Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks

In this paper, we consider the axi-symmetric flow between two infinite stretching disks. By using a similarity transformation, we reduce the governing Navier-Stokes equations to a system of nonlinear ordinary differential equations. We first obtain analytical solutions via a four-term perturbation method for small and large values of the Reynolds number R. Also, we apply the Homotopy Analysis Method (which may be used for all values of R) to obtain analytical solutions. These solutions converge over a larger range of values of the Reynolds number than the perturbation solutions. Our results agree well with the numerical results of Fang and Zhang [22]. Furthermore, we obtain the analytical solutions valid for moderate values of R by use of Homotopy Analysis.

[1]  Tiegang Fang,et al.  Flow between two stretchable disks—An exact solution of the Navier–Stokes equations ☆ , 2008 .

[2]  L. Durlofsky,et al.  The spatial stability of a class of similarity solutions , 1984 .

[3]  Chaoyang Wang.  Exact Solutions of the Steady-State Navier-Stokes Equations , 1991 .

[4]  P. S. Gupta,et al.  Heat and mass transfer on a stretching sheet with suction or blowing , 1977 .

[5]  W. H. H. Banks,et al.  Similarity solutions of the boundary-layer equations for a stretching wall , 1983 .

[6]  L. Crane.  Flow past a stretching plate , 1970 .

[7]  B. C. Sakiadis.  Boundary‐layer behavior on continuous solid surfaces: I. Boundary‐layer equations for two‐dimensional and axisymmetric flow , 1961 .

[8]  A. Acrivos,et al.  Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier—Stokes equations with reverse flow , 1981, Journal of Fluid Mechanics.

[9]  M. B. Zaturska,et al.  Flow in a pipe driven by suction at an accelerating wall , 1995 .

[10]  K. Vajravelu,et al.  On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach , 2009 .

[11]  Chang Y. Wang,et al.  Stretching a surface in a rotating fluid , 1988 .

[12]  Henning Rasmussen,et al.  Steady viscous flow between two porous disks , 1970 .

[13]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[14]  M. B. Zaturska,et al.  New solutions for flow in a channel with porous walls and/or non-rigid walls , 2003 .

[15]  Tiegang Fang,et al.  Flow over a stretchable disk , 2007 .

[16]  Chao-Yang Wang,et al.  The three‐dimensional flow due to a stretching flat surface , 1984 .

[17]  Taylan Altan,et al.  Metal Forming : Fundamentals and Applications , 1983 .

[18]  E. G. Fisher,et al.  Extrusion of plastics , 1976 .

[19]  F. White.  Viscous Fluid Flow , 1974 .

[20]  C. Wang,et al.  Slip flow due to a stretching cylinder , 1988 .

[21]  M. B. Zaturska,et al.  Laminar channel flow driven by accelerating walls , 1991, European Journal of Applied Mathematics.

[22]  Imrich Klein,et al.  Engineering principles of plasticating extrusion , 1970 .

[23]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[24]  M. B. Zaturska,et al.  On transition to chaos in two-dimensional channel flow symmetrically driven by accelerating walls , 1990, Journal of Fluid Mechanics.

[25]  H. Kuiken.  On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small , 1981 .

[26]  B. C. Sakiadis.  Boundary‐layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface , 1961 .