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Journal Article•DOI•

Deceleration of a rotating disk in a viscous fluid

Layne T. Watson1, C.Y. Wang•
Virginia Tech1
01 Dec 1979-Physics of Fluids (AIP Publishing)-Vol. 22, Iss: 12, pp 2267-2269
TL;DR: In this paper, it was shown that the Navier-Stokes equations admit similarity solutions which depend on a nondimensional parameter S =α/Ω0, measuring unsteadiness, and the resulting set of nonlinear ordinary differential equations was then integrated numerically.
Abstract: A disk rotating in a viscous fluid decelerates with an angular velocity inversely proportional to time. It is found that the unsteady Navier–Stokes equations admit similarity solutions which depend on a nondimensional parameter S =α/Ω0, measuring unsteadiness. The resulting set of nonlinear ordinary differential equations is then integrated numerically. The special case of S =−1.606 699 corresponds to the decay of rotation of a free, massless disk in a viscous fluid.

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Deceleration of a rotating disk in a viscous fluid
Layne T. Watson and ChangYi Wang
Citation: Physics of Fluids (1958-1988) 22, 2267 (1979); doi: 10.1063/1.862535
View online: http://dx.doi.org/10.1063/1.862535
View Table of Contents: http://scitation.aip.org/content/aip/journal/pof1/22/12?ver=pdfcov
Published by the AIP Publishing
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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
128.173.125.76 On: Wed, 09 Apr 2014 16:53:44
Copyright by the AIP Publishing. Watson, L. T.; Wang, C. Y., "deceleration of a rotating disk in a viscous fluid," Phys.
Fluids 22, 2267 (1979); http://dx.doi.org/10.1063/1.862535
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
128.173.125.76 On: Wed, 09 Apr 2014 16:53:44
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:
128.173.125.76 On: Wed, 09 Apr 2014 16:53:44
Citations
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Book•DOI•
Numerical Continuation Methods

[...]

Eugene L. Allgower, Kurt Georg
01 Jan 1990

1,149 citations

Book•
Introduction to Numerical Continuation Methods

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Eugene L. Allgower, Kurt Georg
01 Jan 1987
TL;DR: The Numerical Continuation Methods for Nonlinear Systems of Equations (NCME) as discussed by the authors is an excellent introduction to numerical continuuation methods for solving nonlinear systems of equations.
Abstract: From the Publisher: Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

889 citations

Proceedings Article•DOI•
Globally convergent homotopy methods: a tutorial

[...]

Layne T. Watson1•
University of Michigan1
01 May 1989-Applied Mathematics and Computation
TL;DR: Homotopy algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for almost all choices of starting point, are globally convergent with probability one and exhibit a large amount of coarse grain parallelism.

123 citations

Journal Article•DOI•
Fluid flow and heat transfer over a rotating and vertically moving disk

[...]

Mustafa Turkyilmazoglu1•
Hacettepe University1
26 Jun 2018-Physics of Fluids
TL;DR: In this paper, the development of the fluid flow and resultant heat transfer caused by a rotating disk moving vertically upward or downward during an unsteady flow motion is studied, and it is observed that the upward and downward motion of the disk exerts an effect similar to that of the injection/suction through the wall, albeit with observable differences.
Abstract: The object of this study is the development of the fluid flow and resultant heat transfer caused by a rotating disk moving vertically upward or downward during an unsteady flow motion. The problem is formulated such that the similarity equations governing the physical phenomenon eventually reduced to those reported in the traditional viscous pumping study of von Karman for a vertically motionless but still rotating disk. The non-rotating disk with the upward or downward motion leads to the formation of a two-dimensional flow over the disk. Otherwise, the rotation and vertical action of the disk sets up a three-dimensional flow over the surface. It is observed that the upward and downward motion of the disk exerts an effect similar to that of the injection/suction through the wall, albeit with observable differences. Moreover, the viscous pumping is found to be a jet-like radial velocity as the disk moves upward fast. Although the downward movement of the disk suppresses the velocity field, a growth in the boundary layer thickness is anticipated, contrary to the traditional wall suction. The temperature field is shown to be highly dependent on the form of the wall temperature, which is maintained at a time-varying function. Moreover, the impact of the vertical wall movement is observed to be overwhelmed by high disk rotations.

99 citations

Journal Article•DOI•
Exact solutions of the Navier-Stokes equations with the linear dependence of velocity components on two space variables

[...]

S. N. Aristov1, D. V. Knyazev1, Andrei D. Polyanin1•
Russian Academy of Sciences1
14 Oct 2009-Theoretical Foundations of Chemical Engineering
TL;DR: In this article, a wide class of two-dimensional and three-dimensional steady-state and non-steady-state flows of a viscous incompressible fluid is considered, where the components of the velocity of a fluid linearly depend on two spatial coordinates.
Abstract: A wide class of two-dimensional and three-dimensional steady-state and non-steady-state flows of a viscous incompressible fluid is considered. It is assumed that the components of the velocity of a fluid linearly depend on two spatial coordinates. The three-dimensional Navier-Stokes equations in this case are reduced to a closed determining system that consists of six equations with partial derivatives of the third and second orders. A brief review of the known exact solutions of this system and the respective flows of a fluid (Couette-Poiseuille, Ekman, Stokes, Karman, and other flows) is given. The cases of reducing a determining system to one or two equations are described. Many new exact solutions of two-dimensional and three-dimensional nonstationary Navier-Stokes equations containing arbitrary functions and arbitrary parameters are derived. Periodic (both in spatial coordinates and in time) and some other solutions that are expressed in terms of elementary functions are described. The problems of the nonlinear stability of solutions are studied. A number of new hydrodynamic problems are considered. A general interpretation of the solutions as the main terms of the Taylor series expansion in terms of radial coordinates is given.

87 citations

References
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Journal Article•DOI•
Über laminare und turbulente Reibung

[...]

T.-H. V. Kármán
01 Jan 1921-Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik

2,029 citations

Journal Article•DOI•
The flow due to a rotating disc

[...]

William G. Cochran1•
St. John's College1
01 Jul 1934
TL;DR: In this paper, the steady motion of an incompressible viscous fluid due to an infinite rotating plane lamina was considered, and it was shown that the equations of motion and continuity are satisfied by taking
Abstract: 1. The steady motion of an incompressible viscous fluid, due to an infinite rotating plane lamina, has been considered by Karman. If r, θ, z are cylindrical polar coordinates, the plane lamina is taken to be z = 0; it is rotating with constant angular velocity ω about the axis r = 0. We consider the motion of the fluid on the side of the plane for which z is positive; the fluid is infinite in extent and z = 0 is the only boundary. If u, v, w are the components of the velocity of the fluid in the directions of r, θ and z increasing, respectively, and p is the pressure, then Karman shows that the equations of motion and continuity are satisfied by taking

932 citations

Journal Article•DOI•
Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation

[...]

Kenneth M. Brown1, John E. Dennis2•
University of Minnesota1, Cornell University2
01 Aug 1971-Numerische Mathematik
TL;DR: In this article, the authors give two derivative-free computational algorithms for nonlinear least squares approximation, which are finite difference analogues of the Levenberg-Marquardt and Gauss methods.
Abstract: In this paper we give two derivative-free computational algorithms for nonlinear least squares approximation. The algorithms are finite difference analogues of the Levenberg-Marquardt and Gauss methods. Local convergence theorems for the algorithms are proven. In the special case when the residuals are zero at the minimum, we show that certain computationally simple choices of the parameters lead to quadratic convergence. Numerical examples are included.

381 citations

Journal Article•DOI•
On the flow due to a rotating disk

[...]

Edward R. Benton1•
National Center for Atmospheric Research1
01 Apr 1966-Journal of Fluid Mechanics
TL;DR: In this paper, the von Karman problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution.
Abstract: The von Karman (1921) rotating disk problem is extended to the case of flow started impulsively from rest; also, the steady-state problem is solved to a higher degree of accuracy than previously by a simple analytical-numerical method which avoids the matching difficulties in Cochran's (1934) well-known solution. Exact representations of the non-steady velocity field and pressure are given by suitable power-series expansions in the angle of rotation, Ωt, with coefficients that are functions of a similarity variable. The first four equations for velocity coefficient functions are solved exactly in closed form, and the next six by numerical integration. This gives four terms in the series for the primary flow and three terms in each series for the secondary flow.The results indicate that the asymptotic steady state is approached after about 2 radians of the disk's motion and that it can be approximately obtained from the initial-value, time-dependent analysis. Furthermore, the non-steady flow has three phases, the first two of which are accurately and fully described with the terms computed. During the first-half radian (phase 1), the velocity field is essentially similar in time, with boundary-layer thickening the only significant effect. For 0·5 [lsim ] Ωt [lsim ] 1·5 (phase 2), boundary-layer growth continues at a slower rate, but simultaneously the velocity profiles adjust towards the shape of the ultimate steady-state profiles. At about Ωt = 1·5, some flow quantities overshoot the steady-state values by small amounts. In analogy with the ‘Greenspan-Howard problem’ (1963) it is believed that the third phase (Ωt > 1·5) consists of a small amplitude decaying oscillation about the steady-state solution.

378 citations

Journal Article•DOI•
Unsteady flows in a semi-infinite contracting or expanding pipe

[...]

Shigeo Uchida1, Hiroshi Aoki1•
Nagoya University1
07 Sep 1977-Journal of Fluid Mechanics
TL;DR: In this article, an exact solution of the Navier-Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow.
Abstract: Physiological pumps produce flows by alternate contraction and expansion of the vessel. When muscles start to squeeze its wall the valve at the upstream end is closed and that at the downstream end is opened, and the fluid is pumped out in the downstream direction. These systems can be modelled by a semi-infinite pipe with one end closed by a compliant membrane which prevents only axial motion of the fluid, leaving radial motion completely unrestricted. In the present paper an exact similar solution of the Navier–Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow. In a contracting pipe the effects of viscosity are limited to a thin boundary layer attached to the wall, which becomes thinner for higher Reynolds numbers. In an expanding pipe the flow adjacent to the wall is highly retarded and eventually reverses at Reynolds numbers above a critical value. The pressure gradient along the axis of pipe is favourable for a contracting wall, while it is adverse for an expanding wall in most cases. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained in full. The results of the present theory may be applied to the unsteady flow produced by a certain class of forced contractions and expansions of a valved vein or a thin bronchial tube.

204 citations

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Über laminare und turbulente Reibung

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01 Jan 1921-Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik
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The flow due to a rotating disc

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On the flow due to a rotating disk

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