scholarly journals A Study on the Convergence of Series Solution of Non-Newtonian Third Grade Fluid with Variable Viscosity: By Means of Homotopy Analysis Method

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
R. Ellahi
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Viscosity ◽  
Nonlinear Problems ◽  
Third Grade ◽  
Series Solutions ◽  
Analysis Method ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
Grade Fluid ◽  
Physical Features

This work is concerned with the series solutions for the flow of third-grade non-Newtonian fluid with variable viscosity. Due to the nonlinear, coupled, and highly complicated nature of partial differential equations, finding an analytical solution is not an easy task. The homotopy analysis method (HAM) is employed for the presentation of series solutions. The HAM is accepted as an elegant tool for effective solutions for complicated nonlinear problems. The solutions of (Hayat et al., 2007) are developed, and their convergence has been discussed explicitly for two different models, namely, constant and variable viscosity. An error analysis is also described. In addition, the obtained results are illustrated graphically to depict the convergence region. The physical features of the pertinent parameters are presented in the form of numerical tables.

Homotopy Analysis Method

2017 ◽  
pp. 173-226
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Nonlinear Partial Differential Equations ◽  
Analytic Method ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
And Control

In this chapter, the analytic solution of nonlinear partial differential equations arising in heat transfer is obtained using the newly developed analytic method, namely the Homotopy Analysis Method (HAM). The homotopy analysis method provides us with a new way to obtain series solutions of such problems. This method contains the auxiliary parameter provides us with a simple way to adjust and control the convergence region of series solution. By suitable choice of the auxiliary parameter, we can obtain reasonable solutions for large modulus.

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.

Flow of a Non-Newtonian Nanofluid Between Coaxial Cylinders with Variable Viscosity

2012 ◽  
Vol 67 (5) ◽  
pp. 255-261 ◽  
Author(s):  
Muhammad Yousaf Malik ◽  
Azad Hussain ◽  
Sohail Nadeem
Keyword(s):  
Differential Equations ◽  
Newtonian Fluid ◽  
Homotopy Analysis Method ◽  
Variable Viscosity ◽  
Nonlinear Differential Equations ◽  
Analysis Method ◽  
Governing Equations ◽  
Coaxial Cylinders ◽  
Homotopy Analysis ◽  
Physical Features

In the present paper, we have focused our attention to highlight the study of a non-Newtonian nanofluid between coaxial cylinders with variable viscosity. The governing equations of the non- Newtonian fluid with variable viscosity along with energy and nanoparticles are given. The coupled nonlinear differential equations are solved analytically with the help of the homotopy analysis method (HAM). The convergence of the solution is discussed through h-curves. The physical features of pertinent parameters are discussed by plotting graphs.

Analytic Solution of the Sharma-Tasso-Olver Equation by Homotopy Analysis Method

2010 ◽  
Vol 65 (4) ◽  
pp. 285-290 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Mahnaz Ashtiani ◽  
Esmail Babolian
Keyword(s):  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Homotopy Analysis ◽  
Convergence Control ◽  
And Control ◽  
Convergence Control Parameter

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the kink solution of the Sharma-Tasso-Olver equation. The homotopy analysis method is one of the analytic methods and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ħwhich gives us a simple way to adjust and control the convergence region of series solution. “Due to this reason, it seems reasonable to rename ħthe convergence-control parameter” [1].

scholarly journals Approximate Analytic Solution for the KdV and Burger Equations with the Homotopy Analysis Method

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mojtaba Nazari ◽  
Faisal Salah ◽  
Zainal Abdul Aziz ◽  
Merbakhsh Nilashi
Keyword(s):  
Homotopy Analysis Method ◽  
Analytic Solution ◽  
Order Approximation ◽  
Soliton Solutions ◽  
Nonlinear Problems ◽  
Approximate Analytic Solution ◽  
Analysis Method ◽  
Convergence Region ◽  
Burgers Equations ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate analytic solution of the Korteweg-de Vries (KdV) and Burgers equations. The homotopy analysis method (HAM) is an analytic technique which provides us with a new way to obtain series solutions of such nonlinear problems. HAM contains the auxiliary parameterħ, which provides us with a straightforward way to adjust and control the convergence region of the series solution. The resulted HAM solution at 8th-order and 14th-order approximation is then compared with that of the exact soliton solutions of KdV and Burgers equations, respectively, and shown to be in excellent agreement.

The Analysis Approach of Boundary Layer Equations of Power-Law Fluids of Second Grade

2008 ◽  
Vol 63 (9) ◽  
pp. 564-570 ◽  
Author(s):  
Saeid Abbasbandy ◽  
Muhammet Yürüsoy ◽  
Mehmet Pakdemirli
Keyword(s):  
Power Law ◽  
Homotopy Analysis Method ◽  
Nonlinear Problems ◽  
Analytic Solutions ◽  
Second Grade ◽  
Analysis Method ◽  
Boundary Layer Equations ◽  
Power Law Fluids ◽  
Homotopy Analysis ◽  
Constant Coefficients

A powerful analytic technique for nonlinear problems, the homotopy analysis method (HAM), is employed to give analytic solutions of power-law fluids of second grade. For the so-called secondorder power-law fluids, the explicit analytic solutions are given by recursive formulas with constant coefficients. Also, for the real power-law index in a quite large range an analytic approach is proposed. It is demonstrated that the approximate solution agrees well with the finite difference solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear equations of the power-law fluids of second grade.

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