On the analytic solutions of the nonhomogeneous Blasius problem

Allan, Fathi M.; Syam, Muhammed I.

doi:10.1016/j.cam.2004.12.017

Cited by 97 publications

(40 citation statements)

References 18 publications

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“…Even though the problem is almost a century old, recent papers that employ the Blasius problem as an example include [2,1,5,6,11,15,16,21,18,17,23,25,26,27,28,29,30,32,33,34,36].…”

Section: Because All Fluid Flows Must Be Zero At a Solid Boundary Thmentioning

confidence: 99%

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Boyd¹

2008

SIAM Rev.

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Abstract. The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semiinfinite flat plate. The Blasius function is the solution to 2fxxxWe use this famous problem to illustrate several themes. First, although the flow solves a nonlinear partial differential equation (PDE), Toepfer successfully computed highly accurate numerical solutions in 1912. His secret was to combine a Runge-Kutta method for integrating an ordinary differential equation (ODE) initial value problem with some symmetry principles and similarity reductions, which collapse the PDE system to the ODE shown above. This shows that PDE numerical studies were possible even in the precomputer age. The truth, both a hundred years ago and now, is that mathematical theorems and insights are an arithmurgist's best friend, and they can vastly reduce the computational burden. Second, we show that special tricks, applicable only to a given problem, can be as useful as the broad, general methods that are the fabric of most applied mathematics courses: the importance of "particularity." In spite of these triumphs, many properties of the Blasius function f (x) are unknown. We give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician.

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“…Even though the problem is almost a century old, recent papers that employ the Blasius problem as an example include [2,1,5,6,11,15,16,21,18,17,23,25,26,27,28,29,30,32,33,34,36].…”

Section: Because All Fluid Flows Must Be Zero At a Solid Boundary Thmentioning

confidence: 99%

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Boyd¹

2008

SIAM Rev.

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“…The HAM always provides us with a family of solution expressions in the auxiliary parameter the convergence region and rate of each solution might be determined conveniently by the auxiliary parameter Furthermore, the HAM is rather general and contains the homotopy perturbation method (HPM) [12], the Adomian decomposition method (ADM) [14] and δ-expansion method. In recent years, the HAM has been successfully employed to solve many types of nonlinear problems such as the nonlinear equations arising in heat transfer [15], the nonlinear model of diffusion and reaction in porous catalysts [16], the chaotic dynamical systems [17], the non-homogeneous Blasius problem [18], the generalized three-dimensional MHD flow over a porous stretching sheet [19], the wire coating analysis using MHD Oldroyd 8-constant fluid [20], the axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet [21], the MHD flow of a second grade fluid in a porous channel [22], the generalized Couette flow [23], the Glauert-jet problem [24], the Burger and regularized long wave equations [25], the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field [26], the nano boundary layer flows [27], the twodimensional steady slip flow in microchannels [28], and other problems. All of these successful applications verified the validity, effectiveness and flexibility of the HAM.…”

Section: Introductionmentioning

confidence: 99%

Analytical approximate solutions for two-dimensional viscous flow through expanding or contracting gaps with permeable walls

2009

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Abstract:In this paper, the problem of laminar, isothermal, incompressible and viscous flow in a rectangular domain bounded by two moving porous walls, which enable the fluid to enter or exit during successive expansions or contractions is solved analytically by using the homotopy analysis method (HAM). Graphical results are presented to investigate the influence of the nondimensional wall dilation rate α and permeation Reynolds number Re on the velocity, normal pressure distribution and wall shear stress. The obtained solutions, in comparison with the numerical solutions, demonstrate remarkable accuracy. The present problem for slowly expanding or contracting walls with weak permeability is a simple model for the transport of biological fluids through contracting or expanding vessels.

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“…Liao [17][18][19][20][21][22] employed the basic ideas of the homotopy in topology to propose a general analytical method for non-linear problems, namely homotopy analysis method (HAM). This method has been successfully applied to solve many types of nonlinear problems [23][24][25][26][27][28]. The model under study in the present paper is of the fourth grade fluid type, and we have applied the Optimal Homotopy Asymptotic Method in order to analyze the nonlinear behaviour of thin film flow down a vertical cylinder.…”

Section: Introductionmentioning

confidence: 99%

Optimal homotopy asymptotic method with application to thin film flow

2008

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Abstract:A new approximate analytical technique to address for non-linear problems, namely Optimal Homotopy Asymptotic Method (OHAM) is proposed and has been applied to thin film flow of a fourth grade fluid down a vertical cylinder. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The results reveal that the proposed method is very accurate, effective and easy to use. PACS

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On the analytic solutions of the nonhomogeneous Blasius problem

Cited by 97 publications

References 18 publications

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Analytical approximate solutions for two-dimensional viscous flow through expanding or contracting gaps with permeable walls

Optimal homotopy asymptotic method with application to thin film flow

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