Self-similar solutions of three-dimensional laminar magnetohydrodynamic boundary-layer equations

Self-similar solutions of three-dimensional boundary-layer equations of an incompressible fluid in ordinary hydrodynamics were considered in [1–3] et al. The present work looks for self-similar solutions of three-dimensional magnetohydrodynamic boundary-layer equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 10–17, July–August, 1968.     

Invariant solutions of the three-dimensional laminar boundary-layer equations on developing surfaces

Exact solutions play an important part in boundary-layer theory. The traditional path to obtaining these solutions is to decrease the number of independent variables in the boundary layer equations. The aim of this paper is to describe the structure of the solutions for which such a simplification is possible.     

Asymptotic solution of the three-dimensional multicomponent laminar boundary-layer equations with large suction

The flow of a multicomponent compressible gas in a three-dimensional laminar boundary layer is investigated for large values of the suction parameter. Asymptotic expressions are derived for the profiles of the velocities, temperatures, and concentration of the components across the boundary layer, as well as for the friction, heat-, and mass-transfer coefficients on the surface of the body.The authors wish to thank G. A. Tirskii for a discussion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 41–45, July–August, 1975.     

Asymptotic solutions of non-self-similar boundary-layer equations

In this paper asymptotic solutions near the outer boundary are found for the non-self-similar laminar boundary-layer equations in an incompressible fluid and a compressible gas. It is shown that the nature of the asymptotic solution is determined by the form of the initial velocity and enthalpy profiles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 29–34, November–December, 1976.In conclusion, the author is grateful to N. M. Belyanin and F. A. Slobodkina for attention to the research and discussion of the results.     

Some self-similar solutions of Grad's equations

It is shown that the self-similar solutions of the Navier-Stokes and Burnett equations found earlier by the authors [1–9] can be extended to the case of two-dimensional flows of a weakly rarefied gas described by Grad's equations. Examples are given of numerical realization of self-similar solutions for flow in an expanding planar channel. It is found that there are appreciable differences between the behavior of the self-similar solutions of the Navier-Stokes, Burnett, and Grad equations in the neighborhood of a channel wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 88–94, May–June, 1982.     

Second-order boundary-layer effects for the unsteady laminar incompressible three-dimensional stagnation-point flow

All the second-order boundary-layer effects on the unsteady laminar incompressible flow at the stagnation-point of a three-dimensional body for both nodal and saddle point regions have been studied. It has been assumed that the free-stream velocity, wall temperature and mass transfer vary arbitrarily with time. The effect of the Prandtl number has been taken into account. The partial differential equations governing the flow have been derived for the first time and then solved numerically unsteady free-stream velocity distributions, the nature of the using an implicit finite-difference scheme. It is found that the stagnation point and the mass transfer strongly affect the skin friction and heat transfer whereas the effects of the Prandtl number and the variation of the wall temperature with time are only on the heat transfer. The skin friction due to the combined effects of first- and second-order boundary layers is less than the skin friction due to, the first-order boundary layers whereas the heat transfer has the opposite behaviour. Suction increases the skin friction and heat transfer but injection does the opposite.

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Grenzschichteffekte zweiter Ordnung für laminare inkompressible dreidimensionale Staupunktströmung

Zusammenfassung Es wurden Grenzschichteffekte zweiter Ordnung bei instationärer laminarer inkompressibler Strömung am Staupunkt eines dreidimensionalen Körpers, sowohl für knotenartige als auch für sattelförmige Bereiche studiert. Dabei wurde angenommen, daß die Freistrahlgeschwindigkeit, die Wandtemperatur und der Stoffübergang beliebig mit der Zeit variieren. Der Einfluß Prandtlzahl wurde mit in Betracht gezogen. Die partiellen Differentialgleichungen welche die Strömung beschreiben, wurden zum ersten Mal abgeleitet und dann numerisch gelöst, wobei ein implizites finites Differenzverfahren verwendet wurde.Es wurde gefunden, daß die Verteilung der instationären Freistrahlgeschwindigkeit die Natur des Staupunktes und der Stofftransport die Wandreibung und den Wärmeübergang stark beeinflussen, während die Prandtl-Zahl und zeitlich veränderliche Wandtemperaturen sich nur auf den Wärmeübergang auswirken. Die Wandreibung infolge der kombinierten Einflüsse von Grenzschichten erster und zweiter Ordnung ist kleiner als diejenige infolge der Grenzschichten erster Ordnung, während der Wärmetransport umgekehrtes Verhalten zeigt. Eine Absaugung erhöht die Wandreibung und den Wärmeübergang Zublasen hat umgekehrte Wirkung.

     

Semi-similar solutions to the three-dimensional laminar boundary layer

The theory of semi-similar solutions is developed for and applied to the problem of three-dimensional laminar boundary layer flow. A number of specific examples are calculated. Particular attention is given to certain flows in which separation is approached and the nature of three-dimensional laminar boundary layer separation is inferred from the behavior of these solutions close to separation. Two types of separation are observed: singular separation characterized by the vanishing of the total shear along the line of separation and ordinary separation characterized by limiting streamlines which become parallel to the line of separation.     

Similarity solutions of a class of laminar three-dimensional boundary layer equations of power law fluids

An analysis of the possibility of finding similarity solutions to the three-dimensional, steady, incompressible, boundary layer equations in rectangular coordinates for a power law fluid is investigated. It is found that, in general, the two components of the mainstream flow must differ by at most a multiplicative constant and that these components are powers or exponentials of the x'-coordinate.

By assuming small cross-flows, the cross flow component may be generalized and found to be representable by a polynomial in the through flow variable, x'.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

Uniqueness of positive solutions of semilinear equations in ℝn

Archive for Rational Mechanics and Analysis -     

Mechanism of three-dimensional boundary-layer receptivity

Boundary-layer receptivity is always a hot issue in laminar-turbulent transition. Most actual laminar-turbulent transitions belong to three-dimensional flows. An infinite back-swept flat-plate boundary layer is a typical three-dimensional flow. Study of its receptivity is important both in theory and applications. In this paper, a freestream turbulence model is established. A modified fourth-order Runge-Kutta scheme is used for time marching, and compact finite difference schemes are used for space discretization. On these bases, whether unsteady cross-flow vortices can be excited in the three-dimensional boundary layer(the infinite back-swept flat-plate boundary layer) by free-stream turbulence is studied numerically. If so, effects of the level and the direction of free-stream turbulence on the three-dimensional boundary-layer receptivity are further studied. Differences of the three-dimensional boundary-layer receptivity are then discussed by considering the non-parallel effect, influence of the leading-edge stagnation point of the flat plate, and variation of the back-swept angle separately. Intensive studies on the three-dimensional boundary-layer receptivity will benefit the development of the hydrodynamic stability theory, and provide a theoretical basis for prediction and control of laminar-turbulent transition.     

On the existence of self-similar solutions of the equations governing unsteady flow through a porous medium

In this paper the conditions for the existence of self-similar solutions of the equations governing unsteady flows through a porous medium are presented and discussed. The first two sections deal with the case of non-Newtonian fluids of power-law behavior; the third section analyzes the case of non-Darcy gas flows. The boundary and initial conditions occuring currently in a large class of fluid mechanics problems, of practical interest in engineering, are considered.     

Non-Newtonian laminar boundary-layer heat transfer in stirred tanks

 A laminar boundary-layer heat transfer in a jacketed, stirred tank with non-Newtonian fluids has been experimentally and theoretically examined. Heat transfer rates from jacket to process liquid through the stirred tank wall were measured for different large-scale impeller designs and rheological properties. On the basis of the laminar boundary-layer theory, we developed a theoretical correlation for non-Newtonian laminar tank-side film heat transfer coefficients in stirred tanks. The correlation was derived using von Karman's integral technique. Satisfactory agreement between the predictions of the proposed correlation and the present experimental data for viscous Newtonian and non-Newtonian laminar heat transfer in stirred tanks was obtained. Received on 20 February 2001 / Published online: 29 November 2001     

Nonsimilar incompressible laminar boundary-layer flows in micropolar fluids

Summary The solution of the steady laminar incompressible nonsimilar boundary-layer problem for micropolar fluids over two-dimensional and axisymmetric bodies has been presented. The partial differential equations governing the flow have been transformed into new co-ordinates having finite range. The resulting equations have been solved numerically using implicit finite-difference scheme. The computations have been carried out for a cylinder and a sphere. The results indicate that the separation in micropolar fluids occurs at earlier streamwise locations as compared to Newtonian fluids. The skin friction and velocity profiles depend on the shape of the body and are almost insensitive to microrotation or coupling parameter, provided the coupling parameter is small. On the other hand, the microrotation profiles and microrotation gradient depend on the microrotation parameter and they are insensitive to the coupling parameter.

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Zusammenfassung Es wird die Lösung des stationären Grenzschichtproblems inkompressibler mikropolarer Flüssigkeiten für den Fall der Nichtähnlichkeit bei zweidimensionalen und achsensymmetrischen Körpern vorgelegt. Die dem Problem zugrunde liegenden partiellen Differentialgleichungen werden durch Einführung neuer Koordinaten auf ein endliches Gebiet transformiert. Die so erhaltenen Gleichungen werden mit Hilfe eines impliziten Differenzenverfahrens numerisch gelöst. Die Rechnung wird für den Zylinder und die Kugel durchgeführt. Die Ergebnisse zeigen, daß die Grenzschichtablösung früher erfolgt als bei vergleichbaren newtonschen Flüssigkeiten. Wandreibung und Geschwindigkeitsprofile hängen von der Gestalt des Körpers ab und sind nahezu unempfindlich gegen Mikrorotation und Kopplungsparameter, vorausgesetzt, daß der letztere klein ist. Dagegen hängen das Profil und der Gradient der Mikrorotation vom Parameter der Mikrorotation ab und sind ebenfalls unempfindlich gegen die Kopplungsparameter.

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With 6 figures