinuity in the velocity profile

It is proposed to investigate the stability of a plane axisymmetric flow with an angular velocity profile (r) such that the angular velocity is constant when r < r_O – L and r > r_O + L but varies monotonically from ₁ to ₂ near the point r_O, the thickness of the transition zone being small L r_O, whereas the change in velocity is not small ¦₂ – ₁¦ ₂, ₁. Obviously, as L O short-wave disturbances with respect to the azimuthal coordinate (k=m/r_O 1/r_O) will be unstable with a growth rate-close to the Kelvin—Helmholtz growth rate. In the case L=O (i.e., for a profile with a shear-discontinuity) we find the instability growth rate _O and show that where the thickness of the discontinuity L is finite (but small) the growth rate does not differ from _O up to terms proportional to kL 1 and 1/m 1. Using this example it is possible to investigate the effect of rotation on the flow stability. It is important to note that stabilization (or destabilization) of the flow in question by rotation occurs only for three-dimensional or axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–114, January–February, 1985.     

Unsteady supersonic flow around blunt bodies with detached shock wave

The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances.The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M=1.50– and dimensionless oscillation frequency l/V=0–1.0. A comparison is made with the results obtained from the Newton theory.     

Flow around stepped bodies with strong compression in a shock layer

This article discusses plane and axisymmetric flows of a nonviscous ideal gas around bodies of stepped form, forming with a Mach number M= and an adiabatic indexN1. The greatest amount of attention is paid to the case where there is no Newtonian free layer, but the shock layer is detached at great distances from the nose of the body.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 4, pp. 104–112, July–August, 1973.     

Application of the integral diffusion equations to the investigation of turbulent transport

In recent years there have appeared several experimental studies [1–5] which have shown that there are cases of turbulent flow with an asymmetric distribution of the flow velocity and in which at the point where the velocity derivative is zero the turbulent shear stress is not zero. This raises the question of the connection of the Reynolds stress tensor with the characteristics of the average flow. The relationships used in the usual mixing length theory connect the shear stress with the local value of the flow velocity derivative and are not consistent with the experimental results mentioned above. These relationships are based on the assumption that the mixing length is small in comparison with the characteristic length of the flow. Experiment shows that this assumption is not justified [6].Thus, turbulent diffusion refers to the case of diffusion with a large mean free path. In addition to the concept of gradient diffusion, there is also the concept of bulk convection or integral diffusion [10], which means a transfer mechanism in which the shear stress is not expressed in terms of the velocity gradient. The generalization of mixing length theory proposed in [11–14] is based on the very simple kinetic equation which was used for the examination of turbulent transfer problems in [8] and which is encountered in the treatment of transport problems in gases, neutron diffusion, and radiative energy transfer.The proposed generalization of mixing length theory employs an analogy with the indicated processes and permits the derivation of formulas which are valid for large mean free paths. In the case of small mean free paths the obtained relationships lead to the relationships for diffusion in a continuous medium and, in particular, to the relationships of the Prandtl mixing length theory. The integral diffusion model is a phenomenological semiempirical theory in which empirical constants and several hypotheses common in mixing length theory are used. A very general analysis of the expression for the shear stress leads to the conclusion that if the flow is asymmetric over a distance comparable with the mixing length the points at which the velocity derivative and the turbulent shear stress are zero do not coincide [12]. Hence, it is to be hoped that the integral diffusion model will allow treatment of the above questions, which cause difficulty in the case of ordinary mixing length theory. Incompressible turbulent flow is considered.     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ Greek Letters V /V volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m²     

Viscous flow of carbon dioxide over a blunted cone

This paper conducts a numerical investigation of viscous nonequilibrium flow over a spherically blunted cone of hypersonic carbon dioxide (Re_S = 10–10⁵), where Re_S = VL/_S. From the results obtained for the distribution of gas-dynamic and thermochemical parameters in the shock layer the basic flow laws are elucidated and estimates are made of the boundaries for existence of various flow regimes within the framework of continuum theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–105, July–August, 1978.     

Calculation of intense blowing on a blunt body and an airfoil

Various aspects of the problem of intense blowing through the surface of bodies have, been theoretically studied by a number of authors, within the framework of inviscid flow theory. A detailed bibliography on this topic is given, e.g., in [1, 2]. The well-known approaches to solution of this problem have a limited area of application. For example, asymptotic methods can be used for hypersonic flow regimes only at relatively low levels of the blown gas momentum ( = ² = _ov_o ²/ V ² 1). The same limitation applies to the numerical method of straight lines [2]. The forward Eulerian calculation schemes [3, 4] smear the contact discontinuity severely, and cannot handle the case where the blown gas and the gas in the incident flow have different thermodynamic properties (_o ). This paper presents results of a numerical investigation of supersonic flow over two-dimensional and axisymmetric bodies with intense blowing on the forward surface, performed using a time-dependent finite-difference method [5] with an explicit definition of the contact interface between the two cases. The calculations encompass a family of elliptic cylinders with semiaxis ratio 0.5 4, a flat-face cylinder, and a flat plate with rounding near the midsection, with variations in the blowing law, the incident flow Mach number M (3 M 10), the adiabatic indices, and the blowing parameter 0 0.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–124, January–February, 1977.In conclusion, the authors thank T. S. Novikov and I. D. Sandomirskii, who took part In the present calculations.     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

Hydrodynamic instability of the contact zone between accelerated gases

An experimental apparatus for investigating Rayleigh-Taylor instability in the transition layer between two gases at accelerations g 10⁵g₀ (g₀ is the acceleration of gravity) is described. The constantly acting acceleration is communicated to the contact zone by the compression wave formed ahead of a flame front. The linear stage of development is investigated together with the effect of the thickness of the contact zone. It is shown that on the interval 0.3 < <- ( is the wavelength of the disturbance at the edge of the contact zone) the rate of growth of the perturbation amplitude 0.5₀, where ₀ is the amplitude growth rate for media separated by an interface with a discontinuous change of density.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–21, November–December, 1991.     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.     

Turbulent heat and mass transfer for Pr ≫ 1 and law of turbulent diffusion damping at a solid wall

The question of determining the law of damping for the turbulent diffusion coefficient at a smooth wall according to data on mass and heat transfer for Pr 1 is discussed. It is proved that the hypothesis that this law is determined by the first member of the Taylor series expansion of , namely, / = yⁿ ₊ is valid in the Pr range from 10³ to 10⁵ only under the assumption that the subsequent terms in the expansion have smaller coefficients. A statistical analysis of electrochemical and other experiments devoted to this problem shows that apparently n = 3, but singularities in the experimental results do not permit making a final conclusion. Requirements on a conclusive experiment are formulated on the basis of the analysis made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–175, March–April, 1977.     

Two-Phase Models of Formation of Cavitating Spalls in a Liquid

The dynamics of the structure of a liquid layer structure (with microbubbles of a free gas) behind a rarefaction wave front is studied numerically using the two-phase Iordansky–Kogarko–van Wijngaarden model and the frozen mass-velocity field model. An analysis of the initial stage of cavitation by the Iordansky–Kogarko–van Wijngaarden model showed that tensile stresses behind the rarefaction wave front relax quickly and the mass-velocity field in the cavitation zone turns out to be frozen. This effect is used to describe the late stage of the development of the cavitation zone. These models were combined to study the formation of cavitating spalls in a free-surface liquid under shock-wave loading.     

Particle-in-a-box study of the evolution of an ion-acoustic shock wave

In an analysis of a one-dimensional numerical model of a nonisothermal plasma it is shown that an ion-acoustic shock wave of subcritical amplitude separates a soliton from the shock front after the reversing stage. This process is accompanied by turbulent flow behind the front and by trapping of ions in potential wells. The numerical particle-in-a-box method is being used widely to study plasma phenomena. One field in which this method has been found fruitful is in the study of a nonisothermal plasma, characterized by an ion-acoustic wave branch.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 3–5, May–June, 1971.The authors thank R. Z. Sagdeev for support and interest in this study.