The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Boyd, John P.

doi:10.1137/070681594

Cited by 61 publications

(73 citation statements)

References 33 publications

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“…From equation it can also be shown that

ψ = 2 \frac{d^{2} f}{d η^{2}},

thus yielding ψ 0 = 2 κ , where the constant κ ≈ 0.322 is the second derivative of the Blasius function at the origin,

d^{2} f (0) / d η^{2}

. The value of κ is generally obtained by means of the well‐established noniterative Töpfer method [ Töpfer , ], and it was given to 16 digits precision by Boyd []. Further, in this work, we will improve this precision by a similar method.…”

Section: Estimation Of the Radius Of Convergence Through Complex‐planmentioning

confidence: 99%

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An exact series and improved numerical and approximate solutions for the Boussinesq equation

Chor

Dias

Zárate

2013

Water Resources Research

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[1] We derive a series solution for the nonlinear Boussinesq equation in terms of the similarity variable of the Boltzmann transformation in a semi-infinite domain. The first few coefficients of the series have been known for a long time, having been obtained by a truncated inversion of the series solution of the Blasius equation, but no direct recurrence relation was known for the complete series representing the solution of the Boussinesq equation. The series turns out to have a finite radius of convergence, which we estimate with a numerical complex-plane integration method that identifies the singularities of the solution when the equation is extended to the complex plane. The homogeneous condition at the origin produces a singularity which complicates numerical solutions with RungeKutta methods. We present two variable transformations that circumvent the problem and that are best suited to the complex-variable and the real-variable versions of the equation, respectively. Using those tools, an approximate solution accurate to 1.75 Â 10 À10 and valid for the entire positive real axis is then developed by matching a Pade approximant of the exact series and an asymptotic solution (to overcome the restriction imposed by the finite radius of convergence of the series), along the same lines of the expression proposed by Hogarth and Parlange (1999). The accuracies of all of the existing and the newly proposed solutions are obtained.

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“…From equation it can also be shown that

ψ = 2 \frac{d^{2} f}{d η^{2}},

thus yielding ψ 0 = 2 κ , where the constant κ ≈ 0.322 is the second derivative of the Blasius function at the origin,

d^{2} f (0) / d η^{2}

Section: Estimation Of the Radius Of Convergence Through Complex‐planmentioning

confidence: 99%

“…Figure presents |Ф( ζ )| in the complex plane. Notice that |Ф( ζ )| does not blow up in the region analyzed, which is different from the behavior of the complex‐valued Blasius function, which has poles in the complex plane [ Boyd , ].…”

Section: Estimation Of the Radius Of Convergence Through Complex‐planmentioning

confidence: 99%

An exact series and improved numerical and approximate solutions for the Boussinesq equation

Chor

Dias

Zárate

2013

Water Resources Research

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“…Asaithambi [33] found this number correct to nine decimal positions as 0.332057336. In 2008, Boyd [34] reported 0.33205733621519630 as f (0) in the Blasius equation.…”

Section: The Blasius Problemmentioning

confidence: 99%

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

Parand

Nikarya

Rad

et al. 2012

Zeitschrift Für Naturforschung A

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In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on R and is convergent for any x ∈ R. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

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“…Very recently, Tajvidi et al [24] has used a modified rational Legendre method, to show that γ = 0.33209 with a 0.009% relative error. By the fourth-order Runge-Kutta method γ is determined γ «0.33205733621 5 19630, where all the sixteen decimal places are believed correct [8]. A fully analytical solution (i.e.…”

Section: Power Series Solutionsmentioning

confidence: 99%

Similarity solution of boundary layer flows for non-Newtonian fluids

Bognár¹

2009

International Journal of Nonlinear Sciences and Numerical Simulation

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The present paper deals with the analysis of similarity solutions of the two-dimensional boundary layer flow of a power-law non-Newtonian fluid past a semi-infinite flat plate. The boundary value problem of the momentum equation is converted into an initial value problem using a Töpfer-like transformation. The dimensionless wall gradient is determined numerically. The existence of the power series solutions for the problem is presented and the convergence radius of the proposed solutions is estimated.

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The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Cited by 61 publications

References 33 publications

An exact series and improved numerical and approximate solutions for the Boussinesq equation

An exact series and improved numerical and approximate solutions for the Boussinesq equation

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

Similarity solution of boundary layer flows for non-Newtonian fluids

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