scholarly journals Solving Prandtl-Blasius Boundary Layer Equation Using Maple

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Boundary Layer ◽  
Analytic Solution ◽  
Boundary Layer Equation ◽  
Approximate Analytic Solution ◽  
Convergence Radius ◽  
Classic Problem ◽  
Runge Kutta Method ◽  
Blasius Boundary Layer ◽  
Blasius Equation ◽  
Maple Code

A solution for the Prandtl-Blasius equation is essential to all kinds of boundary layer problems. This paper revisits this classic problem and presents a general Maple code as its numerical solution. The solutions were obtained from the Maple code, using the Runge-Kutta method. The study also considers convergence radius expanding and an approximate analytic solution is proposed by curve fitting. Similarly, the study resolves some boundary layer related problems and provide relevant Maple codes for these.

scholarly journals Solving Prandtl-Blasius boundary layer equation using Maple

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Boundary Layer ◽  
Analytic Solution ◽  
Boundary Layer Equation ◽  
Approximate Analytic Solution ◽  
Convergence Radius ◽  
Classic Problem ◽  
Runge Kutta Method ◽  
Blasius Boundary Layer ◽  
Blasius Equation ◽  
Maple Code

A solution for the Prandtl-Blasius equation is essential to all kinds of boundary layer problems. This paper revisits this classic problem and presents a general Maple code as its numerical solution. The solutions were obtained from the Maple code, using the Runge-Kutta method. The study also considers convergence radius expanding and an approximate analytic solution is proposed by curve fitting.

An Initial Value Method for the Solution of a Class of Nonlinear Equations in Fluid Mechanics

1970 ◽  
Vol 92 (3) ◽  
pp. 503-508 ◽  
Author(s):  
T. Y. Na
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Porous Medium ◽  
Fluid Mechanics ◽  
Boundary Layer Equation ◽  
Physical Parameters ◽  
Point Boundary ◽  
Trial And Error ◽  
Initial Value ◽  
Blasius Boundary Layer

An initial value method is introduced in this paper for the solution of a class of nonlinear two-point boundary value problems. The method can be applied to the class of equations where certain physical parameters appear either in the differential equation or in the boundary conditions or both. Application of this method to two problems in Fluid Mechanics, namely, Blasius’ boundary layer equation with suction (or blowing) and/or slip and the unsteady flow of a gas through a porous medium, are presented as illustrations of this method. The trial-and-error process usually required for the solution of such equations is eliminated.

scholarly journals Numerical Method in Two Dimensional Boundary Layer Theory for Incompressible Viscous Fluid

2013 ◽  
Vol 38 ◽  
pp. 61-73
Author(s):  
MA Haque
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Initial Value Problem ◽  
Shooting Method ◽  
Boundary Layer Equation ◽  
Incompressible Viscous Fluid ◽  
Initial Value ◽  
Box Scheme ◽  
Runge Kutta Method ◽  
Dimensional Boundary

In this paper laminar flow of incompressible viscous fluid has been considered. Here two numerical methods for solving boundary layer equation have been discussed; (i) Keller Box scheme, (ii) Shooting Method. In Shooting Method, the boundary value problem has been converted into an equivalent initial value problem. Finally the Runge-Kutta method is used to solve the initial value problem. DOI: http://dx.doi.org/10.3329/rujs.v38i0.16549 Rajshahi University J. of Sci. 38, 61-73 (2010)

scholarly journals Extension of Blasius Newtonian Boundary Layer to Blasius Non-Newtonian Boundary Layer

2021 ◽  
Vol 9 (2) ◽  
pp. 35-41
Author(s):  
Manisha Patel ◽  
Hema Surati ◽  
M G Timol
Keyword(s):  
Boundary Layer ◽  
Similarity Solutions ◽  
Power Law Model ◽  
Prandtl Model ◽  
Blasius Boundary Layer ◽  
Blasius Equation ◽  
Boundary Layer Problems ◽  
The One ◽  
Graphical Presentation ◽  
Theory Technique

Blasius equation is very well known and it aries in many boundary layer problems of fluid dynamics. In this present article, the Blasius boundary layer is extended by transforming the stress strain term from Newtonian to non-Newtonian. The extension of Blasius boundary layer is discussed using some non-newtonian fluid models like, Power-law model, Sisko model and Prandtl model. The Generalised governing partial differential equations for Blasius boundary layer for all above three models are transformed into the non-linear ordinary differewntial equations using the one parameter deductive group theory technique. The obtained similarity solutions are then solved numerically. The graphical presentation is also explained for the same. It concludes that velocity increases more rapidly when fluid index is moving from shear thickninhg to shear thininhg fluid.MSC 2020 No.: 76A05, 76D10, 76M99

An Approximate Method for the Solution of a Class of Nonlinear Equations in Fluid Mechanics and Magnetohydrodynamics

1973 ◽  
Vol 95 (3) ◽  
pp. 439-444
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Fluid Mechanics ◽  
Approximate Method ◽  
Boundary Layer Equation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Exact Results ◽  
Point Boundary ◽  
Blasius Boundary Layer

An approximate method has been developed to obtain a closed form solution of a class of nonlinear two-point boundary value problems. Application of this method to three problems in Fluid Mechanics and Magnetohydrodynamics, namely, Blasius boundary layer equation with suction in Newtonian and non-Newtonian fluids, and Falkner-Skan boundary layer equation with magnetic field, are presented as illustration of the method. The present results in the presence of suction or magnetic field differ from the exact results by about 1 to 2 percent.

scholarly journals Asymptotic Approximant for the Falkner–Skan Boundary Layer Equation

2020 ◽  
Vol 73 (1) ◽  
pp. 36-50 ◽  
Author(s):  
E R Belden ◽  
Z A Dickman ◽  
S J Weinstein ◽  
A D Archibee ◽  
E Burroughs ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Boundary Layer Equation ◽  
Power Series Solution ◽  
Closed Form Solutions ◽  
Shear Rates ◽  
Wide Range ◽  
Blasius Boundary Layer ◽  
Layer Flow ◽  
Appl Math

Summary We demonstrate that the asymptotic approximant applied to the Blasius boundary layer flow over a flat plat (Barlow et al., Q. J. Mech. Appl. Math. 70 (2017) 21–48.) yields accurate analytic closed-form solutions to the Falkner–Skan boundary layer equation for flow over a wedge having angle $\beta\pi/2$ to the horizontal. A wide range of wedge angles satisfying $\beta\in[-0.198837735, 1]$ are considered, and the previously established non-unique solutions for $\beta<0$ having positive and negative shear rates along the wedge are accurately represented. The approximant is used to determine the singularities in the complex plane that prescribe the radius of convergence of the power series solution to the Falkner–Skan equation. An attractive feature of the approximant is that it may be constructed quickly by recursion compared with traditional Padé approximants that require a matrix inversion. The accuracy of the approximant is verified by numerical solutions, and benchmark numerical values are obtained that characterize the asymptotic behavior of the Falkner–Skan solution at large distances from the wedge.

New approximate solutions of the Blasius equation

2015 ◽  
Vol 25 (7) ◽  
pp. 1590-1599 ◽  
Author(s):  
Lazhar Bougoffa ◽  
Abdul-Majid Wazwaz
Keyword(s):  
Exact Solution ◽  
Design Methodology ◽  
Analytic Solution ◽  
Approximate Solutions ◽  
Second Step ◽  
Approximate Analytic Solution ◽  
Content Type ◽  
Blasius Problem ◽  
Blasius Equation ◽  
The Given

Purpose – The purpose of this paper is to propose a reliable treatment for studying the Blasius equation, which arises in certain boundary layer problems in the fluid dynamics. The authors propose an algorithm of two steps that will introduce an exact solution to the equation, followed by a correction to that solution. An approximate analytic solution, which contains an auxiliary parameter, is obtained. A highly accurate approximate solution of Blasius equation is also provided by adding a third initial condition y ' ' (0) which demonstrates to be quite accurate by comparison with Howarth solutions. Design/methodology/approach – The approach consists of two steps. The first one is an assumption for an exact solution that satisfies the Blasius equation, but does not satisfy the given conditions. The second step depends mainly on using this assumption combined with the given conditions to derive an accurate approximation that improves the accuracy level. Findings – The obtained approximation shows an enhancement over some of the existing techniques. Comparing the calculated approximations confirm the enhancement that the derived approximation presents. Originality/value – In this work, a new approximate analytical solution of the Blasius problem is obtained, which demonstrates to be quite accurate by comparison with Howarth solutions.

Acoustic receptivity and evolution of two-dimensional and oblique disturbances in a Blasius boundary layer

2001 ◽  
Vol 432 ◽  
pp. 69-90 ◽  
Author(s):  
RUDOLPH A. KING ◽  
KENNETH S. BREUER
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Acoustic Wave ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Surface Waviness ◽  
S Wave ◽  
Boundary Layer Instability ◽  
Blasius Boundary Layer ◽  
Tollmien Schlichting

An experimental investigation was conducted to examine acoustic receptivity and subsequent boundary-layer instability evolution for a Blasius boundary layer formed on a flat plate in the presence of two-dimensional and oblique (three-dimensional) surface waviness. The effect of the non-localized surface roughness geometry and acoustic wave amplitude on the receptivity process was explored. The surface roughness had a well-defined wavenumber spectrum with fundamental wavenumber kw. A planar downstream-travelling acoustic wave was created to temporally excite the flow near the resonance frequency of an unstable eigenmode corresponding to kts = kw. The range of acoustic forcing levels, ε, and roughness heights, Δh, examined resulted in a linear dependence of receptivity coefficients; however, the larger values of the forcing combination εΔh resulted in subsequent nonlinear development of the Tollmien–Schlichting (T–S) wave. This study provides the first experimental evidence of a marked increase in the receptivity coefficient with increasing obliqueness of the surface waviness in excellent agreement with theory. Detuning of the two-dimensional and oblique disturbances was investigated by varying the streamwise wall-roughness wavenumber αw and measuring the T–S response. For the configuration where laminar-to-turbulent breakdown occurred, the breakdown process was found to be dominated by energy at the fundamental and harmonic frequencies, indicative of K-type breakdown.

The laminar boundary-layer equation: a method of solution by means of an automatic computer

1955 ◽  
Vol 51 (2) ◽  
pp. 320-332 ◽  
Author(s):  
D. C. F. Leigh
Keyword(s):  
Boundary Layer ◽  
Asymptotic Solution ◽  
Laminar Boundary Layer ◽  
Iterative Process ◽  
Boundary Layer Equation ◽  
Algebraic Equations ◽  
Automatic Computer ◽  
Linear Algebraic Equations ◽  
Method Of Solution ◽  
Incompressible Boundary

ABSTRACTA method, very suitable for use with an automatic computer, of solving the Hartree-Womersley approximation to the incompressible boundary-layer equation is developed. It is based on an iterative process and the Choleski method of solving a simultaneous set of linear algebraic equations. The programming of this method for an automatic computer is discussed. Tables of a solution of the boundary-layer equation in a region upstream of the separation point are given. In the upstream neighbourhood of separation this solution is compared with Goldstein's asymptotic solution and the agreement is good.

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