Micropolar flow past a stretching sheet
01 Nov 1985-Zeitschrift für Angewandte Mathematik und Physik (Springer International Publishing)-Vol. 36, Iss: 6, pp 845-853
TL;DR: In this paper, the flow of an incompressible, constant density micropolar fluid past a stretching sheet is studied using a globally convergent homotopy method in conjunction with a least change secant update quasi-Newton algorithm.
Abstract: This paper studies the flow of an incompressible, constant density micropolar fluid past a stretching sheet. The governing boundary layer equations of the flow are solved numerically using a globally convergent homotopy method in conjunction with a least change secant update quasi-Newton algorithm. The flow pattern depends on three non-dimensional parameters. Some interesting results are illustrated graphically and discussed.
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TL;DR: Probability one homotopy algorithms as mentioned in this paper are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical.
Abstract: Probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one These methods are theoretically powerful, and if constructed and implemented properly, are robust, numerically stable, accurate, and practical The concomitant numerical linear algebra problems deal with rectangular matrices, and good algorithms require a delicate balance (not always achieved) of accuracy, robustness, and efficiency in both space and time The author's experience with globally convergent homotopy algorithms is surveyed here, and some of the linear algebra difficulties for dense and sparse problems are discussed
155 citations
TL;DR: In this paper, an analysis is presented to study dual nature of solution of mass transfer with first order chemical reaction in boundary layer stagnation-point flow over a stretching/shrinking sheet.
Abstract: In this paper, an analysis is presented to study dual nature of solution of mass transfer with first order chemical reaction in boundary layer stagnation-point flow over a stretching/shrinking sheet. The governing equations are transformed into a set of self-similar ordinary differential equations by similarity transformations. The transformed equations are solved numerically using very efficient shooting method. The study reveals that the dual solutions of velocity and concentration exist for certain values of velocity ratio parameter (the ratio of stretching/shrinking rate and straining rate). The concentration boundary layer thickness decreases with increasing values of Schmidt number and reaction-rate parameter for both solutions.
144 citations
Cites background from "Micropolar flow past a stretching s..."
...[4], Sankara and Watson [5], Chen and Char [6], Vajravelu and Rollins [7] and Chamkha [8] by considering different types of fluid and various physical conditions....
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TL;DR: In this article, the effects of thermal radiation on the flow of micropolar fluid and heat transfer past a porous shrinking sheet is investigated and self-similar ODEs are obtained using similarity transformations from the governing PDEs and are then solved numerically by very efficient shooting method.
Abstract: The effects of thermal radiation on the flow of micropolar fluid and heat transfer past a porous shrinking sheet is investigated. The self-similar ODEs are obtained using similarity transformations from the governing PDEs and are then solved numerically by very efficient shooting method. The analysis reveals that for the steady flow of micropolar fluid, the wall mass suction needs to be increased. Dual solutions of velocity and temperature are obtained for several values of the each parameter involved. For increasing values of the material parameter K, the velocity decreases for first solution, whereas, for second solution it increases. Due to increase of thermal radiation, the temperature and thermal boundary layer thickness reduce in both solutions and also the heat transfer from the sheet enhances with thermal radiation.
139 citations
Cites background from "Micropolar flow past a stretching s..."
...[11] extended the work of Sankara and Watson [10] by considering the mass suction or injection through the porous sheet....
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...An important contribution in micropolar flow dynamics was made by Sankara and Watson [10], when they investigated the flow of micropolar fluids past a stretching sheet....
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TL;DR: In this article, an analysis is made to study boundary layer flow and heat transfer over an exponentially shrinking sheet using similarity transformations in exponential form, the governing boundary layer equations are transformed into self-similar nonlinear ordinary differential equations, which are then solved numerically using a very efficient shooting method.
Abstract: An analysis is made to study boundary layer flow and heat transfer over an exponentially shrinking sheet. Using similarity transformations in exponential form, the governing boundary layer equations are transformed into self-similar nonlinear ordinary differential equations, which are then solved numerically using a very efficient shooting method. The analysis reveals the conditions for the existence of steady boundary layer flow due to exponential shrinking of the sheet and it is found that when the mass suction parameter exceeds a certain critical value, steady flow is possible. The dual solutions for velocity and temperature distributions are obtained. With increasing values of the mass suction parameter, the skin friction coefficient increases for the first solution and decreases for the second solution.
119 citations
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.
Abstract: The basic theory for probability one globally convergent homotopy algorithms was developed in 1976, and since then the theory, algorithms, and applications have considerably expanded. These are algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for almost all choices of starting point. Thus they are globally convergent with probability one. They are applicable to Brouwer fixed point problems, certain classes of zero-finding problems, unconstrained optimization, linearly constrained optimization, nonlinear complementarity, and the discrezations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements. A mathematical software package, HOMPACK, exists that implements several different strategies and handles both dense and sparse problems. Homotopy algorithms are closely related to ODE algorithms, and make heavy use of ODE techniques. Homotopy algorithms for some classes of nonlinear systems, such as polynomial systems, exhibit a large amount of coarse grain parallelism. These and other topics are discussed in a tutorial fashion.
112 citations
References
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TL;DR: In this paper, a plastischem material fliesst aus einem Spalt with einer Geschwindigkeit, die proportional zum Abstand vom Spalt ist.
Abstract: Eine Platte aus plastischem Material fliesst aus einem Spalt mit einer Geschwindigkeit, die proportional zum Abstand vom Spalt ist. Eine exakte Losung der Grenzschichtgleichungen fur die von der Platte erzeugte Luftbewegung wird gegeben. Oberflachenreibung und Warmeleitungskoeffizient werden berechnet.
2,851 citations
TL;DR: In this article, the boundary layer flow over a semi-infinite flat plate is studied and the partial differential equations of motion are reduced to 2 couple differential equations and numerical solutions for different values of the parameters are obtained.
Abstract: Theory of microplar fluid and its application to low concentration suspension flow is discussed. The boundary layer flow over a semi-infinite flat plate is studied. It is observed that when the micro-inertia is not constant, it is possible to obtain self-similar solutions. The partial differential equations of motion are then reduced to 2 couple differential equations. The numerical solutions for different values of the parameters are obtained. The variation of the velocity, micro-rotation, shear and couple stresses are plotted and discussed.
482 citations
01 Aug 1980
TL;DR: A pack of FORTRAN subprograms for the numerical solution of nonlinear equations and nonlinear least-squares problems and this report provides an overview of the algorithms and software in the package.
Abstract: MINPACK-1 is a pack of FORTRAN subprograms for the numerical solution of nonlinear equations and nonlinear least-squares problems. This report provides an overview of the algorithms and software in the package, and includes the documentation and program listings.
376 citations
TL;DR: Chow, Mallet-Paret, and Yorke have recently proposed an algorithm for computing Brouwer fixed points of C^2 maps as discussed by the authors, and a numerical implementation of that algorithm is presented here.
Abstract: Chow, Mallet-Paret, and Yorke have recently proposed in abstract terms an algorithm for computing Brouwer fixed points of C^2 maps. A numerical implementation of that algorithm is presented here. Careful attention has been paid to computational efficiency, accuracy, and robustness. Some convergence theorems and results of numerical tests are also included.
157 citations
TL;DR: In this paper, two different boundary conditions for the spin are considered: vanishing spin and vanishing surface moment, and the equations of motion are reduced to dimensionless forms which include three dimensionless parameters, and integrated numerically by a Runge-Kutta method.
Abstract: Plane and axially symmetric flows of a micropolar fluid, in contact with an infinite plate, and tending to potential flow at infinity, with a stagnation point on the plate, are considered Two different boundary conditions for the spin are considered: (a), vanishing spin; and (b), vanishing surface moment The equations of motion are reduced to dimensionless forms which include three dimensionless parameters, and integrated numerically by a Runge—Kutta method Results are presented both in tabular and graphical form, and the effects of the values of the parameters on the flow are discussed
155 citations
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