Numerical analysis of simulation accuracy for hypersonic heat transfer in subsonic jets of dissociated nitrogen

One-dimensional problems of the flow in a boundary layer of finite thickness on the end face of a model and in a thin viscous shock layer on a sphere are solved numerically for three regimes of subsonic flow past a model with a flat blunt face exposed to subsonic jets of pure dissociated nitrogen in an induction plasmatron [1] (for stagnation pressures of (10⁴–3·10⁴) N/m² and an enthalpy of 2.1·10⁷ m²/sec²) and three regimes of hypersonic flow past spheres with parameters related by the local heat transfer simulation conditions [2, 3]. It is established that given equality of the stagnation pressures, enthalpies and velocity gradients on the outer edges of the boundary layers at the stagnation points on the sphere and the model, for a model of radius R_m=1.5·10^–2 m in a subsonic jet the accuracy of reproduction of the heat transfer to the highly catalytic surface of a sphere in a uniform hypersonic flow is about 3%. For surfaces with a low level of catalytic activity the accuracy of simulation of the nonequilibrium heat transfer is determined by the deviations of the temperatures at the outer edges of the boundary layers on the body and the model. For this case the simulation conditions have the form: dU_e/dx=idem, p₀=idem, T_e=idem. At stagnation pressuresP ₀2·10⁴ N/m² irrespective of the catalycity of the surface the heat flux at the stagnation point and the structure of the boundary layer near the axis of symmetry of models with a flat blunt face of radius R_m1.5·10^–2 m exposed to subsonic nitrogen jets in a plasmatron with a discharge channel radius R_c=3·10^–2 m correspond closely to the case of spheres in hypersonic flows with parameters determined by the simulation conditions [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–143, March–April, 1990.     

Nearly Spherically Symmetric Expanding Fronts in a Bistable Reaction-Diffusion Equation

We consider nearly spherically symmetric expanding fronts in the scalar bistable reaction-diffusion equation on R^N. As t, the front is known to look more and more like a sphere under the rescaling of the radius to unity. In this paper we prove that, if the initial state is spherically symmetric and approximated by a one-dimensional traveling wave with a sufficiently large radius, then the solution is approximated uniformly for all t0 without the rescaling of the radius by the one-dimensional traveling wave with the speed of V=c–(N–1), where c>0 is the speed of the one-dimensional traveling wave solution and the mean curvature of the sphere. We further show that, if the initial state is a slightly perturbed one from the spherical front, the difference between the actual front and the expanding sphere hardly grows or decays for all t0, although the relative magnitude of the perturbation to the radius of the sphere decreases to zero.     

Theoretical analysis of heat and mass transfer through eccentric spherical fluid shells at large peclet number

A mathematical analysis of the transport of heat or mass through eccentric spherical closed fluid shells at large Peclet numbers is presented. The bispherical coordinates system is utilized to solve the flow of heat or mass between the two spherical boundaries. The analysis also includes an approximate solution for the motion, if exists, of the inner sphere within the external one. These solutions are then employed in analysing the problem of the evaporation of three-phase liquid drop. The general analysis presented here might prove useful also in the general case of soap bubble (as in foam) or in liquid-liquid systems.Nomenclature a constant - A constant (=a/R ₀) - C _D drag coefficient - g gravitational acceleration - G density ratio (= ₁/ _v) - h , h , h linearizing factors of bispherical coordinates - k thermal conductivity of fluid enclosed between the shells - unit vector in the direction of gravitational acceleration - K drag constant (=16.0 to 24.0) - l distance between the image doublet and outer sphere center - L _r longtitudal reference scale (=R ₀) - P* pressure of the system - q heat flux through the sphere area - q _av average rate of heat absorption in the case of concentric spheres (=q _c/t _c) - q _n heat flux per unit area of a sphere - Q dimensionless heat flux (=q/q _c) - r _i radius of interior sphere - r ₀ radius of exterior sphere - Re, c characteristic Reynolds number (= 2R ₀(R ₀/t _c)/) - R ₁ radius of liquid mother drop - R ₀ initial value of R ₁ - R _v radius of inner vapour bubble - s distance between centers of spheres - S dimensionless distance between centers of spheres (=s/R ₀) - t time - t _c time required for complete evaporation in the case of concentric system - t _r reference scale for time (=t _c) - T temperature - T _i temperature of inner spherical surface - T ₀ temperature of outer spherical surface - T* saturation temperature corresponding to P* - T temperature of undisturbed field - T temperature driving force (T–T*) - u _c translatory velocity of inner bubble (u _c=u _c ) - V volume of inner bubble - x cartesian coordinate - y cartesian coordinate - z cartesian coordinate - dimensionless radius (=r/R ₀ or R/R ₀) - point doublet strength - ₁ first image of - ₁ density of fluid enclosed between the shells (liquid) - _v density of fluid enclosed in the inner sphere (vapour) - ₁ kinematic viscosity of fluid enclosed between the shells (liquid) - heat of evaporation - bispherical coordinate, liquid viscosity in eq. (25) - bispherical coordinate - bispherical coordinate - dimensionless time (=t/t _c)     

Diffusion to a chain of drops (bubbles) at large péclet numbers

The problem of diffusion of a substance, dissolved in a flow, to absorbing drops (bubbles) moving one after another in a viscous incompressible fluid is investigated. An approximate analytic expression is obtained for the differential and integral flows of the substance to the surface of each drop with consideration of the changes of the concentration and velocity fields due to the presence of other drops. A chain of spherical drops of equal radius arranged on the axis of a uniform forward flow is examined. It is shown that if the distance between drops, referred to the radius of the drops, satisfies the inequality 1lP^1/2 (P is the Péclet number), then the integral inflow of the substance to the surface of the second drop of the chain is 2.41 times less than the integral inflow to the first (the drops are enumerated along the flow); the total diffusion flow to the surface of an arbitrary drop with number k is determined by the expression I_k=I₁[k^1/2 – (k–1)^1/2], where I_k is the total flow to the first drop of the chain. The case of diffusion interaction of a solid particle and drop is examined. It is shown that for particles moving one after another with the same velocity in a fluid at rest the presence of a drop before the solid particle leads to a marked decrease of the total diffusion flow of the solid particle [by O(P^1/6) times], whereas the presence of a solid particle before a drop does not affect (in the main approximation with respect to the characteristic diffusion parameter) the total flow of the latter.I _k=I _i[k ^1/2–(k–1)^1/2]Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–69, January–February, 1978.     

Stability of spherical Couette flow in thick layers when the inner sphere revolves

A natural generalization of cylindrical Couette flow is the flow of a viscous incompressible liquid between two concentric spheres rotating about the same axis with different angular velocities. As has often been noted, spherical Couette flow is, despite the apparent similarity, considerably more complex than cylindrical flow. It consists of differential rotation about the axis and one- or two-eddy circulation (depending on the ratio between the angular velocities of the two spheres = ₂/₁) in the meridional plane and depends significantly on the Reynolds number Re = ₁r ₁ ² and the relative thickness of the layer = (r₂-r₁)/r₁ (₁, ₂ and r₁, r₂ are the angular velocities and radii of the inner and outer spheres, respectively. The investigation of spherical Gouette flow and its stability has begun relatively recently (within the last 10 years) and has evidently been stimulated by applied problems associated with astro- and geophysics. Because of the great computational difficulties encountered in investigating such flow theoretically, experimental investigations have yielded more extensive and interesting results [1–8], although all the published results refer to the case of rotation of one inner sphere ( = 0).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–15, March–April, 1978.It remains to thank S. A. Shcherbakov for help in organizing automatic input of the signals to the BÉSM-6 computer.     

Axisymmetric convection in a spherical layer: Transitions between steady-state regimes

The steady-state convective motions of a viscous fluid occupying a spherical layer R₁ r R₂, R₂/R₁=1.2 are studied. The non-deformable boundaries of the layer are assumed to be free of shear stresses. At the outer boundary the constant temperature and at the inner boundary the constant heat flux are given. The system of equations in the Boussinesq approximation is solved by the Galerkin method with time stabilization on the assumption of axial and equatorial symmetry. It is shown that at the point Ra=Ra_c the state of mechanical equilibrium loses stability and steady symmetrical supercritical bifurcation is observed. The modes most unstable in the linear sense determine the form of convection when Ra > Ra_c and the supercriticality is not too great. At Rayleigh numbers Ra_c < Ra < 200Ra_c there exists a set of steady-state solutions with different spatial structures. The realization of solutions of a particular type depends on the supercriticality and the initial conditions. The evolution of the solutions with variation of the Rayleigh number is investigated. The changes in the spatial kinetic energy spectra and the integral heat fluxes upon transition from one branch of the solutions to another and with variation of the supercriticality are analyzed. As the supercriticality increases, despite the excitation of more and more new small-scale modes, the large-scale motions begin to make an ever greater contribution to the total energy. The results obtained can be used for constructing hydrodynamic models of the global motions in the atmospheres of giant planets, the convective envelopes of stars, and in the depths of the earth's mantle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–24, November–December, 1989.     

On the static and dynamic equilibrium of concentrated loads in linear acoustics and linear elasticity

The displacement field in an unbounded linear elastic fluid subjected to a time-dependent point force is obtained by using integral transform techniques. Differentiation of the displacement field yields the pressure field. It is shown that the pressure on the surface of a spherical ball B _r of radius r centered at the point where the force is applied is statically equivalent in the limit as r0 to only one-third of the force. The remaining two-thirds are carried by the inertia terms. It is also shown, by an independent reasoning, that a point force cannot be carried in static equilibrium by a linear elastic fluid.The displacement field corresponding to an unbounded isotropic linear-elastic solid subjected to a time-dependent point force (the Stokes solution) is also obtained by using integral transform techniques. As is well-known, the tractions of the Stokes solution on the surface of a spherical ball B _r are statically equivalent in the limit as r0 to the force itself; consequently, the inertia terms do not contribute to the dynamic equilibrium of B _r. The contrast between the response of a fluid and that of an isotropic solid under the action of a point force is discussed.     

Numerical investigation of convective heat transfer in liquid hydrocarbons stored in underground reservoirs

In planning for the underground storing of liquid hydrocarbons and calculating the technological parameters of underground reservoirs formed by leeching from a rock salt massif, it is necessary to understand the hydrodynamic and heat transfer processes produced by natural convection. The paper is devoted to numerical study of the initial stage of convective heat transfer in a vertical cylindrical cavity completely filled with a liquid hydrocarbon. It is assumed that at the initial time the temperature of the liquid, which is at rest, is homogeneous, Convective flow develops in part due to the initial temperature difference between the liquid and the massif and partly due to the geothermal temperature gradient in the latter. The problem is regarded as coupled, the convective process in the liquid being determined simultaneously with the solution of the heat-conduction problem in the surrounding rock. The Grashof numbers characterizing the intensity of the real process are very large: G lO¹²–lO¹⁵. A numerical solution was obtained for moderate Grashof numbers G lO⁷–lO¹¹. The asymptotic dependences for the integral characteristics of the unsteady process obtained in a series of calculations were extrapolated to the real values of the Grashof number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–148, March–April, 1981.     

Collision of water drops with a plane water surface

The dependence of the outcome of the collision of uncharged water drops with a plane water surface on the impact angle , the velocity v₁ and the radius r₁ of the drops has been investigated experimentally. The impact parameters were varied over the intervals: v₁=0.40–1.05 m/sec, r₁=75–150m, and =16-85°. The method employed made it possible to avoid having to monitor the individual high-speed impact process. A stream of drops, produced in a vibrating reed type monodisperse droplet generator, was directed at the target. The impact parameters were measured by means of pulsed illumination. The results are expressed in the form of the dependence of the rebound probability and the coalescence coefficient E_S on the impact parameters. The existence of alternating conditional rebound-coalescence-rebound zones for different impact angles is established, together with a decrease in E_S with increase in r₁ and v₁. The data obtained generalize the results of previous experiments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 165–168, May–June, 1990.     

Quenching of a solid sphere in oil

Transient heat conduction in a solid sphere quenched in oil is studied. Specifically, the influence of radius and thermal conductivity of the sphere (Biot number) on the surface temperature is investigated. The problem is extremely nonlinear because the heat transfer coefficient is a strong function of the surface temperature. In the analysis, the transient heat conduction equation and boundary conditions are transformed into a singular nonlinear Volterra Integral equation of the second kind for the surface temperature. This equation is solved numerically by a modified successive approximation method and a separable kernel approximation. Results for a sphere initially at 1144 K and cooled to 317 K are obtained over a wide range of Biot numbers and are compared with the infinite thermal conductivity approximation. The maximum difference between the infinite thermal conductivity approximation and the exact results is 24 K, 64 K, and 144 K for Biot numbers 0.1, 1.0, 10.0, respectively.

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Zusammenfassung Es wird die nichtstationäre Wärmeleitung in einer in Öl abgeschreckten Kugel untersucht unter besonderer Berücksichtigung des Einflusses von Radius und Wärmeleitfähigkeit der Kugel (Biot-Zahl) auf die Oberflächentemperatur. Das Problem ist hochgradig nichtlinear wegen der starken Abhängigkeit des Wärmeübergangskoeffizienten von der Oberflächentemperatur. Wärmeleitgleichung und Randbedingung werden in eine einzelne nichtlineare Volterra-Integralgleichung zweiter Art für die Oberflächentemperatur transformiert, die numerisch durch eine modifizierte schrittweise Approximation mit getrennter Näherung für den Kern gelöst wird. Ergebnisse für eine Anfangstemperatur der Kugel von 1144 K bei Abkühlung auf 317 K in einem weiten Bereich der Biot-Zahl werden mit der Näherung für unendliche Wärmeleitfähigkeit verglichen. Die größte Temperaturdifferenz zwischen dieser Näherung und den genauen Resultaten beträgt 24 K, 64 K und 144 K für die entsprechenden Biot-Zahlen 0,1, 1,0 und 10.

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Nomenclature g dimensionless surface heat flux - h heat transfer coefficient - h_max maximum heat transfer coefficient - k thermal conductivity - Bi Biot number, h_max R/k - r₁ one dimensional spherical coordinate - R radius of the sphere - r dimensionless spherical coordinate, r₁/R - T temperature - T_f fluid temperature - T_i initial temperature of the sphere - T_s surface temperature of the sphere - t dimensionless time t₁/R² or Fourier number - t₁ time - u dimensionless temperature, T/T_i - u_s, y dimensionless surface temperature, T_s/T_i - thermal diffusivity - _m constant defined by Eq. (16) - _c T_f/T_i - _k eigenvalue of tan _k=_k     

Stress state of an elastic ball in a spherically symmetric gravitational field

We consider a spherically symmetric static problem of general relativity whose solution was obtained in 1916 by Schwarzschild for a metric form of a special type. This solution determines the metric coefficients of the exterior and interior Riemannian spaces generated by a gravitating solid ball of constant density and includes the so-called gravitational radius r _g. For a ball of outer radius R=r _g, the metric coefficients are singular, and hence the radius r _g is traditionally assumed to be the radius of the event horizon of an object called a black hole. The solution of the interior problem obtained for an incompressible ideal fluid shows that the pressure at the ball center increases without bound for R=9/8r _g, which is traditionally used for the physical justification of the existence of black holes. The discussion of Schwarzschild’s traditional solution carried out in this paper shows that it should be generalized with respect to both the geometry of the Riemannian space and the elastic medium model. In this connection, we consider the general metric form of a spherically symmetric Riemannian space and prove that the solution of the corresponding static problem exists for a broad class of metric forms. A special metric form based on the assumption that the gravitation generating the Riemannian space inside a fluid ball or an elastic ball does not change the ball mass is singled out from this class. The solution obtained for the special metric form is singular with respect to neither the metric coefficients nor the pressure in the fluid ball and the stresses in the elastic ball. The obtained solution is compared with Schwarzschild’s traditional solution.     

Steady rotatory flow of Bingham material between two concentric spheres

Summary Steady flow of Bingham material between two concentric rotating spheres has been investigated taking into account the constitutive equations given by Oldroyd. Motions in the elastic as well as in the flow region have been discussed. A critical value of the Bingham number has been found below which the flow takes place in the whole region and above which the elastic and flow regions occur side by side.Nomenclature r, , space co-ordinates - u _r ,u ,u velocity components - density - modulus of rigidity (constant) - bulk modulus - e _ii , the dilatation - ₁ coefficient of viscosity (constant) - e _ik strain tensor - d _ik rate of strain tensor - p _ik stress tensor primes denote deviatoric components of tensors, e.g. - p _ik p _ik +p _k ⁱ , p=–1/3p _ii - yield value (constant) - D/Dt materiaal derivative with regard to time following the particle - R ₁ radius of the inner sphere - R ₂ radius of the outer sphere - R radius of the yield surface (spherical) - ₁ velocity of the inner sphere - B Bingham number     

Drag of a sphere in an unsteady flow of viscous fluid at small reynolds numbers

An unsteady flow of viscous incompressible fluid past a sphere is investigated. The values of the inertial and unsteady terms in the Navier-Stokes equations are characterized by translational (R) and vibrational (R_k) Reynolds numbers, which are assumed small. The solution is constructed in the form of an expansion with respect to max(R, R _k ^1/2 ) by the method of matched asymptotic expansions. A correction to the Stokes force, correct to o[max(R, R _k ^1/2 )], is calculated. It is shown that the result depends strongly on the ratio R/R _k ^1/2 and goes over into the well-known equations for the cases R 0, R_k 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 11–16, January–February, 1988.     

Rheological changes occurring during the storage of the system sodium dodecyl sulphate/1-hexadecanol/water

Summary The physical decomposition of the system formed when sodium dodecyl sulphate, 1-hexadecanol and water are heated, mixed and cooled could be achieved at an accelerated rate by cycling the storage temperature between 5–7 °C and 20–25 °C. This decomposition caused the formation of pearliness and the systems became progressively more fluid with loss in rigidity.The changes in the rheological properties of the systems were investigated at 25 °C using a modifiedWeissenberg Rheogoniometer, anEpprecht Rheomat 15 viscometer, and a concentric cylinder air turbine viscometer.The systems are referred to by a code R_x where x is the molar ratio of 1-hexadecanol to sodium dodecyl sulphate. The systems R₁–R₁₀ were examined during temperature cycling for a maximum period of eight days. During this period, the systems R₁–R₄ progressively increased in consistency when examined in continuous shear using anEpprecht Rheomat 15, concentric cylinder viscometer. System R₅ was also examined using anEpprecht Rheomat 15, but after increasing in consistency up to the fourth day, thereafter decreased in consistency. Systems R₃ to R₁₀ were investigated in creep mode. For R₃ and R₄, the compliance progressively decreased over a period of eight days, whereas for systems R₅–R₈ compliance was found to pass through a minimum about four days after preparation and the start of temperature cycling. Thereafter it increased. For R₉ and R₁₀, the creep compliance at one hour increased continuously during the eight day period.

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Zusammenfassung Beschleunigte Zersetzung des Systems, das beim Erwärmen, Mischen und Kühlen von Natriumdodecylsulfat, 1-Hexadekanol und Wasser entsteht, wurde durch zyklische Temperaturänderungen zwischen 5–7 und 20–25 °C bewirkt. Durch Zersetzung wurde das System perlmutterartig verfärbt und unter Verlust an Festigkeit fließfähiger.Die Änderungen im rheologischen Verhalten wurden mit einem modifiziertenWeissenberg-Rheogoniometer, einem Rheomat-15-Viskosimeter, und einem Luftturbinen-Viskosimeter mit konzentrischen Zylindern untersucht.Auf die verschiedenen Systeme wird mit einem Code R_x Bezug genommen, wobei x das molare Verhältnis von 1-Hexadekanol zu Natriumdodecylsulfat ist. Die Systeme R₁–R₁₀ wurden über einen Zeitraum von 8 Tagen unter dem Einfluß von zyklischen Temperaturänderungen untersucht. Innerhalb dieses Zeitraumes nahmen die flüssigen Systeme R₁–R₄, wie kontinuierliche Scherexperimente mit dem Rheomat 15 Viskosimeter (konzentrische Zylinder) zeigten, beständig an Konsistenz zu. System R₅ wurde auch mit dem Rheomat 15 untersucht; es zeigte aber, nach einer Konsistenzzunahme bis zum 4. Tag, eine Abnahme an Konsistenz. Die Systeme R₆–R₁₀, die mehr Festkörper-Charakter hatten, wurden mit Hilfe von Kriechtesten untersucht. Die dynamische Nachgiebigkeit bei 1 Std. ging durch ein Minimum 4 Tage nach der Herstellung bei Lagerung mit zyklischer Temperaturänderung. Danach nimmt sie zu. Für R₉–R₁₀ nahm die dynamische Nachgiebigkeit bei 1 Std. kontinuierlich über die 8 Tage zu.

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Paper presented to the British Society of Rheology Conference on Rheology in Medicine and Pharmacy, London, April 14–15, 1970.     

Wall visualization of swirling decaying flow using a dot-paint method

This paper deals with the visualization of swirling decaying flow in an annular cell fitted with a tangential inlet. A wall visualization method, the so-called dot-paint method, allows the determination of the flow direction on both cylinders of the cell. This study showed the complex structure of the flow field just downstream of the inlet, where a recirculation zone exists, the effects of which are more sensitive on the inner cylinder. The flow structure can be considered as three-dimensional in the whole entrance section. The swirl number and the entrance length were estimated using the measured angle of the streamlines. Experimental correlations of these two parameters, taking into account the Reynolds number and the axial distance from the tangential inlet, are given.List of symbols e = R ₂ – R ₁ thickness of the annular gap (m) - L _ax entrance length of axial flow on the outer cylinder (m) - L _ti length of the three-dimensional flow region on the inner cylinder (m) - L _to length of the three-dimensional flow region on the outer cylinder (m) - Q _v volumetric flowrate in the annular cell (m³s^–) - r radial position (m) - R ₁ external radius of the inner cylinder (m) - R ₂ internal radius of the outer cylinder (m) - Re=2eU _m /v Reynolds number - Sn swirl number - T time average resulting velocity (m s^–) - u time average axial velocity component (ms ^–) - average velocity in the annulus (m s^–) - w time average tangential velocity component (m s^–) - x axial location from the tangential inlet (m) - _e diameter of the tangential inlet (m) - streak angle with respect to the horizontal (degree) - angle with respect to the tangential inlet axis (degree) - gn kinematic viscosity of the working liquid (m²s^–)     

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.     

Consecutive chemical reactions in an annular reactor with non-newtonian flow

The purpose of this paper is to analyze the homogeneous consecutive chemical reactions carried out in an annular reactor with non-Newtonian laminar flow. The fluids are assumed to be characterized by a Ostwald-de Waele (powerlaw) model and the reaction kinetics is considered of general order. Effects of flow pseudoplasticity, dimensionless reaction rate constants, order of reaction kinetics and ratio of inner to outer radii of reactor on the reactor performances are examined in detail.Nomenclature c _A concentration of reactant A, g.mole/cm³ - c _B concentration of reactant B, g.mole/cm³ - c _A0 inlet concentration of reactant A, g.mole/cm³ - C ₁ dimensionless concentration of A, c _A/c _A0 - C ₂ dimensionless concentration of B, c _B/c _A0 - C ₁ dimensionless bulk concentration of A - C ₂ dimensionless bulk concentration of B - D _A molecular diffusivity of A, cm²/sec - D _B molecular diffusivity of B, cm²/sec - k _A first reaction rate constant, (g.mole/cm³)^1–m /sec - k _B second reaction rate constant, (g.mole/cm³)^1–n /sec - K ₁ dimensionless first reaction rate constant, k _A r ₀ ² c _A0 ^m–1 /D _A - K ₂ dimensionless second reaction rate constant, k _B r ₀ ² c _A0 ^n–1 /D _B - K apparent viscosity, dyne(sec) ^m /cm² - m order of reaction kinetics - n order of reaction kinetics - P pressure, dyne/cm² - r radial coordinate, cm - r _i radius of inner tube, cm - r _max radius at maximum velocity, cm - r _o radius of outer tube, cm - R dimensionless radial coordinate, r/r _o - s reciprocal of rheological parameter for power-law model - u local velocity, cm/sec - u _max maximum velocity, cm/sec - u bulk velocity, cm/sec - U dimensionless velocity, u/u - z axial coordinate, cm - Z dimensionless axial coordinate, zD _A/r ₀ ² /u - ratio of molecular diffusivity, D _B/D _A - ratio of inner to outer radius of reactor, r _i/r _o - ratio of radius at maximum velocity to outer radius, r _max/r _o     

Asymptotic solutions of the problem of heat and mass transfer of particles in a fluid

Some questions related to asymptotic analysis (as P , where P is the Péclet number) of problems involving heat and mass transfer of particles in a fluid are considered. The first part of the paper investigates the stationary convective diffusion of a solute to a particle near its front critical point (incidence point). An explicit expression is obtained for the concentration in the region of the front critical point of a solid or liquid particle around which a Stokes flow occurs. In the second part of the paper, a unified formula is obtained for the concentration distribution behind the particle. Appropriate limits in this formula determine the concentration in the mixing region and the inner and convective boundary-layer regions of the diffusion wake. In the final part of the paper, a study is made of the diffusion to a chain of absorbing solid spheres of equal radius a at distances1, 1 1/a P^1/3, from each other on the axis of an oncoming Stokes flow; an integral equation is obtained for the local diffusion flux when a chemical reaction with arbitrary kinetics takes place on the surfaces of the spheres. A certain heterogeneity of the material in the paper is due to the investigation in it of various questions that arise in the solution of more general problems (see, for example, [1–14]) which have not been considered hitherto.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 1, pp. 89–96, January–February, 1979.     

Instabilities of the Stewartson layer Part 2. Supercritical mode transitions

We investigate, both experimentally and numerically, the fluid flow in a spherical shell with radius ratio r_i/r_o=2/3. Both spheres rotate about a common axis, with _i>_o. The basic state consists of a Stewartson layer situated on the tangent cylinder, the cylinder parallel to the axis of rotation and touching the inner sphere. If the differential rotation is sufficiently large, non-axisymmetric instabilities arise, with the wavenumber of the most unstable mode increasing with increasing overall rotation. In the increasingly supercritical regime, a series of mode transitions occurs in which the wavenumber decreases again. The experimental and numerical results are in good agreement regarding this basic sequence of mode transitions, and the numerics are then used to study some of the finer details of the solutions that could not be observed in the experiment.     

Pressure pulsations on the surface of a sphere in a subsonic stream

Many data are available on the drag C_x and the distribution of the static pressure over the surface of a sphere [1, 2]. However, there are virtually no data on pulsations of the pressure over the surface of a sphere. In the present paper, the results are given of an investigation of the total and spectral levels of the pressure pulsations at different points of the surface of a sphere at M 0.5–1.0 and Re (1.7–2.7)·.10⁶. It was found that the strongest pressure pulsations occur on the side in the region of the angle 90°. In this region at M 0.6–0.8 the relative total level o/q where q is the velocity head in the oncoming stream, reaches values 0.18–0.22. It was established that at M = 0.7–0.9 narrow-band maxima occur in the spectra of the pressure pulsations at frequencies Sh fD/V = 0.2–0.3. Data are also presented on the pulsations of the base pressure behind a spherical segment with short cylindrical and conical trailing edges.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 164–168, September–October, 1981.