The Navier-Stokes Equations: A Classification of Flows and Exact Solutions (London Mathematical Society Lecture Note Series)

The Navier-Stokes equations were firmly established in the 19th Century as the system of nonlinear partial differential equations which describe the motion of most commonly occurring fluids in air and water, and since that time exact solutions have been sought by scientists. Collectively these solutions allow a clear insight into the behavior of fluids, providing a vehicle for novel mathematical methods and a useful check for computations in fluid dynamics, a field in which theoretical research is now dominated by computational methods. This 2006 book draws together exact solutions from widely differing sources and presents them in a coherent manner, in part by classifying solutions via their temporal and geometric constraints. It will prove to be a valuable resource to all who have an interest in the subject of fluid mechanics, and in particular to those who are learning or teaching the subject at the senior undergraduate and graduate levels. Cover......Page 1 LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES 334......Page 2 The Navier–Stokes equations: a classification of flows and exact solutions......Page 4 Copyright - ISBN: 9780521681629......Page 5 Contents......Page 6 Preface......Page 10 1 Scope of the book......Page 12 2.1 Plane Couette–Poiseuille flow......Page 22 2.2 Beltrami flows and their generalisation......Page 26 2.2.2 Flow due to a stretching plate......Page 28 2.2.4 The asymptotic suction profile......Page 30 2.3.1 The classical Hiemenz (1911) solution......Page 31 2.3.2 Oblique stagnation-point flows......Page 34 2.3.3 Two-fluid stagnation-point flow......Page 37 2.4.1 Parallel-sided channels......Page 39 2.4.2 Non-parallel-sided channels......Page 43 2.5.1 A corner flow......Page 49 2.5.2 A swept stagnation flow......Page 50 2.5.3 Vortices in a stagnation flow......Page 51 2.5.4 Three-dimensional stagnation-point flow......Page 53 3.1 Circular pipe flow......Page 56 3.3 Beltrami flows and their generalisation......Page 61 3.4.1 The classical Homann (1936) solution......Page 64 3.4.2 Stagnation on a circular cylinder......Page 67 3.4.3 Flow inside a porous or stretching tube......Page 73 3.5 Rotating-disk flows......Page 79 3.5.1 The one-disk problem......Page 80 3.5.2 The two-disk problem......Page 84 3.6 Ekman flow......Page 88 3.7.1 The round jet......Page 89 3.7.2 The Burgers vortex......Page 93 3.7.3 The influence of boundaries......Page 94 4 Unsteady flows bounded by plane boundaries......Page 100 4.1 The oscillating plate......Page 101 4.2 Impulsive flows......Page 102 4.2.1 Applied body force......Page 104 4.2.2 Applied shear stress......Page 105 4.2.3 Diffusion of a vortex sheet......Page 106 4.3 More general flows......Page 107 4.4 The angled flat plate......Page 109 4.5 Unsteady plate stretching......Page 111 4.6 Beltrami flows and their generalisation......Page 112 4.7.1 Transverse oscillations......Page 117 4.7.2 Orthogonal oscillations......Page 120 4.7.3 Superposed shear flows......Page 124 4.7.4 Three-dimensional stagnation-point flow......Page 125 4.7.5 Rotational three-dimensional stagnation-point flow......Page 127 4.7.6 Flow at a rear stagnation point......Page 129 4.8.1 Fixed boundaries......Page 130 4.8.2 Squeeze flows......Page 132 4.8.3 Periodic solutions......Page 135 5.1.1 Impulsive pipe flow......Page 139 5.1.2 Periodic pipe flow......Page 140 5.1.3 Pulsed pipe flow......Page 141 5.1.4 The effects of suction or injection on periodic flow......Page 143 5.1.5 Pipes with varying radius......Page 147 5.1.6 Impulsive cylinder flows......Page 149 5.3 Stagnation-point flows......Page 153 5.3.1 The Homann flow against an oscillating plate......Page 154 5.3.2 Oblique stagnation-point flow......Page 157 5.3.3 Unsteady stagnation on a circular cylinder......Page 159 5.4 Squeeze flows......Page 162 5.4.1 Constant force......Page 163 5.4.2 Prescribed gap width......Page 165 5.5.1 Self-similar flows......Page 167 5.5.2 Rotating disk in a counter-rotating fluid......Page 172 5.5.3 Non-axisymmetric flows......Page 175 5.5.4 An Ekman flow......Page 178 5.6.1 Single-cell vortices......Page 180 5.6.2 Multi-cell vortices......Page 183 5.6.3 The influence of boundaries......Page 188 References......Page 192 Index......Page 206 "This book draws together exact solutions from widely differing sources and presents them in a coherent manner, in part by classifying solutions via their temporal and geometric constraints. It will prove to be a valuable resource to all who have an interest in the subject of fluid mechanics, and in particular to those who are learning or teaching the subject at senior undergraduate and graduate levels."--Jacket This 2006 book details exact solutions to the Navier-Stokes equations for senior undergraduates and graduates or research reference