Formulas for the heat and diffusion fluxes to a catalytic surface in a chemically nonequilibrium multicomponent boundary layer

On the basis of an asymptotic expansion of the solution of the equations of a multicomponent chemically nonequilibrium boundary layer for large Schmidt numbers, formulas are obtained for the heat flux and the diffusion fluxes of the reaction products and chemical elements on a surface with arbitrary catalytic activity. The results are compared with well-known analytic and numerical solutions. The comparison reveals the high accuracy of the formulas proposed. The results of calculating the diffusional separation of the mixture due to the selectivity of the catalytic properties of the surface with respect to recombination of oxygen and nitrogen atoms are presented. Values of the reduction of the convective heat fluxes due to the catalytic properties of the surface are obtained over a wide range of conditions in the free stream.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 171–176, March–April, 1996.     

Development of two-dimensional perturbations on the surface of a tangential discontinuity

A study is made of the asymptotic behavior at long times of initially localized small two-dimensional perturbations of the interface of two fluids in the presence of a tangential discontinuity of the velocity; surface tension is taken into account. The development of one-dimensional perturbations was considered earlier in [1]. The asymptotic behavior of the perturbed region is found, i.e., in the xyt space there is found a cone with apex at the origin such that perturbations tend to infinity with increasing t along rays within the cone, while perturbations tend to zero along the remaining rays. Conditions are found under which the instability of the tangential discontinuity is not absolute, i.e., when these conditions are satisfied, flows with tangential discontinuity of the velocity can take place. These conditions, like the shape of the cone, do not depend on the magnitude of the surface tension.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 12–16, May–June, 1979.     

Consideration of longitudinal diffusion in flow within a channel

The stationary problem of convective diffusion in a channel with absorbent walls is considered. It is assumed that a Poiseuille flow exists. Two methods are employed in the solution, the method of separation of variables, and the method of expansion in eigenfunctions of the corresponding problem with piston profile (expansion method). It is established by comparison with independently obtained solutions for high Peclet number that for the first eigenfunctions and eigenvalues the expansion method gives satisfactory results over the entire Peclet-number range. For approximate calculation of subsequent eigenfunctions and eigenvalues a modification of the smooth asymptotic expansion method is used. The results are used to calculate matter flow density on the wall, to evaluate the length of the entrance region, and to obtain an analytical expression for the limiting Nusselt number in terms of the Peclet number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 63–73, November–December, 1973.     

Boundary layer in the neighborhood of the intersection of a concave cylindrical surface and a plane

In studies devoted to the theoretical and experimental investigation of longitudinal flow of a viscous fluid past corner regions, a corner formed by the intersection of two planes is usually considered [1–3]. In contrast, the present paper is concerned with the flow in the neighborhood of the line of intersection of a plane and a concave cylindrical surface (see Fig. 1). The asymptotic behavior of the Navier-Stokes equations at large Re is investigated for such a flow. Estimates are obtained for the velocity and characteristic scales of the flow. It is shown that curvature of one of the surfaces qualitatively changes the pattern of the longitudinal flow of a viscous fluid past a corner. The development of a three-dimensional boundary layer on a plane in the domain of influence of a concave cylindrical surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 160–165, March–April, 1981.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Stability of a steady-state discontinuity of a fluid layer on the surface of an immiscible fluid

A new stable structure of the three-phase system formed by a gas, a horizontal liquid layer with a free upper surface and an underlying immiscible liquid substrate is investigated experimentally and theoretically. When the upper layer has a greater surface tension than the lower layer and its thickness is fairly small, a local deformation of its surface can lead to the development of a steady-state concentric discontinuity within whose limits the lower layer os in contact with the gas. The conditions of stability of such a phase system with a steady-state discontinuity are studied and the dependences of the discontinuity parameters on the vessel diameter, the upper layer thickness, and the liquid surface tensions are obtained for various pairs of liquids. The formulation of the analytic problem of the layer discontinuity is discussed. The experimental data are compared with the results of calculations carried out for a model of a discontinuity in an infinite layer.     

Calculated characteristics of the anode region of a longitudinal discharge including diffusion

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–5, September–October, 1990.     

On the driving traction acting on a surface of discontinuity within a continuum in the presence of electromagnetic fields

The notion of the driving traction of Knowles [6] is generalized to account for the interaction of a continuum with electromagnetic fields, and the expression for the generalized driving traction is reduced to that of Abeyaratne and Knowles [15] in the absence of electromagnetic fields. The admissibility condition implied by the second law of thermodynamics is discussed and the results are consistent with those obtained in [23] where a thermoelastic polarizable material undergoing a bulk phase transition is considered.     

Flow of a gas behind a Chapman-Jouguet detonation wave in the vicinity of a line of discontinuity of a surface wave curvature

This paper discusses questions of constructing a solution of the gasdynamic equations near a line of curvature discontinuity at the surface of a detonation wave, propagating under Chapman—Jouguet conditions. It describes the construction of the solution in two cases: in a flow arising with the initiation of a detonation along a half-plane in a quiescent homogeneous combustible gas and in a flow arising with the initiation of a detonation along a half-line under these same conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–126, January–February, 1978.     

Diffusion separation of chemical elements on a catalytic surface

An asymptotic expansion of the solution, for large Schmidt numbers, of the system of equations of a chemically nonequilibrium multicomponent boundary layer on the catalytic surface of a blunt body [1] is used to obtain expressions for the diffusion fluxes of the reaction products and chemical elements and the heat flux as functions of the gradients of the reaction product concentrations, chemical element concentrations and enthalpy across the boundary layer. It is shown that when the body is exposed to a supersonic air flow, the diffusion separation of the chemical element oxygen depends importantly on the atom concentration at the outer edge of the boundary layer and the nature of the homogeneous and heterogeneous catalytic reactions. If the surface promotes the rapid recombination of oxygen atoms and is chemically neutral with respect to nitrogen atoms, then an excess of the chemical element oxygen is formed on the body. Otherwise we get an enhanced concentration of the element nitrogen. As distinct from the case of an ideally catalytic wall [2–4], on a surface possessing the property of catalytic selectivity the diffusion separation of chemical elements takes place even when only atoms are present at the outer edge of the boundary layer. On a chemically neutral surface diffusion separation may be caused by homogeneous recombination reactions between oxygen and nitrogen atoms if their rate constants are essentially different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–121, July–August, 1988.     

Three-dimensional separation in the neighborhood of roughness on the surface of an axisymmetric body

Steady-state viscous incompressible fluid flow past an axisymmetric slender body is considered at high Reynolds numbers in the regime with vanishing surface friction in a certain cross-section. In a small neighborhood of this cross-section interaction between the boundary layer flow and the external irrotational stream develops. In order to study the structure of the three-dimensional flow with local separation zones it is assumed that there is three-dimensional roughness on the surface of the body with the scale of the interaction zone. For this zone a numerical solution of the problem is obtained and its nonuniqueness is established. The surface friction line (limiting streamline) patterns with their inherent features are constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–79, May–June, 1995.Thus, on the basis of the asymptotic marginal separation theory it is possible to obtain fairly simple solutions describing flows with a complex surface friction line structure.     

Exact solution of the kinetic equation in the problem of nonisothermal flow in the neighborhood of a weakly curved surface

An analytic method for solving the half-space boundary value problem for the inhomogeneous Boltzmann equation with the collision operator in the form of an elliptico-statistical model (the ES-model of the Boltzmann equation) is proposed for the problem of nonisothermal rarefied gas flow in the neighborhood of a curved surface. An exact analytic expression is derived for the thermal slip of a monatomic gas along the surface of a rigid spherical aerosol particle. A numerical value of the gas-kinetic coefficient which takes into account the effect of the curvature of the surface on the thermal slip coefficient is obtained. A comparison with published data is carried out. Moscow, Arkhangelsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 165–173, March–April, 1998.     

Slip boundary conditions on a catalytic surface in a multicomponent gas flow

The boundary conditions for the velocity slip and temperature and concentration jumps on the surface of a body in a rarefied multicomponent gas flow are obtained. The mathematical treatment is given in detail because of the need to examine critically some previous results which disagree with each other in spite of the fact that the initial premises and the methods of solution were the same. The results of this study, which are given in a convenient form, represent the boundary conditions for both the simplified and the complete Navier-Stokes equations in problems of hypersonic rarefied gas flow past bodies with a catalytically active surface.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–168, January–February, 1996.     

Propagation of discontinuity waves in complex rheological media with viscous properties

The propagation of discontinuity waves of various order in rheological media is examined. It is assumed that the region of discontinuity of values can be represented by an intermediate layer of infinitesimal thickness. By means of this representation, results can be obtained for a rather wide class of continuous media with viscous properties, which generalize Duhem's results. The first integrals of the laws of momentum and energy conservation are obtained, which hold inside the intermediate layer at a shock wave.It is shown that when viscosity elements are introduced in a special way into the rheological model of a continuous medium, discontinuity waves of any order are propagated in the medium, and that at the surface of a strong discontinuity in a heat-conducting medium, the temperature is continuous. Additional conditions for strain discontinuities at the viscosity elements are obtained. For certain inclusions of the viscosity elements into the rheological model discontinuity waves do not propagate; instead there is merely a weak discontinuity surface which acts as an interface between the flow region of the continuous medium and the region in the state of rest. Contact discontinuities can occur in any continuous medium.The possible existence of a geometrical discontinuity surface in a viscous gas was examined first by Duhem [1]. He established that singluar strong-discontinuity surfaces cannot take place in a viscous gas. However, if one assumes that the velocity and temperature are continuous in the passage through a singular surface, only contact discontinuities are possible [2].     

Turbulent boundary layer on a catalytic surface in a nonequilibrium dissociating gas

The majority of the studies which consider the flow of a dissociating gas in a turbulent boundary layer are devoted to the investigation of either frozen or equilibrium flows on a flat plate.The frozen turbulent boundary layer has been studied by Dorrance [1], Kutateladze and Leont'ev [2], and Lapin and Sergeev [3]. A study of the effect of catalytic recombination processes at the plate surface on the heat transfer in a frozen turbulent boundary layer was made by Lapin [4].Kosterin and Koshmarov [5], Ginzburg [6], Dorrance [7], and Lapin [8] have studied the turbulent boundary layer on a plate in equilibrium dissociating gas.The calculation of the heat transfer in a turbulent boundary layer on a catalytic plate surface with nonequilibrium dissociation was made by Kulgein [9]. In this study the nonequilibrium nature of the dissociation process was taken into account only in the laminar sublayer, while the flow in the turbulent core was considered frozen. The solution was found numerically using a computer by means of a laborious iteration process.The present paper reports a method for calculating the turbulent boundary layer on a flat catalytic plate with arbitrary dissociation rate. The method, constructed using the assumptions customary for turbulent boundary layer theory, is a successive approximation method. Good convergence of the method is assured by the fact that the effect of the nonequilibrium nature of the dissociation process on the parameter distribution in the boundary layer and, consequently, on the friction and heat transfer may be allowed for merely by finding corrections, usually relatively small, to the distribution of these parameters in the equilibrium or frozen flows. The basis of the study is the two-layer scheme of the turbulent boundary layer. The Prandtl and Schmidt numbers and also their turbulent analogs are taken equal to unity. As the model of the dissociating gas we use the Lighthill model of the ideal dissociating gas [10], extended by Freeman [11] to nonequilibrium flows.     

Stability of Thermocapillary Flow in a Flat Layer with Allowance for the Soret Effect

The stability of thermocapillary two-component liquid flow is studied taking into account thermal diffusion. An explicit expression is obtained to construct neutral Marangoni numbers under the assumption of monotonicity of perturbations. The thermocapillary and hydrodynamic instability mechanisms are considered. It is shown that plane perturbations are the greatest hazard to the stability of return flow.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 86–92, September–October, 2005.