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Stream function

About: Stream function is a(n) research topic. Over the lifetime, 4212 publication(s) have been published within this topic receiving 87652 citation(s).

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19 Jun 1986-
Abstract: I. Mathematical Foundation of the Stokes Problem.- 1. Generalities on Some Elliptic Boundary Value Problems.- 1.1. Basic Concepts on Sobolev Spaces.- 1.2. Abstract Elliptic Theory.- 1.3. Example 1: Dirichlet's Problem for the Laplace Operator.- 1.4. Example 2: Neumann's Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- s Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- 2. Theory of the Steady-State Navier-Stokes Equations.- 2.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 2.2. The Stream Function Formulation of the Homogeneous Problem..- 3. Approximation of Branches of Nonsingular Solutions.- 3.1. An Abstract Framework.- 3.2. Approximation of Branches of Nonsingular Solutions.- 3.3. Application to a Class of Nonlinear Problems.- 3.4. Non-Differentiable Approximation of Branches of Nonsingular Solutions.- 4. Numerical Analysis of Centered Finite Element Schemes.- 4.1. Formulation in Primitive Variables: Methods Using Discontinuous Pressures.- 4.2. Formulation in Primitive Variables: the Case of Continuous Pressures.- 4.3. Mixed Incompressible Methods: the "Stream Function-Vorticity" Formulation.- 4.4. Remarks on the "Stream Function-Gradient of Velocity Tensor" Scheme.- 5. Numerical Analysis of Upwind Schemes.- 5.1. Upwinding in the Stream Function-Vorticity Scheme.- 5.2. Error Analysis of the Upwind Scheme.- 5.3. Approximating the Pressure with the Upwind Scheme.- 6. Numerical Algorithms.- 2.11. General Methods of Descent and Application to Gradient Methods.- 2.12. Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations.- 2.13. Newton's Method and the Continuation Method.- References.- Index of Mathematical Symbols.

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Topics: Finite element method (54%), Dirichlet problem (54%), Numerical analysis (53%) ...read more

5,336 Citations


Journal ArticleDOI: 10.1098/RSTA.1953.0006
Michael S. Longuet-Higgins1Institutions (1)
Abstract: It was shown by Stokes that in a water wave the particles of fluid possess, apart from their orbital motion, a steady second-order drift velocity (usually called the mass-transport velocity). Recent experiments, however, have indicated that the mass-transport velocity can be very different from that predicted by Stokes on the assumption of a perfect, non-viscous fluid. In this paper a general theory of mass transport is developed, which takes account of the viscosity, and leads to results in agreement with observation. Part I deals especially with the interior of the fluid. It is shown that the nature of the motion in the interior depends upon the ratio of the wave amplitude a to the thickness $\delta $ of the boundary layer: when a$^{2}$/$\delta ^{2}$ is small the diffusion of vorticity takes place by viscous 'conduction'; when a$^{2}$/$\delta ^{2}$ is large, by convection with the mass-transport velocity. Appropriate field equations for the stream function of the mass transport are derived. The boundary layers, however, require separate consideration. In part II special attention is given to the boundary layers, and a general theory is developed for two types of oscillating boundary: when the velocities are prescribed at the boundary, and when the stresses are prescribed. Whenever the motion is simple-harmonic the equations of motion can be integrated exactly. A general method is described for determining the mass transport throughout the fluid in the presence of an oscillating body, or with an oscillating stress at the boundary. In part III, the general method of solution described in parts I and II is applied to the cases of a progressive and a standing wave in water of uniform depth. The solutions are markedly different from the perfect-fluid solutions with irrotational motion. The chief characteristic of the progressive-wave solution is a strong forward velocity near the bottom. The predicted maximum velocity near the bottom agrees well with that observed by Bagnold.

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Topics: Stokes boundary layer (63%), Stokes drift (61%), Boundary layer (57%) ...read more

1,139 Citations


Journal ArticleDOI: 10.1016/0045-7930(73)90027-3
C. Taylor1, Paul Hood1Institutions (1)
01 Jan 1973-Computers & Fluids
Abstract: The finite element discretisation technique is used to effect a solution of the Navier- Stokes equations. Two methods of formulation are presented, and a comparison of the effeciency of the methods, associated with the solution of particular problems, is made. The first uses velocity and pressure as field variables and the second stream function and vorticity. It appears that, for contained flow problems the first formulation has some advantages over previous approaches using the finite elemental method[1,2].

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1,081 Citations


Abstract: Tropical convective heating is balanced on the large scale by the adiabatic cooling of ascent. The horizontal divergence of the wind above this heating may be viewed as driving the upper tropospheric rotational wind field. A vorticity equation model is used to diagnose this relationship. It is shown that because of the advection of vorticity by the divergent component of the flow, the Rossby wave source can be very different from the simple −fD source often used. In particular, an equatorial region of divergence situated in easterly winds can lead to a Rossby wave source in the subtropical westerlies where it is extremely effective. This part of the source can be relatively insensitive to the longitudinal position of the equatorial divergence. A divergence field which is asymmetric about the equator can lead to a quite symmetric Rossby wave source. For a steady frictionless flow the Rossby wave source averaged over regions within closed streamfunction or absolute vorticity contours is, under cert...

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Topics: Rossby radius of deformation (65%), Rossby wave (64%), Vorticity (58%) ...read more

988 Citations


Open accessProceedings ArticleDOI: 10.1109/CVPR.2001.990497
01 Dec 2001-
Abstract: Image inpainting involves filling in part of an image or video using information from the surrounding area. Applications include the restoration of damaged photographs and movies and the removal of selected objects. We introduce a class of automated methods for digital inpainting. The approach uses ideas from classical fluid dynamics to propagate isophote lines continuously from the exterior into the region to be inpainted. The main idea is to think of the image intensity as a 'stream function for a two-dimensional incompressible flow. The Laplacian of the image intensity plays the role of the vorticity of the fluid; it is transported into the region to be inpainted by a vector field defined by the stream function. The resulting algorithm is designed to continue isophotes while matching gradient vectors at the boundary of the inpainting region. The method is directly based on the Navier-Stokes equations for fluid dynamics, which has the immediate advantage of well-developed theoretical and numerical results. This is a new approach for introducing ideas from computational fluid dynamics into problems in computer vision and image analysis.

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Topics: Inpainting (66%), Image restoration (57%), Stream function (51%) ...read more

954 Citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20222
2021113
2020104
2019143
201892
2017128

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Topic's top 5 most impactful authors

Tasawar Hayat

89 papers, 3.2K citations

Sohail Nadeem

24 papers, 1K citations

Abdul Majeed Siddiqui

22 papers, 322 citations

Dharmendra Tripathi

21 papers, 677 citations

Mikhail A. Sheremet

19 papers, 784 citations

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