Asymptotics of solutions to boundary layer equations for a generalized newtonian medium under a close to symmetric external flow

We study the qualitative behavior of solutions to the boundary layer equations of the generalized Newtonian fluid model proposed by O. A. Ladyzhenskaya. Under certain conditions on an external flow, we show that the solution to the boundary layer equations asymptotically tends to the automodel one. Bibliography: 8 titles.     

The boundary layer analysis for stokes equations on a half space

The time dependent Stokes equations on a half space are considered. We decompose the solution of these equations in an inviscid solution: a boundary layer solution and a correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction is shown to be of the same order of magnitude of the square root of the viscosity.     

Junction layer analysis in one-dimensional steady-state Euler-Poisson equations

We study the quasi-neutral limit in one-dimensional steady-state Euler-Poisson equations with junction layers. Typically, the junction layer phenomenon occurs in a ballistic diode of a semiconductor device where the doping profile is a discontinuous function. We derive the junction layer equations and prove the existence of their solutions which decay exponentially. Finally, we justify the quasi-neutral limit with junction layers by giving uniform error estimates.     

A model of unsteady flow separation from a cylinder without boundary layer equations

Presented at the seminar of the Department of Computational Mathematics, May 26, 1983.     

Minimum error solutions of boundary layer equations

The minimum error solutions of boundary layer equations in the least square sense have been studied by employing the Euler-Lagrange equations. To test the method a class of problems,i.e., boundary layer on a flat plate, Hiemenz flow, boundary layer on a moving sheet and boundary layer in non-Newtonian fluids have been studied. The comparison of the results with approximate methods, like Karman-Pohlhuasen, local potential and other variational methods, shows that the present predictions are invariably better.     

Exact analytical solution for a thermal boundary layer in a saturated porous medium

The forced convection thermal boundary layer in a porous medium as an analytically tractable special case of a mixed convection problem is considered. It is shown that some general features of the mixed convection solutions reported recently by other authors [B. Brighi, J.-D. Hoernel, On the concave and convex solutions of mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett. (published online, 2005); M. Guedda, Multiple solutions of mixed convection boundary layer approximations in a porous medium, Appl. Math. Lett. (published online, 2005)] can already be recovered from this exactly solvable case.     

Mixed boundary value problems for steady-state magnetohydrodynamic equations of viscous incompressible fluid

The inhomogeneous boundary value problem for the steady-state magnetohydrodynamic equations of viscous incompressible fluid under the Dirichlet conditions for the velocity and mixed boundary conditions for the electromagnetic field is considered. Sufficient conditions for the data that ensure the global solvability of this problem and the local uniqueness of its solution are found.     

Some Fredholm equations with boundary layer resonance

Certain singularly perturbed differential equations which exhibit boundary layer resonance are difficult to solve by the application of standard asymptotic methods. After reformulation as a singularly perturbed integral equation and treatment by a recently developed asymptotic methodology, the desired solution is obtained in a straightforward manner.     

The Crocco transformation for the three-dimensional Prandtl boundary layer equations

The three-dimensional fluid flow of a steady or unsteady moving incompressible continuous medium is considered. Several Crocco-type transformations are applied which reduce the system of three equations of the Prandtl boundary layer to a system of two strongly coupled parabolic equations.     

Asymptotic solution of the unsteady boundary layer equations

Summary An extension of the Meksyn asymptotic method to unsteady boundary layers in laminar, incompressible flow is investigated. The results indicate that unsteady boundary layers can be calculated by the Meksyn asymptotic method with comparable accuracy to that obtained for steady flows. Several differences from the well developed steady-flow application exist and require further work before general problems can be treated. The calculation technique is more straight-forward for cases involving acceleration because three or four terms in the expansions may then yield sufficient accuracy. The form of the governing equation required by the Meksyn method indicates that it is most useful for unsteady stagnation boundary layers since some basic unsteady flows are not directly accessible in their simplest form from that equation. The effect of unsteadiness on the rate of asymptotic convergence is assessed by detailed comparison of a similar solution for unsteady, stagnation flow with analogous results from the Falkner-Skan equation and of reliable numerical results for both cases.

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Résumé On étudie une extension de la méthode asymptotique de Meksyn aux couches limites instables des écoulements laminaires de fluides incompressibles. Les résultats montrent que les couches limites instables peuvent être calculées à l'aide de la méthode asymptotique de Meksyn avec une précision comparable à celle obtenue pour les écoulements stables. Plusieurs différences existent par rapport à l'application, bien mise au point, aux écoulements stables; elles demandent encore du travail avant que les problèmes généraux puissent être traités. La méthode de calcul est plus directe dans les cas impliquant une accélération, car 3 ou 4 termes dans les développements assurent alors une précision suffisante. La forme de l'équation principale nécessaire à la méthode de Meksyn indique qu'elle est très utile pour les couches limites instables au repos; en effet, certains écoulements instables de base ne peuvent être atteints directement dans leur forme la plus simple à partir de cette équation. L'effet de l'instabilité sur la vitesse de convergence asymptotique est établi grâce à une comparaison détaillée d'une solution analogue pour un écoulement instable stagnant avec les résultats semblables obtenus par l'équation Falkner-Skan, et des résultats numériques sûrs obtenus dans les deux cas.

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This work was supported by the Energy Research and Development Administration.     

A class of boundary-value problems for a system of nonlinear equations with a free boundary

The method of fixed domains is applied to derive conditions which ensure unique solvability of boundary-value problems for a system of nonlinear evolution equations with a partially free boundary.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 83–92, 1988.     

A note on the ill-posedness of shear flow for the MHD boundary layer equations

For the two-dimensional Magnetohydrodynamics(MHD)boundary layer system,it has been shown that the non-degenerate tangential magnetic field leads to the well-posedness in Sobolev spaces and high Reynolds number limits without any monotonicity condition on the velocity field in our previous works.This paper aims to show that sufficient degeneracy in the tangential magnetic field at a non-degenerate critical point of the tangential velocity field of shear flow indeed yields instability as for the classical Prandtl equations without magnetic field studied by G′erard-Varet and Dormy(2010).This partially shows the necessity of the non-degeneracy in the tangential magnetic field for the stability of the boundary layer of MHD in 2D at least in Sobolev spaces.     

A problem with normal derivatives in boundary conditions for a system of differential equations

We seek for a solution to a system of differential equations, using linear relations connecting normal derivatives of the desired functions at the domain boundary.     

Similar solutions of the boundary layer equations for power law fluids

Zusammenfassung Grenzschichten in nicht-newtonschen Medien werden für den Fall berechnet, dass die Geschwindigkeit der Potentialströmung, ausgehend von einem Staupunkt, sich nach einem Potenzgesetz entlang der Oberfläche ändert.Als spezielle Fälle werden behandelt: Strömungen 1) an einem Keil, 2) entlang einer ebenen Platte, 3) in einem konvergenten Kanal, 4) in der Nähe einer ebenen Staulinie.     

A finite time singularity of the boundary layer equations of natural convection

Summary The unsteady natural convection flow about a horizontal circular cylinder at whose surface the heat flux is constant is considered. A numerical solution of the equations valid in the vicinity of the uppermost generator of the cylinder displays singular behaviour after a finite time. It is demonstrated that the breakdown of the solution is consistent with a triple layer structure characterised by an inertia dominated central region.

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Zusammenfassung Wir untersuchen die nicht-stationäre natürliche Konvektion entlang eines horizontalen Zylinders mit konstantem Oberflächenwärmefluß. Eine in der Umgebung des obersten Generators des Zylinders gültige numerische Lösung führt nach endlicher Zeit zu einem singulären Verhalten. Es wird gezeigt, daß das Zusammenbrechen der Lösung konsistent ist mit einer Dreischichtstruktur in der der mittlere Bereich von der Trägheit dominiert wird.

     

Equations of magnetohydrodynamic boundary layer with an injection of a dilatable medium

Methods of boundary layer theory are efficient when studying problems of fluid dynamics of multicomponent media with pronounced boundaries between different components. For example, such problems arise when placing screens isolating the main flow of a medium from the surface past which the medium flows, when investigating the mixing layer on the boundary of a submerged jet, when considering thin layers of plasma generated on the surface of a body entering dense layers of the atmosphere at a high velocity, and so on. All such problems are characterized by the presence of an internal boundary on which the rheological and electromagnetic properties of the continuous medium can change. This leads to various diffraction problems for systems of quasilinear partial differential equations. In the present paper, we consider one such problem.     

Numerical simulation of the turbulent boundary layer equations via a Runge-Kutta algorithm

This paper addresses a hybrid computational procedure for the step-by-step calculation of momentum transfer in turbulent boundary layer flows along flat plates. The proposed procedure relies on a modified method of lines wherein transversal discretizations are carried out by a “control volume” being infinitesimal in the streamwise direction and finite in the transversal direction of the fluid flow. Using mixing length theory and coarse intervals in the transversal direction, the resulting system of ordinary differential equations of first order may be readily integrated on a personal computer utilizing a fourth-order Runge-Kutta algorithm. In general, a maximum number of sixteen lines is necessary at the trailing edge of the flat plate for a typical calculation. As a consequence, computing time and storage for each run were very small when compared to other finite-difference methods. Furthermore, to validate the hybrid procedure involving the method of lines and control volumes (MOLCV), comparisons with experimental data have been done in terms of both velocity distributions and local skin friction coefficients. For all cases tested, the proposed methodology predicts the growth of the boundary layer of gases correctly.