Journal Article•
General Formulations of Navier-Stokes Exact Solutions for Rotating Flow Systems with Variable Viscosity
TL;DR: In this article, an analytical solution of the Navier-Stokes equations for the case of laminar flows in rotating systems with variable viscosity fluids, aiming to provide reference solutions for the validation of numerical or empirical prediction models for such flows.
Abstract: Flows of variable viscosity fluids have many industrial applications in fluid mechanics and in engineering such as pump flow for high viscosity fluids. In most cases the fluid viscosity is mainly temperature dependent. Numerical investigation of such flows involves the solution of the Navier-Stokes equations with an extra difficulty arising from the fact that the viscosity is not constant over the flow field. This article presents an analytical solution of the Navier-Stokes equations for the case of laminar flows in rotating systems with variable viscosity fluids, aiming to provide reference solutions for the validation of numerical or empirical prediction models for such flows. In the present method, the analytical solution of the flow field is achieved by expressing the flow variables by using combination of Bessel and exponential functions. It is shown that the proposed solution satisfies the governing equations.
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01 Jan 1965TL;DR: This book discusses ODEs, Partial Differential Equations, Fourier Series, Integrals, and Transforms, and Numerics for ODE's and PDE's, as well as numerical analysis and potential theory, and more.
Abstract: PART A: ORDINARY DIFFERENTIAL EQUATIONS (ODE'S). Chapter 1. First-Order ODE's. Chapter 2. Second Order Linear ODE's. Chapter 3. Higher Order Linear ODE's. Chapter 4. Systems of ODE's Phase Plane, Qualitative Methods. Chapter 5. Series Solutions of ODE's Special Functions. Chapter 6. Laplace Transforms. PART B: LINEAR ALGEBRA, VECTOR CALCULUS. Chapter 7. Linear Algebra: Matrices, Vectors, Determinants: Linear Systems. Chapter 8. Linear Algebra: Matrix Eigenvalue Problems. Chapter 9. Vector Differential Calculus: Grad, Div, Curl. Chapter 10. Vector Integral Calculus: Integral Theorems. PART C: FOURIER ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS. Chapter 11. Fourier Series, Integrals, and Transforms. Chapter 12. Partial Differential Equations (PDE's). Chapter 13. Complex Numbers and Functions. Chapter 14. Complex Integration. Chapter 15. Power Series, Taylor Series. Chapter 16. Laurent Series: Residue Integration. Chapter 17. Conformal Mapping. Chapter 18. Complex Analysis and Potential Theory. PART E: NUMERICAL ANALYSIS SOFTWARE. Chapter 19. Numerics in General. Chapter 20. Numerical Linear Algebra. Chapter 21. Numerics for ODE's and PDE's. PART F: OPTIMIZATION, GRAPHS. Chapter 22. Unconstrained Optimization: Linear Programming. Chapter 23. Graphs, Combinatorial Optimization. PART G: PROBABILITY STATISTICS. Chapter 24. Data Analysis: Probability Theory. Chapter 25. Mathematical Statistics. Appendix 1: References. Appendix 2: Answers to Odd-Numbered Problems. Appendix 3: Auxiliary Material. Appendix 4: Additional Proofs. Appendix 5: Tables. Index.
3,643 citations
TL;DR: In this article, the authors present an approach for ODE's Phase Plane, Qualitative Methods, and Partial Differential Equations (PDE's) to solve ODE problems.
Abstract: PART A: ORDINARY DIFFERENTIAL EQUATIONS (ODE'S). Chapter 1. First-Order ODE's. Chapter 2. Second Order Linear ODE's. Chapter 3. Higher Order Linear ODE's. Chapter 4. Systems of ODE's Phase Plane, Qualitative Methods. Chapter 5. Series Solutions of ODE's Special Functions. Chapter 6. Laplace Transforms. PART B: LINEAR ALGEBRA, VECTOR CALCULUS. Chapter 7. Linear Algebra: Matrices, Vectors, Determinants: Linear Systems. Chapter 8. Linear Algebra: Matrix Eigenvalue Problems. Chapter 9. Vector Differential Calculus: Grad, Div, Curl. Chapter 10. Vector Integral Calculus: Integral Theorems. PART C: FOURIER ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS. Chapter 11. Fourier Series, Integrals, and Transforms. Chapter 12. Partial Differential Equations (PDE's). Chapter 13. Complex Numbers and Functions. Chapter 14. Complex Integration. Chapter 15. Power Series, Taylor Series. Chapter 16. Laurent Series: Residue Integration. Chapter 17. Conformal Mapping. Chapter 18. Complex Analysis and Potential Theory. PART E: NUMERICAL ANALYSIS SOFTWARE. Chapter 19. Numerics in General. Chapter 20. Numerical Linear Algebra. Chapter 21. Numerics for ODE's and PDE's. PART F: OPTIMIZATION, GRAPHS. Chapter 22. Unconstrained Optimization: Linear Programming. Chapter 23. Graphs, Combinatorial Optimization. PART G: PROBABILITY STATISTICS. Chapter 24. Data Analysis: Probability Theory. Chapter 25. Mathematical Statistics. Appendix 1: References. Appendix 2: Answers to Odd-Numbered Problems. Appendix 3: Auxiliary Material. Appendix 4: Additional Proofs. Appendix 5: Tables. Index.
2,257 citations
TL;DR: In this paper, the Navier-Stokes equations are used as a standard for checking the accuracy of approximate methods, whether they are numerical, asymptotic, or empirical.
Abstract: 1. The solutions represent fundamental fluid-dynamic flows. Also, owing to the uniform validity of exact solutions, the basic phenomena described by the Navier-Stokes equations can be more closely studied. 2. The exact solutions serve as standards for checking the accuracies of the many approximate methods, whether they are numerical, asymp totic, or empirical. Current advances in computer technology make the complete numerical integration of the Navier-Stokes equations more feasible. However, the accuracy of the results can only be ascertained by a comparison with an exact solution.
328 citations
"General Formulations of Navier-Stok..." refers methods in this paper
...Wang [2] has given an excellent review of these solutions of the Navier-Stokes equations....
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72 citations
TL;DR: A simple boundary value problem is studied that clearly delineates the difference between the solution to these equations and those due to the equations that are referred to as “Darcy Law” and finds the maximum value of the vorticity occurs at the boundary near which the fluid is less viscous in virtue of the pressure being lower.
52 citations
"General Formulations of Navier-Stok..." refers methods in this paper
...Some authors [16-18] have used inverse methods where some a priori conditions were assumed about the flow variables and have found some exact solutions....
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