scholarly journals On the Use of Recursive Evaluation of Derivatives and Padé Approximation to Solve the Blasius Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Asai Asaithambi
Keyword(s):  
Differential Equation ◽  
Computational Effort ◽  
Navier Stokes ◽  
Series Solutions ◽  
Third Order ◽  
Navier Stokes Equation ◽  
Order Problem ◽  
Algorithmic Differentiation ◽  
Blasius Problem ◽  
Recursive Evaluation

The Blasius problem is one of the well-known problems in fluid mechanics in the study of boundary layers. It is described by a third-order ordinary differential equation derived from the Navier-Stokes equation by a similarity transformation. Crocco and Wang independently transformed this third-order problem further into a second-order differential equation. Classical series solutions and their Padé approximants have been computed. These solutions however require extensive algebraic manipulations and significant computational effort. In this paper, we present a computational approach using algorithmic differentiation to obtain these series solutions. Our work produces results superior to those reported previously. Additionally, using increased precision in our calculations, we have been able to extend the usefulness of the method beyond limits where previous methods have failed.

Mechanics of a Free-Surface Liquid Film Flow

1987 ◽  
Vol 54 (4) ◽  
pp. 951-954 ◽  
Author(s):  
Cyrus K. Aidun
Keyword(s):  
Differential Equation ◽  
Free Surface ◽  
Film Flow ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equation ◽  
Higher Order Terms ◽  
Liquid Film Flow ◽  
Surface Liquid ◽  
Similar Equation

The mechanics of a free surface viscous liquid curtain flowing steadily between two vertical guide wires under the influence of gravity is investigated. The Navier-Stokes equation is integrated over the film thickness and an integro-differential equation is derived for the average film velocity. An approximate nonlinear differential equation, attributed to G. I. Taylor, is obtained by neglecting the higher order terms. An analytical solution is obtained for a similar equation which neglects the surface tension effects and the results are compared with the experimental measurements of Brown (1961).

scholarly journals Numerical Approximate Solution to Flow over a Flat Plate ( the Balusis Problem) By Perturbation Iteration- Algorithm

2021 ◽  
Author(s):  
Abeer Jasim
Keyword(s):  
Continuity Equation ◽  
Perturbation Technique ◽  
Navier Stokes ◽  
Iteration Algorithm ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Homotopy Perturbation ◽  
Blasius Problem ◽  
Layer Flow ◽  
Homotopy Perturbation Technique

This paper applied perturbation iteration- algorithm (PI-A) to solve the problem of the incompressible two-dimensional laminar boundary layer flow over a flat plat as wall as called the Blasius problem (BP). BP is governed by Navier- Stokes equation (NSE) and continuity equation which were transformed into ordinary differential equation using similarity transforms. The results presented are tabulated for similarity stream function and can be seen highly of accuracy through comparable with that obtained by Ganji et al.[3] who studied BP using Homotopy perturbation technique (HPT) , Research results for the same problem using the variationally This paper applied perturbation iteration- algorithm (PI-A) to solve the problem of the incompressible two-dimensional laminar boundary layer flow over a flat plat as wall as called the Blasius problem (BP). BP is governed by Navier- Stokes equation (NSE) and continuity equation which were transformed into ordinary differential equation using similarity transforms. The results presented are tabulated for similarity stream function and can be seen highly of accuracy through comparable with that obtained by Ganja et al.[3] who studied BP using Homotopy perturbation technique (HPT) , Research results for the same problem using the variational iteration technique (VIT) before Aiyesimi and Niyi[5] and results numerical by Blasius[1]. Finally, The method that is efficient and widely applicable for solving ODE.

scholarly journals Solusi Persamaan Burgers Inviscid dengan Metode Pemisahan Variabel

2021 ◽  
Vol 4 (2) ◽  
pp. 88
Author(s):  
Hisyam Ihsan ◽  
Syafruddin Side ◽  
Muhammad Iqbal
Keyword(s):  
Differential Equation ◽  
Stokes Equation ◽  
Burgers Equation ◽  
Equation System ◽  
Separation Method ◽  
Navier Stokes ◽  
Variable Separation ◽  
Partial Differential ◽  
Navier Stokes Equation ◽  
Variable Separation Method

Penelitian ini mengkaji tentang solusi persamaan Burgers Inviscid dengan metode pemisahan variabel. Tujuan dari penelitian ini adalah untuk mengetahui penyederhanaan sistem persamaan Navier-Stokes menjadi persamaan Burgers Inviscid, menemukan solusi persamaan Burgers Inviscid dengan metode pemisahan variabel, dan melakukan simulasi solusi persamaan dengan menggunakan software Maple18. Persamaan Burgers muncul sebagai penyederhanaan model yang rumit dari sistem persamaan Navier-Stokes. Persamaan Burgers adalah persamaan diferensial parsial hukum konservasi dan merupakan masalah hiperbolik, yaitu representasi nonlinier paling sederhana dari persamaan Navier-Stokes. Metode pemisahan variabel merupakan salah satu metode klasik yang efektif digunakan dalam menyelesaikan persamaan diferensial parsial dengan mengasumsikan  untuk mendapatkan komponen x dan t. Kemudian akan dilakukan subtitusi pada persamaan diferensial, sehingga dengan cara ini akan didapatkan solusi persamaan diferensial parsial.Kata Kunci: Persamaan Burgers Inviscid, metode pemisahan variabel, persamaan Navier-StokesThis study examines the solution of Burgers Inviscid equation with variable separation method. The purpose of this study was to find out the simplification of the Navier-Stokes equation system into the Burgers Inviscid equation, find a solution to the Burgers Inviscid equation with the variable separation method, and simulate equation solutions using Maple18 software. The Burgers equation emerged as a complicated simplification of the Navier-Stokes equation system. The Burgers equation is a partial differential equation of conservation law and is a hyperbolic problem, i.e. the simplest nonlinear representation of the Navier-Stokes equation. The variable separation method is one of the classic methods that is effectively used in solving partial differential equations assuming  to obtain the x and t components. Then there will be substitutions to differential equations, so that in this way there will be a partial differential equation solution.Keywords: Burgers Inviscid Equation, variable separation method, Navier-Stokes equations.

scholarly journals ANALYTICAL SOLUTION OF NAVIER STOKES EQUATION REDUCED TO THE EQUATION OF THIRD ORDER TO STUDY THE MOTION OF LIQUID IN A FLAT PIPE.

2020 ◽  
Vol 82 (02) ◽  
pp. 563-569
Author(s):  
Kamolkhon Karimov ◽  
◽  
Mukhiddin Khudjaev ◽  
Erkin Nematov ◽  
Tohir Hojibekov ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Third Order ◽  
Navier Stokes Equation

scholarly journals Conservation of Total Vorticity for a 2D Stochastic Navier Stokes Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Peter M. Kotelenez ◽  
Bradley T. Seadler
Keyword(s):  
Differential Equation ◽  
Stokes Equation ◽  
Continuum Limit ◽  
Empirical Process ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Spatially Correlated ◽  
State Dependent ◽  
Point Measure ◽  
Stochastic Term

We consider point vortices whose positions satisfy a stochastic ordinary differential equation on perturbed by spatially correlated Brownian noise. The associated signed point measure-valued empirical process turns out to be a weak solution to a stochastic Navier-Stokes equation (SNSE) with a state-dependent stochastic term. As the number of vortices tends to infinity, we obtain a smooth solution to the SNSE, and we prove the conservation of total vorticity in this continuum limit.

A series solution of the Falkner–Skan equation using the crocco–wang transformation

2017 ◽  
Vol 28 (11) ◽  
pp. 1750139 ◽  
Author(s):  
Asai Asaithambi
Keyword(s):  
Differential Equation ◽  
Friction Coefficient ◽  
Newton’S Method ◽  
Newton's Method ◽  
Series Solution ◽  
Skin Friction Coefficient ◽  
Point Boundary ◽  
Third Order ◽  
Order Problem ◽  
The Third

A direct series solution for the Falkner–Skan equation is obtained by first transforming the problem using the Crocco–Wang transformation. The transformation converts the third-order problem to a second-order two-point boundary value problem. The method first constructs a series involving the unknown skin-friction coefficient [Formula: see text]. Then, [Formula: see text] is determined by using the secant method or Newton’s method. The derivative needed for Newton’s method is also computed using a series derived from the transformed differential equation. The method is validated by solving the Falkner–Skan equation for several cases reported previously in the literature.

scholarly journals Mathematic simulation of cutting unloading from the bunker

2019 ◽  
Vol 10 (7) ◽  
pp. 758 ◽  
Author(s):  
Serhii Yermakov ◽  
Taras Hutsol ◽  
Oleh Ovcharuk ◽  
Iryna Kolosiuk
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Velocity Field ◽  
Boundary Conditions ◽  
Gaseous Medium ◽  
Navier Stokes ◽  
Bulk Materials ◽  
Navier Stokes Equation ◽  
Planting Material ◽  
The Mathematical Model

The peculiarities of cutting movement at unloading them from the hopper are described. The analysis of the scientific researches on bulk materials movement and bridging is given. To develop the mathematical model of cutting unloading the layer should be described as a pseudoliquid, that consists of discrete components (cuttings) and gaseous medium (air). The Navier-Stokes equation can be applied to the process of cutting unloading and velocity field. The equation of pseudoliquid motion is a nonlinear integral and differential equation. The initial and boundary conditions for speed of cutting movement are identified. As a result of research has been theoretically obtained a formula, that evaluates the rate of planting material unloading, the adequacy of which has already been partially tested in experimental experiments carried out by the authors on the way to creating an automatic planting machine.

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