A reliable treatment of a homotopy analysis method for two-dimensional viscous flow in a rectangular domain bounded by two moving porous walls

2010 ◽  
Vol 11 (3) ◽  
pp. 1502-1512 ◽  
Author(s):  
Saeed Dinarvand ◽  
Mohammad Mehdi Rashidi
Keyword(s):  
Viscous Flow ◽  
Homotopy Analysis Method ◽  
Rectangular Domain ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Porous Walls

A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate

1999 ◽  
Vol 385 ◽  
pp. 101-128 ◽  
Author(s):  
SHI-JUN LIAO
Keyword(s):  
Viscous Flow ◽  
Flat Plate ◽  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Two Dimensional ◽  
Analysis Method ◽  
Numerical Result ◽  
Homotopy Analysis ◽  
Potential Possibility ◽  
Valid Solution

We apply a new kind of analytic technique, namely the homotopy analysis method (HAM), to give an explicit, totally analytic, uniformly valid solution of the two-dimensional laminar viscous flow over a semi-infinite flat plate governed by f‴(η)+αf(η)f″(η)+β[1−f′2(η)]=0 under the boundary conditions f(0)=f′(0)=0, f′(+∞)=1. This analytic solution is uniformly valid in the whole region 0[les ]η<+∞. For Blasius' (1908) flow (α=1/2, β=0), this solution converges to Howarth's (1938) numerical result and gives a purely analytic value f″(0)=0.332057. For the Falkner–Skan (1931) flow (α=1), it gives the same family of solutions as Hartree's (1937) numerical results and a related analytic formula for f″(0) when 2[ges ]β[ges ]0. Also, this analytic solution proves that when −0.1988[les ]β0 Hartree's (1937) family of solutions indeed possess the property that f′→1 exponentially as η→+∞. This verifies the validity of the homotopy analysis method and shows the potential possibility of applying it to some unsolved viscous flow problems in fluid mechanics.

scholarly journals A Novel Numerical Technique for Two-Dimensional Laminar Flow between Two Moving Porous Walls

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Zodwa G. Makukula ◽  
Precious Sibanda ◽  
Sandile S. Motsa
Keyword(s):  
Homotopy Analysis Method ◽  
Numerical Technique ◽  
Rectangular Domain ◽  
Two Dimensional ◽  
Analysis Method ◽  
Computationally Efficient ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Analysis ◽  
Successive Linearisation Method ◽  
Spectral Homotopy Analysis Method

We investigate the steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain that is bounded by two permeable surfaces. The governing fourth-order nonlinear differential equation is solved by applying the spectral-homotopy analysis method and a novel successive linearisation method. Semianalytical results are obtained and the convergence rate of the solution series was compared with numerical approximations and with earlier results where the homotopy analysis and homotopy perturbation methods were used. We show that both the spectral-homotopy analysis method and successive linearisation method are computationally efficient and accurate in finding solutions of nonlinear boundary value problems.

scholarly journals Dynamics with Cattaneo–Christov heat and mass flux theory of bioconvection Oldroyd-B nanofluid

2020 ◽  
Vol 12 (8) ◽  
pp. 168781402093046 ◽  
Author(s):  
Noor Saeed Khan ◽  
Qayyum Shah ◽  
Arif Sohail
Keyword(s):  
Mass Transfer ◽  
Porous Media ◽  
Differential Equations ◽  
Mass Flux ◽  
Homotopy Analysis Method ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Transfer Flow ◽  
Insight Into

Entropy generation in bioconvection two-dimensional steady incompressible non-Newtonian Oldroyd-B nanofluid with Cattaneo–Christov heat and mass flux theory is investigated. The Darcy–Forchheimer law is used to study heat and mass transfer flow and microorganisms motion in porous media. Using appropriate similarity variables, the partial differential equations are transformed into ordinary differential equations which are then solved by homotopy analysis method. For an insight into the problem, the effects of various parameters on different profiles are shown in different graphs.

Numerical Results of a Flow in a Third Grade Fluid between Two Porous Walls

2009 ◽  
Vol 64 (1-2) ◽  
pp. 59-64 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Tasawar Hayat ◽  
Rahmat Ellahi ◽  
Saleem Asghar
Keyword(s):  
Numerical Solution ◽  
Homotopy Analysis Method ◽  
Series Solution ◽  
Material Parameter ◽  
Third Grade ◽  
Analysis Method ◽  
Third Grade Fluid ◽  
Homotopy Analysis ◽  
Grade Fluid ◽  
Porous Walls

Series solution for a steady flow of a third grade fluid between two porous walls is given by the homotopy analysis method (HAM). Comparison with the existing numerical solution is shown. It is found that, unlike the numerical solution, the present series solution holds for all values of the material parameter of a third grade fluid.

scholarly journals Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

2021 ◽  
Vol 54 (1) ◽  
pp. 11-24
Author(s):  
Atanaska Georgieva
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Homotopy Analysis Method ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

scholarly journals Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

2020 ◽  
Vol 4 (1) ◽  
pp. 9
Author(s):  
Atanaska Georgieva ◽  
Snezhana Hristova
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Homotopy Analysis Method ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

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