Analytical solutions of a coupled nonlinear system arising in a flow between stretching disks. (English) Zbl 1277.76024
Summary: 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. Furthermore, we obtain the analytical solutions valid for moderate values of \(R\) by use of homotopy analysis.
MSC:
| 76D99 | Incompressible viscous fluids |
| 65N99 | Numerical methods for partial differential equations, boundary value problems |
Keywords:
Navier-Stokes equations; similarity solution; stretching disk; analytical solution; numerical solutionReferences:
| [1] | Altan, T.; Oh, S.; Gegel, H., Metal Forming Fundamentals and Applications (1979), American Society of Metals: American Society of Metals Metals Park, OH |
| [2] | Fisher, E. G., Extrusion of Plastics (1976), Wiley: Wiley New York |
| [3] | Tadmor, Z.; Klein, I., Engineering principles of plasticating extrusion. Engineering principles of plasticating extrusion, Polymer Science and Engineering Series (1970), Van Norstrand Reinhold: Van Norstrand Reinhold New York |
| [4] | Sakiadis, B. C., Boundary-layer behavior on continuous solid surface: I. Boundary layer equations for two-dimensional and axi-symmetric flow, AIChE J., 7, 26-28 (1961) |
| [5] | Sakiadis, B. C., Boundary-layer behavior on continuous solid surface: II. Boundary layer on continuous flat surface, AIChE J., 7, 221-225 (1961) |
| [6] | Wang, C. Y., Exact solutions of the steady state Navier-Stokes equations, Annu. Rev. Fluid Mech., 23, 159 (1991) · Zbl 0717.76033 |
| [7] | Crane, L. J., Flow past a stretching plate, Z. Angew. Math. Phys. (ZAMP), 21, 645-647 (1970) |
| [8] | Kuiken, H. K., On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small, IMA J. Appl. Math., 27, 387-405 (1981) · Zbl 0472.76045 |
| [9] | Banks, W. H.H., Similarity solutions of the boundary-layer equations for a stretching wall, J. Mech. Theor. Appl., 2, 375-392 (1983) · Zbl 0538.76039 |
| [10] | Gupta, P. S.; Gupta, A. S., Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng., 55, 744-746 (1977) |
| [11] | Wang, C. Y., Stretching a surface in a rotating fluid, J. Appl. Math. Phys. (ZAMP), 39, 177-185 (1988) · Zbl 0642.76118 |
| [12] | Wang, C. Y., The three-dimensional flow due to a stretching flat surface, Phys. Fluids, 27, 1915-1917 (1984) · Zbl 0545.76033 |
| [13] | Fang, T., Flow over a stretchable disk, Phys. Fluids, 19, 128105 (2007) · Zbl 1182.76239 |
| [14] | Brady, J. F.; Acrivos, A., Steady flow in a channel or tube with an accelerating surface velocity: an exact solution to the Navier-Stokes equations with reverse flow, J. Fluid Mech., 112, 127-150 (1981) · Zbl 0491.76037 |
| [15] | Wang, C. Y., Fluid flow due to a stretching cylinder, Phys. Fluids, 31, 466-468 (1988) |
| [16] | Durlofskyl, L.; Brady, J. F., The spatial stability of a class of similarity solutions, Phys. Fluids, 27, 5, 1068-1076 (1984) |
| [17] | Watson, E. B.B.; Banks, W. H.H.; Zaturska, M. B.; Drazin, P. G., On transition to chaos in two-dimensional channel flow symmetrically driven by accelerating walls, J. Fluid Mech., 212, 451 (1990) · Zbl 0692.76034 |
| [18] | Watson, P.; Banks, W. H.H.; Zaturska, M. B.; Drazin, P. G., Laminar channel flow driven by accelerating walls, Eur. J. Appl. Math., 2, 359 (1991) · Zbl 0744.76041 |
| [19] | Zaturska, M. B.; Banks, W. H.H., Flow in a pipe driven by suction and accelerating wall, Acta Mech., 110, 1-4, 111-121 (1995) · Zbl 0851.76016 |
| [20] | Zaturska, M. B.; Banks, W. H.H., New solutions for flow in a channel with porous walls and/or non-rigid walls, Fluid Dyn. Res., 33, 1-2, 57-71 (2003) · Zbl 1032.76532 |
| [21] | Rasmusse, H., Steady viscous flow between two porous disks, J. Appl. Math. Phys. (ZAMP), 21, 2, 187-195 (1970) |
| [22] | Fang, T.; Zhang, J., Flow between two stretchable disks - An exact solution of the Navier-Stokes equations, Int. Commun. Heat Mass Transfer, 35, 892-895 (2008) |
| [23] | White, F. M., Viscous Fluid Flow (1991), McGraw-Hill Company: McGraw-Hill Company New York |
| [24] | Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method (2003), Chapman & Hall CRC Press: Chapman & Hall CRC Press Boca Raton |
| [25] | Liao, S. J., On the Homotopy Analysis Method for nonlinear problems, Appl. Math. Comput., 147, 499-513 (2004) · Zbl 1086.35005 |
| [26] | Van Gorder, R. A.; Vajravelu, K., 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, Commun. Nonlinear Sci. Numer. Simul., 14, 4078-4089 (2009) · Zbl 1221.65208 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
