LECTURES IN ELEMENTARY FLUID DYNAMICS

1 LECTURES IN ELEMENTARY . FLUID DYNAMICS : Physics, Mathematics and Applications J. M. McDonough Departments of Mechanical Engineering and Mathematics University of Kentucky, Lexington, KY 40506-0503. c 1987, 1990, 2002, 2004, 2009. Contents 1 Introduction 1. Importance of Fluids .. 1. Fluids in the pure sciences .. 2. Fluids in technology .. 3. The Study of Fluids .. 4. The theoretical approach .. 5. Experimental FLUID DYNAMICS .. 6. Computational FLUID DYNAMICS .. 6. Overview of Course .. 8. 2 Some Background: Basic Physics of Fluids 11.

2 The Continuum Hypothesis .. 11. Definition of a FLUID .. 13. Shear stress induced deformations .. 13. More on shear stress .. 15. FLUID Properties .. 15. Viscosity .. 15. Thermal conductivity .. 23. Mass diffusivity .. 23. Other FLUID properties .. 24. Classification of Flow Phenomena .. 29. Steady and unsteady flows .. 29. Flow dimensionality .. 29. Uniform and non-uniform flows .. 31. Rotational and irrotational flows .. 32. Viscous and inviscid flows .. 36. Incompressible and compressible flows .. 37.

3 Laminar and turbulent flows .. 37. Separated and unseparated flows .. 40. Flow Visualization .. 41. Streamlines .. 41. Pathlines .. 43. Streaklines .. 44. Summary .. 45. i ii CONTENTS. 3 The Equations of FLUID Motion 47. Lagrangian & Eulerian Systems; the Substantial Derivative .. 47. The Lagrangian viewpoint .. 48. The Eulerian viewpoint .. 49. The substantial derivative .. 49. Review of Pertinent Vector Calculus .. 52. Gauss's theorem .. 53. Transport theorems .. 56. Conservation of Mass the continuity equation.

4 58. Derivation of the continuity equation .. 58. Other forms of the differential continuity equation .. 60. Simple application of the continuity equation .. 61. Control volume (integral) analysis of the continuity equation .. 61. Momentum Balance the Navier Stokes Equations .. 69. A basic force balance; Newton's second law of motion .. 69. Treatment of surface forces .. 73. The Navier Stokes equations .. 78. Analysis of the Navier Stokes Equations .. 80. Mathematical structure .. 80. Physical interpretation.

5 81. Scaling and Dimensional Analysis .. 83. Geometric and dynamic similarity .. 83. Scaling the governing equations .. 85. Dimensional analysis via the Buckingham theorem .. 91. Physical description of important dimensionless parameters .. 98. Summary .. 99. 4 Applications of the Navier Stokes Equations 101. FLUID Statics .. 101. Equations of FLUID statics .. 102. Buoyancy in static fluids .. 108. Bernoulli's Equation .. 109. Derivation of Bernoulli's equation .. 110. Example applications of Bernoulli's equation.

6 113. Control-Volume Momentum Equation .. 116. Derivation of the control-volume momentum equation .. 116. Application of control-volume momentum equation .. 118. Classical Exact Solutions to N. S. Equations .. 122. Couette flow .. 122. Plane Poiseuille flow .. 124. Pipe Flow .. 126. Some terminology and basic physics of pipe flow .. 126. The Hagen Poiseuille solution .. 129. Practical Pipe Flow Analysis .. 133. Summary .. 158. List of Figures Mean Free Path and Requirements for Satisfaction of Continuum Hypothesis; (a).

7 Mean free path determined as average of distances between collisions; (b) a volume too small to permit averaging required for satisfaction of continuum hypothesis.. 12. Comparison of deformation of solids and liquids under application of a shear stress;. (a) solid, and (b) liquid.. 14. Behavior of things that flow ; (a) granular sugar, and (b) coffee.. 14. Flow between two horizontal, parallel plates with upper one moving at velocity U .. 16. Physical situation giving rise to the no-slip condition.. 17. Structure of water molecule and effect of heating on short-range order in liquids; (a).

8 Low temperature, (b) higher temperature.. 20. Effects of temperature on molecular motion of gases; (a) low temperature, (b) higher temperature.. 21. Diffusion of momentum initial transient of flow between parallel plates; (a) very early transient, (b) intermediate time showing significant diffusion, (c) nearly steady- state profile.. 21. Interaction of high-speed and low-speed FLUID parcels.. 22. Pressure and shear stress.. 26. Surface tension in spherical water droplet.. 27. Capillarity for two different liquids.

9 28. Different types of time-dependent flows; (a) transient followed by steady state, (b). unsteady, but stationary, (c) unsteady.. 30. Flow dimensionality; (a) 1-D flow between horizontal plates, (b) 2-D flow in a 3-D. box, (c) 3-D flow in a 3-D box.. 30. Uniform and non-uniform flows; (a) uniform flow, (b) non-uniform, but locally uniform flow, (c) non-uniform flow.. 32. 2-D vortex from flow over a step.. 34. 3-D vortical flow of FLUID in a box.. 35. Potential Vortex.. 36. Laminar and turbulent flow of water from a faucet; (a) steady laminar, (b) periodic, wavy laminar, (c) turbulent.

10 38. da Vinci sketch depicting turbulent flow.. 38. Reynolds' experiment using water in a pipe to study transition to turbulence; (a). low-speed flow, (b) higher-speed flow.. 39. Transition to turbulence in spatially-evolving flow.. 39. (a) unseparated flow, (b) separated flow.. 40. Geometry of streamlines.. 42. Temporal development of a pathline.. 44. iii iv LIST OF FIGURES. FLUID particles and trajectories in Lagrangian view of FLUID motion.. 48. Eulerian view of FLUID motion.. 49. Steady accelerating flow in a nozzle.