A numerical solution of Blasius equation on a semi-infinity flat plate

El-Gamel, Mohamed; El-Shenawy, Atallah

doi:10.1007/s40324-017-0145-x

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…The Blasius equation is the well-known third-order nonlinear ordinary differential equation that appeared in several boundary layer problems involving a fluid's two-dimensional laminar viscous flow through a flat plate. The following equation presents it as a governing equation for fluid dynamics [41]:…”

Section: The Blasius Equationmentioning

confidence: 99%

“…Furthermore, the inner product for the left and right sides of the Equations ( 38), ( 40), (42), and (44), respectively, is implemented to obtain the values of 𝑨 = [𝑎 0 𝑎 1 𝑎 2 … 𝑎 𝑛 ] 𝑇 by solving the algebraic system of equations. Then, the desired novel approximate solutions are achieved by applying the initial conditions to Equations ( 37), ( 39), (41), and (43), respectively.…”

Section: Solving the Blasius Equation By The Ecm And I-ecmsmentioning

confidence: 99%

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Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Mahdi Salih,

A. AL-Jawary,

Mustafa Turkyilmazoglu

2023

IHJPAS

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This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

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Section: The Blasius Equationmentioning

confidence: 99%

Section: Solving the Blasius Equation By The Ecm And I-ecmsmentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Mahdi Salih,

A. AL-Jawary,

Mustafa Turkyilmazoglu

2023

IHJPAS

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“…Because of their compact support, numerical methods involving B-spline as a basis function give systems including band matrices. Developments to Bspline function and methods have been implemented in the fundamental work of El-Gamel and his students, 53,54 Caglar and Caglar, 55 Zhang et al, 56 and Goh et al 57 B-spline method was developed to handle a semi-infinity flat plate by El-Gamel and El-Shenawy, 58 and very recently, Cubic B-spline solution to second-order singular linear PDEs was also introduced by El-Gamel and El-Shamy. 59 In the literature, we did not find any previous work for solving nonlinear Jeffery-Hamel fluid flow problems via Bernstein and B-splines to the best of our knowledge.…”

Section: Introductionmentioning

confidence: 99%

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

El‐Shenawy,

El‐Gamel,

Abd El‐Hady

2024

Heat Trans

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The Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation.

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