# The Axisymmetric Motion of a Liquid Film on an Unsteady Stretching Surface

01 Mar 1995-Journal of Fluids Engineering-transactions of The Asme (American Society of Mechanical Engineers)-Vol. 117, Iss: 1, pp 81-85

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TL;DR: In this paper, the viscous flow induced by a shrinking sheet is studied and its existence and uniqueness are proved. Exact solutions, both numerical and in closed form, are found.

Abstract: The viscous flow induced by a shrinking sheet is studied. Existence and (non)uniqueness are proved. Exact solutions, both numerical and in closed form, are found.

496 citations

### Cites methods from "The Axisymmetric Motion of a Liquid..."

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TL;DR: In this paper, the momentum and heat transfer in a laminar liquid film on a horizontal stretching sheet is analyzed by the Homotopy analysis method (HAM), and analytic series solutions are given and compared with numerical results given by other authors.

Abstract: The momentum and heat transfer in a laminar liquid film on a horizontal stretching sheet is analyzed by the Homotopy analysis method (HAM). Analytic series solutions are given and compared with numerical results given by other authors. The good agreement between them shows the effectiveness of HAM to the problem of liquid film on an unsteady stretching surface.

145 citations

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TL;DR: In this paper, the homotopy analysis method (HAM) was applied to the case of two-dimensional and axisymmetric shrinking and the convergence of the obtained series solution was discussed explicitly.

Abstract: This work is concerned with the magnetohydrodynamic (MHD) viscous flow due to a shrinking sheet. The cases of two dimensional and axisymmetric shrinking have been discussed. Exact series solution is obtained using the homotopy analysis method (HAM). The convergence of the obtained series solution is discussed explicitly. The obtained HAM solution is valid for all values of the suction parameter and Hartman number.

135 citations

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TL;DR: In this paper, the problem of momentum and heat transfer in a thin liquid film of power-law fluid on an unsteady stretching surface has been studied, and numerical solutions are obtained for some representative values of the unsteadiness parameter S and the power law index n for a wide range of the generalized Prandtl number.

Abstract: In this paper the problem of momentum and heat transfer in a thin liquid film of power-law fluid on an unsteady stretching surface has been studied. Numerical solutions are obtained for some representative values of the unsteadiness parameter S and the power-law index n for a wide range of the generalized Prandtl number, 0.001 ≤ Pr ≤ 1000. Typical temperature and velocity profiles, the dimensionless film thickness, free-surface temperature, and the surface heat fluxes are presented at selected controlling parameters. The results show that increasing the value of n tends to increase the boundary-layer thickness and broadens the temperature distributions. The free-surface temperature of a shear thinning fluid is larger than that of a Newtonian fluid, but the opposite trend is true for a shear thickening fluid. For small generalized Prandtl numbers, the surface heat flux increases with a decrease in n, but the impacts of n on the heat transfer diminish for Pr greater than a moderate value (approximately 1 ≤ Pr ≤ 10, depending on the magnitude of S).

120 citations

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TL;DR: In this article, the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface is considered and the homotopy analysis method is employed for the analytic solution.

Abstract: This study is concerned with the magnetohydrodynamic (MHD) rotating boundary layer flow of a viscous fluid caused by the shrinking surface. Homotopy analysis method (HAM) is employed for the analytic solution. The similarity transformations have been used for reducing the partial differential equations into a system of two coupled ordinary differential equations. The series solution of the obtained system is developed and convergence of the results are explicitly given. The effects of the parameters M, s and λ on the velocity fields are presented graphically and discussed. It is worth mentioning here that for the shrinking surface the stable and convergent solutions are possible only for MHD flows.

107 citations

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##### 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

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01 Jan 1974

TL;DR: In this paper, the authors define the notion of groups of transformations and prove that a one-parameter group essentially contains only one infinitesimal transformation and is determined by it.

Abstract: 1. Ordinary Differential Equations.- 1.0. Ordinary Differential Equations.- 1.1. Example: Global Similarity Transformation, Invariance and Reduction to Quadrature.- 1.2. Simple Examples of Groups of Transformations Abstract Definition.- 1.3. One-Parameter Group in the Plane.- 1.4. Proof That a One-Parameter Group Essentially Contains Only One Infinitesimal Transformation and Is Determined by It.- 1.5. Transformations Symbol of the Infinitesimal Transformation U.- 1.6. Invariant Functions and Curves.- 1.7. Important Classes of Transformations.- 1.8. Applications to Differential Equations Invariant Families of Curves.- 1.9. First-Order Differential Equations Which Admit a Group Integrating Factor Commutator.- 1.10. Geometric Interpretation of the Integrating Factor.- 1.11. Determination of First-Order Equations Which Admit a Given Group.- 1.12. One-Parameter Group in Three Variables More Variables.- 1.13. Extended Transformation in the Plane.- 1.14. A Second Criterion That a First-Order Differential Equation Admits a Group.- 1.15. Construction of All Differential Equations of First-Order Which Admit a Given Group.- 1.16. Criterion That a Second-Order Differential Equation Admits a Group.- 1.17. Construction of All Differential Equations of Second-Order Which Admit a Given Group.- 1.18. Examples of Application of the Method.- 2. Partial Differential Equations.- 2.0. Partial Differential Equations.- 2.1. Formulation of Invariance for the Special Case of One dependent and Two Independent Variables.- 2.2. Formulation of Invariance in General.- 2.3. Fundamental Solution of the Heat Equation Dimensional Analysis.- 2.4. Fundamental Solutions of Heat Equation Global Affinity.- 2.5. The Relationship Between the Use of Dimensional Analysis and Stretching Groups to Reduce the Number of Variables of a Partial Differential Equation.- 2.6. Use of Group Invariance to Obtain New Solutions from Given Solutions.- 2.7. The General Similarity Solution of the Heat Equation.- 2.8. Applications of the General Similarity Solution of the Heat Equation,.- 2.9. -Axially-Symmetric Wave Equation.- 2.10. Similarity Solutions of the One-Dimensional Fokker-Planck Equation.- 2.11. The Green's Function for an Instantaneous Line Particle Source Diffusing in a Gravitational Field and Under the Influence of a Linear Shear Wind - An Example of a P.D.E. in Three Variables Invariant Under a Two-Parameter Group.- 2.12. Infinite Parameter Groups - Derivation of the Poisson Kernel.- 2.13. Far Field of Transonic Flow.- 2.14. Nonlinear and Other Examples.- 2.15. Construction of Partial Differential Equations Invariant Under a Given Multi-parameter Group.- Appendix. Solution of Quasilinear First-Order Partial Differential Equations.- Bibliography. Part 1.- Bibliography. Part 2.

998 citations

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TL;DR: An exact similarity solution of the Navier-Stokes equations is found in this article, where the solution represents the three-dimensional fluid motion caused by the stretching of a flat boundary.

Abstract: An exact similarity solution of the Navier–Stokes equations is found. The solution represents the three‐dimensional fluid motion caused by the stretching of a flat boundary.

486 citations

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TL;DR: In this article, a similarity transform was used to reduce the Navier-Stokes equations to a nonlinear ordinary differential equation governed by a non-dimensional unsteady parameter.

Abstract: A fluid film lies on an accelerating stretching surface. A similarity transform reduces the unsteady Navier-Stokes equations to a nonlinear ordinary differential equation governed by a nondimensional unsteady parameter. Asymptotic and numerical solutions are found. The results represent rare exact similarity solutions of the unsteady Navier-Stokes equations

405 citations

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TL;DR: In this paper, the fluid flow outside of a stretching cylinder is studied, governed by a third-order nonlinear ordinary differential equation that leads to exact similarity solutions of the Navier-Stokes equations.

Abstract: The fluid flow outside of a stretching cylinder is studied. The problem is governed by a third‐order nonlinear ordinary differential equation that leads to exact similarity solutions of the Navier–Stokes equations. Because of algebraic decay, an exponential transform is used to facilitate numerical integration. Asymptotic solutions for large Reynolds numbers compare well with numerical results. The heat transfer is determined.

231 citations