A note on similarity solutions for boundary layer flows with prescribed heat flux

We study the third‐order nonlinear equation: f′′′ + (m + 2)ff′′ ? (2m + 1)f′² = 0 on (0, ∞), subject to the boundary conditions f(0) = ? γ ∈ ?, f′(∞) = 0 and f′′(0) = ?1. The problem arises in the study of similarity solutions for boundary layer flows with prescribed heat flux. We will address the following two open questions which were proposed by Brighi and Hoernel (Math. Methods Appl. Sci. 2005; 28 : 479–503): The first one is the uniqueness of bounded solutions for and γ>0, and the second one is the structure of solutions for and γ?0. Copyright © 2007 John Wiley & Sons, Ltd.     

On the structure of similarity solutions of a differential equation arising from boundary layer problem

We consider an ordinary differential equation with f(0)=a, f^′(0)=1, f^′(∞):=limt→∞f^′(t)=0, where β is a real constant. The given problem may arise from the study of steady free convection flow over a vertical semi-infinite flat plate in a porous medium, or the study of a boundary layer flow over a vertical stretching wall. In this paper, the structure of solutions for the cases of β?−2 is studied. Combining the results of [B. Brighi, T. Sari, Blowing-up coordinates for a similarity boundary layer equation, Discrete Contin. Dyn. Syst. 5 (2005) 929-948; J.-S. Guo, J.-C. Tsai, The structure of solution for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math. 22 (2005) 311-351; J.-C. Tsai, Similarity solutions for boundary layer flows with prescribed surface temperature, Appl. Math. Lett. 21 (1) (2008) 67-73], we conclude that the given problem may possess at most two types solutions for β∈R. Moreover, multiple solutions are also verified for various pairs of (a,β).     

On similarity solutions for boundary layer flows with prescribed heat flux

This paper is concerned with existence, uniqueness and behaviour of the solutions of the autonomous third‐order non‐linear differential equation f?+(m+2)f f″?(2m+1)f′²=0 on ?⁺ with the boundary conditions f(0)=?γ, f′(∞)=0 and f″(0)=?1. This problem arises when looking for similarity solutions for boundary layer flows with prescribed heat flux. To study solutions we use some direct approach as well as blowing‐up co‐ordinates to obtain a plane dynamical system. Copyright © 2004 John Wiley & Sons, Ltd.     

A note on the similarity equations arising in free convection boundary layers with blowing and suction

The similarity equations for the free convection boundary-layer flow on a vertical plate with prescribed wall temperature and transpiration velocity are considered. The range of existence of solution is discussed first. For blowing this is seen to be independent of the transpiration parameter , depending only on the Prandtl number. For suction this range of existence of solutions is seen to depend on as well. Asymptotic solutions for strong suction and strong blowing are derived and compared with numerical solutions of the similarity equations.     

A note on some similarity solutions of the diffusion equation

Summary A first integral of the diffusion equation exists for similarity solutions when the diffusivity obeys a power or exponential law. Structure of the solutions in both cases and connection to an optimization result are discussed for an arbitrary diffusivity.

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Résumé Une intégrale première de l'équation de diffusion existe pour les solutions similaires quand le coefficient de diffusion obéit une loi de puissance arbitraire ou une loi exponentielle. La structure des solutions est discutée dans les deux cas ainsi que leur relation avec les résultats d'optimisation pour une diffusion arbitraire.

     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

A new set of solutions to a singular second-order differential equation arising in boundary layer theory

The paper studies a boundary value problem of a nonlinear second-order differential equation that governs the laminar flow of Newtonian and non-Newtonian fluids. It presents a new sufficient condition for the existence of solutions to the boundary value problem.     

Existence of solutions of laminar boundary layer equations with decelerating external flows

New results on the existence and nonexistence of solutions of laminar boundary layer equations with decelerating external flows are obtained. Some previous results treated these equations with accelerating or constant external flows. Our approach is to establish a system of two integral equations with singularities which will be proved to be equivalent to the laminar boundary layer equations and prove that the system has solutions by using the Leray–Schauder fixed point theorem and the Helly selection principle.     

Series solutions of nano boundary layer flows by means of the homotopy analysis method

We present here a ‘similar’ solution for the nano boundary layer with nonlinear Navier boundary condition. Three types of flows are considered: (i) the flow past a wedge; (ii) the flow in a convergent channel; (iii) the flow driven by an exponentially-varying outer flows. The resulting differential equations are solved by the homotopy analysis method. Different from the perturbation methods, the present method is independent of small physical parameters so that it is applicable for not only weak but also strong nonlinear flow phenomena. Numerical results are compared with the available exact results to demonstrate the validity of the present solution. The effects of the slip length ?, the index parameters n and m on the velocity profile and the tangential stress are investigated and discussed.     

A common error made in investigation of boundary layer flows

The concept of boundary layer flow, introduced in 1904 by Prandtl, is a popular field in Fluid Mechanics for engineers, physicists and mathematicians. The present work is a critique to many papers published in the last 15 years in the field of boundary layer flow. The critique concerns the shape of velocity, temperature and concentration profiles which are truncated due to small calculation domain used during the numerical solution procedure. These truncated profiles are not compatible with the boundary layer theory and introduce errors in wall shear stress and wall heat transfer values.     

A note on a class of exact solutions in inviscid rotational flows

The inviscid rotational flow field about an infinite cylinder of arbitrary cross-section is determined when the free stream velocity has a component perpendicular to the generators of the cylinder and an arbitrary non-uniform component parallel to the generators.

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Résumé Le champ d'écoulement rotationnel d'un fluide parfait autour d'un cylindre infini et d'un profil arbitraire est determiné quand la vitesse à écoulement libre a un composant perpendiculaire aux générateurs du cylindre et un composant non-uniforme arbitraire parallèle aux générateurs.

     

Nonideal magnetogasdynamic flows in boundary layer approximation

Zusammenfassung In der vorliegenden Arbeit werden schwach konvergierende und schwach divergierende MGD-Strömungen in der Grenzschichtnäherung untersucht. Das Verhalten dieser magnetischen Grenzschicht, charakterisiert durch alleinige Berücksichtigung der magnetischen Viskosität und Vernachlässigung der dynamischen Viskosität, wird durch die starke Kopplung zwischen Magnetfeld und Strömungsfeld bestimmt. Es werden im allgemeinen durch diese Kopplung die MGD-Grenzschichtgleichungen komplizierter, so dass deren analytische Lösung nur mit Hilfe von Ähnlichkeitstransformationen möglich ist. Nur im GrenzfallR _M (magnetische Reynoldszahl sehr gross) erhält man eine analytische Lösung. Mit Hilfe bestimmter Symmetriebedingungen werden die Lösungen auf symmetrische Strömungen eingeschränkt (freier Strahl oder Kanalströmungen). Schliesslich wird noch gezeigt, dass die endliche elektrische Leitfähigkeit die Ursache der freien Turbulenz gewisser Jet-Konfigurationen ist.

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This research has been sponsored in part by the United States Government under Contract No. F61052-67-C-0014.     

A note on the similarity depth

In this note we demonstrate the existence of E0L forms F and G which are n-similar, i.e. $\text{L}$ⁿ(F) = $\text{L}$ⁿ(G) but $\text{L}$ⁿ⁺¹(F)≠$\text{L}$ⁿ⁺¹(G) for n ∈ {2, 3}. This partially solves an open problem from [3].     

Expansion problems in the linear stability of boundary layer flows

Several problems in the linearized stability of boundary layers are examined. They are all treated as perturbations of constant coefficient differential operators. Spectral theory and spectral expansions are developed. Possible anomalies, which might arise for nonparallel boundary layer flows with nonzero transverse component at infinity are also handled.