On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer

Hartree, D. R.

doi:10.1017/s0305004100019575

Cited by 442 publications

(214 citation statements)

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“…Coppel [2] proved the following important theorem on existence and uniqueness. It states that for all nonnegative f ð0Þ and f 0 ð0Þ, the second derivative f 00 ðZÞ is positive, zero or negative throughout the interval 0 r Zr1 according as f given by Hartree [3], Cebeci and Keller [4] and many more, using the well-known shooting technique. Asaithambi [5] has solved the Falkner-Skan equation numerically using finite difference scheme which is different from shooting technique.…”

Section: Introductionmentioning

confidence: 99%

A new exact solution for boundary layer flow over a stretching plate

Kudenatti

2012

International Journal of Non-Linear Mechanics

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a b s t r a c tIn this paper, we give an exact analytical solution of the Falkner-Skan equation for all values of b.Generalized similarity transformations are used to convert the Prandtl's boundary layer equations into a non-linear ordinary differential equation which accounts two important flow parameters: the pressure gradient parameter b and velocity ratio parameter E. Our exact solution method embeds a known closed-form solution for b ¼ À1 as a special case. We also give the Dirichlet's series solution to the problem for E ¼ 0, which is particularly useful when the derivative boundary condition at infinity is zero. We compare the results of both methods with that of direct numerical solution, and found that there is a good agreement between both the results. The results are presented in the form of velocity profiles and skin friction coefficient. Finally, the physical significance of the flow parameters is discussed in detail.

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Section: Introductionmentioning

confidence: 99%

A new exact solution for boundary layer flow over a stretching plate

Kudenatti

2012

International Journal of Non-Linear Mechanics

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“…Fluid properties are idealized as constant and freestream turbulence is considered negligible. For a given exponent m, the Falkner-Skan equation can be solved in a variety of ways [7,9] and for a few values they are tabulated [10][11][12] in terms of β = 2m/(1 + m). However, for the present comment we employed an Excel spread sheet kindly provided by Prof. Frank M. White for tabulations of f{}, f '{}, f "{} and f '''{} plus resulting non-dimensional integral parameters.…”

Section: Open Accessmentioning

confidence: 99%

Entropy Generation in Steady Laminar Boundary Layers with Pressure Gradients

McEligot

Walsh

2014

Entropy

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In an earlier paper in Entropy [1] we hypothesized that the entropy generation rate is the driving force for boundary layer transition from laminar to turbulent flow. Subsequently, with our colleagues we have examined the prediction of entropy generation during such transitions [2,3]. We found that reasonable predictions for engineering purposes could be obtained for flows with negligible streamwise pressure gradients by adapting the linear combination model of Emmons [4]. A question then arises-will the Emmons approach be useful for boundary layer transition with significant streamwise pressure gradients as by Nolan and Zaki [5]. In our implementation the intermittency is calculated by comparison to skin friction correlations for laminar and turbulent boundary layers and is then applied with comparable correlations for the energy dissipation coefficient (i.e., non-dimensional integral entropy generation rate). In the case of negligible pressure gradients the Blasius theory provides the necessary laminar correlations.How can one conveniently predict C f {x} and C d {x} for pure laminar boundary layers with pressure gradients? (The nomenclature we use is defined by Walsh et al. [3].) One approach would be a full computational fluid dynamics (CFD) calculation with the turbulence model suppressed. An alternate approach is to employ the Thwaites integral technique [6,7] which can be programmed on a simple spread sheet. Then, if the local freestream velocity can be approximated by a power law, U ∞ {x} ~ x m , the correlations to be used in the Thwaites approach can be derived from Falkner-Skan solutions [7,8]. OPEN ACCESS

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“…A raft of computational approaches and methodologies have been presented for the solution of the FS equation, see for example Hartree (1937), Asaithambii (1997), Asaithambi (1998Asaithambi ( , 2004bAsaithambi ( , 2005, Abbasbandy (2007), Alizadeh et al (2009) and Zhang and Chen (2009). The most widely used and 'classical' approach to numerical solution is to reduce the boundary value problem to an initial value problem via a shooting method (see Cebeci and Bradshaw (1977); Cebeci and Keller (1971) for a thorough discussion).…”

Section: Paper Received 25 October 2010 Paper Accepted 21 June 2011mentioning

confidence: 99%