Contents. Chapter I icon

Contents. Chapter I



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CONTENTS.

CHAPTER I.

HYDROKINEMATICS.

1. Introduction............ 1

2. Definition of a fluid........... 3

3. Lagrangian method, and Eulerian or flux method..... 4

4. Velocity and acceleration — Lagrangian method..... 4

5-6. „ „ „ — flux method....... 5

7. Analytical lemma........... 7

8-9. The equation of continuity......... 7

10. Definition of the velocity potential, and the forms of Laplace's operator

in polar and cylindrical coordinates....... 8

11. Lagrange's equation of continuity........ 9

12. The bounding surface.......... 9

13. Lines of flow and stream lines........ 10

14-15. Properties of lines of flow in a liquid....... 11

16. Earnshaw's and Stokes' current function...... 11

17. Molecular rotation........... 12

18-19. Formulae of transformation........ 13

Examples............ 15

^ CHAPTER II.

ON THE GENERAL EQUATIONS OF MOTION OF A PERFECT FLUID.

20. Pressure at every point of a fluid is equal in all directions ... 19

21. Equations of motion of a perfect fluid....... 20

22. Pressure is a function of the density....... 21

23. Equations of motion referred to moving axes...... 21

24. Equations satisfied by the components of molecular rotation ... 22

25. Stokes' proof that a velocity potential always exists, if it exists at any

particular instant.......... 22

26. Physical distinction between rotational and irrotational motion .. 23

27. Lagrange's hydrodynamical equations of motion..... 25

28. Weber's transformation.......... 25

29. Proof of theorem that if the pressure is not a function of the density,

vortex motion can be generated or destroyed in a perfect fluid .. 26

30. Cauchy's integrals........... 26

31. Integration of equations of motion when a velocity potential exists .. 27

32. Definition of a vortex line......... 27

33. Clebsch's transformation.......... 28

34. Proof thatis a maximum or minimum . 30

35-36. Energy and least action......... 31

37. Steady motion — Bernoulli's theorem....... 33

38. Conditions of steady motion which is symmetrical with respect to an axis,

or is in two dimensions......... 34

39-40. Steady motion of a liquid — Clebsch's method..... 36

41. Maximum and minimum theorem........ 38

42. Impulsive motion...........
38

43. Motion of a liquid surrounding a sphere which is suddenly annihilated . 39

44. Torricelli's theorem........... 40

45. Application of the hypothesis of parallel sections, to the motion of liquid

flowing out of a vessel......... 41

Examples............. 43

^ CHAPTER III.

ON SOURCES, DOUBLETS AND IMAGES.

46. Velocity potential due to a source or sink...... 48

47. do. due to a doublet........ 49

48. do. due to a doublet sheet....... 49

49. do. due to a source and doublet in two dimensions .. 49

50. Theory of images........... 50

51. Image of a source in a liquid bounded by a fixed plane ....51

52. Image of a source in a sphere......... 51

53. Image of a doublet in a sphere, whose axis passes through the centre of

the sphere............ 53

54. Image of a doublet in a sphere, whose axis is perpendicular to the line

joining it with the centre of the sphere...... 54

55. Image of a source and line sink in a sphere...... 54

56. Image of a source and doublet in a cylinder...... 56

57. Image of a source between two parallel planes...... 56

Examples............. 59

^ CHAPTER IV.

VORTEX MOTION AND CYCLIC IRROTATIONAL MOTION.

58. Statement of problem.......... 62

59. Fundamental properties of vortex filaments...... 62

60. Integration of the equations which determine the components of

molecular rotation in terms of the velocities..... 64

61. Velocity due to a vortex.......... 66

62. Velocity potential due to a vortex........ 66

63. Vortex sheets............ 68

64. Surfaces of discontinuity.......... 68

65. Surfaces of discontinuity possess the properties of vortex sheets .. 68

66. Cyclic and acyclic irrotational motion....... 70

67. Flow and circulation.......... 70

68. Stokes' theorem........... 70

69. Circulation due to a vortex filament is equal to twice the product of its

angular velocity and its cross section...... 71

70. Polycyclic velocity potentials......... 72

71. Stream lines cannot form closed curves unless the motion is cyclic .. 72

72. Circulation is independent of the time....... 72

73. Irrotational motion which is acyclic cannot become cyclic ... 73

74. Reconcileable and irreconcileable lines....... 73

75. Simply and multiply connected regions....... 73

76. A multiply connected region is reducible to a simply connected one . 74

77-78. Reduction of polycyclic velocity potentials to monocyclic functions . 74

79. Vorticity............. 75

80. Green's theorem........... 76

81. Deductions from Green's theorem — Kinetic energy of a liquid .. 77

82. Physical interpretation of Green's theorem...... 78

83. Liquid whose motion is acyclic and irrotational, is reduced to rest if the

motion of the bounding surface is destroyed..... 78

84. Extension of Green's theorem to spaces bounded internally by several closed surfaces........... 78

85. Kinetic energy of a liquid occupying such a region..... 78

86. Adaption of Green's theorem to two-dimensional space .... 79

87. Stokes' theorem a particular case of Green's theorem .... 80

88. Thomson's extension of Green's theorem....... 81

89. Kinetic energy of an infinite liquid occupying a multiply connected space 82

90. Kinetic energy of a liquid contained within a closed surface, is less

when the motion is irrotational and acylic, than if the liquid had any

other possible motion......... 83

91. Kinetic energy when the motion is rotational...... 83

92. Kinetic energy due to two vortex filaments is proportional to the electro-

kinetic energy due to two electric currents..... 84

93. Another expression for the kinetic energy...... 85

94. Kinetic energy in terms of Stokes' current function .... 85

95. Connection between vortex motion and electromagnetism ... 86 Examples............ 89

^ CHAPTER V.

ON THE MOTION OF A LIQUID IN TWO DIMENSIONS.

96. Statement of problem.......... 90

97. Boundary conditions for a cylinder moving in a liquid .... 90

98-101. Conjugate functions, and their properties...... 91

102. Examples of conjugate functions — Circular cylinder moving in an infinite liquid — Initial motion due to a circular cylinder in a liquid bounded by a fixed concentric cylinder — Motion of a liquid contained in a rotating equilateral prism — do. in a rotating elliptic cylinder............ 93

103. Motion of a liquid contained in a rotating rectangular prism .. 96

104-105. Motion of a liquid contained in a rotating sector .... 98

106. Further applications of conjugate functions...... 100

107. Motion of an elliptic cylinder in an infinite liquid..... 100

108. Motion of translation of a cylinder whose cross section is the inverse of

an ellipse with respect to its centre....... 102

109. Expression of results in terms of elliptic functions .... 104

110. Current function due to the rotation of the cylinder .... 105

111. do. when liquid is contained within a cylindrical cavity of this form............ 105

112. Application of results to the theory of conduction of heat and of electrified cylinders.......... 106

113. Motion of a cylinder whose cross section is the inverse of an ellipse with respect to its focus..106

114. Motion of translation of a cylinder whose cross section is a lemniscate. 106

115. Coefficients of cos in the expansion of (1 + 2c cos θ + c2)½ ... 107

116. Motion of rotation of a cylinder whose cross section is a lemniscate . 109

117-119. Motion of a cylinder whose cross section is a lemniscate of Bernoulli 109

120-121. Dipolar coordinates......... 110

122. Motion of two circular cylinders in an infinite liquid .... 112

123. Kinetic energy of an infinite liquid in which two circular cylinders are moving............ 113

124. Expressions for the coefficients of the velocities..... 114

Examples............ 115

^ CHAPTER VI.

DISCONTINUOUS MOTION.

125-127. Statement of problems to be solved...... 120

128-129. Representation of a vector by means of a complex quantity, and the

properties of the latter......... 121

130. Every complex has a differential coefficient...... 122

131. Kirchhoff's method of solving problems of discontinuous motion .. 123

132-133. Transformation by means of complex variables .... 124

134. Particular cases of transformation........ 126

135. Motion of a jet escaping from a slit....... 127

136. Motion of a jet escaping through a small tube..... 129

137. Coefficient of contraction of a jet can never be less than ½ 130

138. Stream of liquid flowing past a rectangular lamina. Pressure on the lamina............ 131

139. Conditions of stable and unstable equilibrium of the lamina ... 134

140. Intrinsic equation of the surface of discontinuity..... 135

Examples............ 135

^ CHAPTER VII.

ON THE KINEMATICS OF SOLID BODIES MOVING IN A LIQUID.

141. Conditions to which the velocity potential must be subject ... 137

142. Boundary conditions for the case of a single solid..... 137

143-144. Velocity potential due to the motion of a sphere .... 138

145. do. due to the motion of the solid formed by the revolution

of two spheres cutting orthogonally...... 139

146. do. due to the initial motion of two concentric spheres . 140

147. do. due to the motion of an ellipsoid..... 140

148. do. due to the motion of an ellipsoid of revolution .. 143

149. do. due to the motion of a circular disc .... 144

150. do. due to liquid contained in a rotating ellipsoidal cavity . 145

151. do. due to liquid contained between two confocal ellipsoids. 145

152. Magnetic potential of a spherical bowl....... 146

153. Velocity potential due to the motion of liquid about a spherical bowl . 147

154. do. due to a source situated on the axis of the bowl .. 149

155. do. due to the motion of the bowl in an infinite liquid . 149

156. Electro-static potential of a bowl placed in a field of force symmetrical

with respect to the axis of the bowl...... 150

157. Electro-static potential when the bowl is placed in a uniform field of force parallel to the axis.152

158. Interpretation of the result......... 153

159. Electro-static potential when the bowl is placed in a uniform field of force perpendicular to a plane containing the axis ....154

160. Current function due to the motion of a solid of revolution parallel to its axis............ 155

Examples............ 156

^ CHAPTER VIII.

ON THE GENERAL EQUATIONS OF MOTION OF A SYSTEM OF SOLID BODIES MOVING IN A LIQUID.

161. The motion can be determined by Lagrange's equations ... 159

162. Acyclic motion........... 159

163. Kinetic energy is a homogeneous quadratic function of the velocities of

the solids alone.......... 160

164. Proof that Lagrange's equations can be employed..... 161

165. Impulse of the motion.......... 162

166. Hamiltonian equations.......... 163

167. Kirchhoff's equations.......... 164

168-9. Geometrical equations......... 165

170. Cyclic motion........... 167

171. Kinetic energy is the sum of a homogeneous quadratic function of the

velocities of the solids and a similar function of the circulations . 169

172. Expressions for the generalised components of momentum due to the

cyclic motion........... 170

173. The modified Lagrangian function........ 171

174. Interpretation of the result. Product of the circulation and density is

a generalised component of momentum...... 175

175. The flux through an aperture relative to the solid is, the generalised

velocity corresponding to the product of the density and the circulation through that aperture....... 176

176. Modified function for a single solid having one aperture ... 177

177-178. Modified function for a system of cylinders..... 178

179. Explanation of results.......... 181

^ CHAPTER IX.

ON THE MOTION OF A SINGLE SOLID IN AN INFINITE LIQUID.

180. General expression for kinetic energy....... 182

181. Kinetic energy due to the motion of an ellipsoid and of a solid of

revolution........... 183

182. Motion of a sphere under the action of gravity..... 183

183. Motion may become unstable owing to the formation of a hollow .. 185

184. Initial motion of a sphere, when the liquid is enclosed within a con-

centric spherical envelope........ 185

185. Motion of a circular cylinder when there is circulation ....186

186. Determination of the motion by means of Lagrange's equations .. 188

187. Motion of an elliptic cylinder........ 189

188. Motion of a cylinder whose cross section is a curve such as a cardioid . 193

189. Motion of an ellipsoid.......... 193

190. Stability and instability of steady motion parallel to an axis .. 194

191. Calculation of the coefficients of inertia of an ellipsoid .... 195

192. Motion of an ellipsoid when two of its axes remain in a plane .. 195

193. Motion of a ring-shaped solid of revolution through whose aperture

there is circulation.......... 196

194. Motion of a ring produced by an impulsive couple about a diameter . 197

195. Angular motion expressible in terms of the time by means of elliptic

functions............ 199

196. Explanation of results by means of general principles .... 202

197. Steady motion and stability of a ring moving parallel to its axis .. 202

198. Steady motion and stability when the centre of inertia describes a circle 203

199-200. Helicoidal steady motion........ 205

201. Stability of helicoidal steady motion....... 207

202. Expressions for the kinetic energy in the case of an isotropic helicoids and other solids.......... 208

203. Three directions of permanent translation for every solid ... 209

204. Wrenches and screws.......... 210

205. Infinite number of steady motions when the impulse consists of a twist about a screw........... 210

206. Motion of a solid is determinate when the impulse consists of a couple 212

Examples............ 213

^ CHAPTER X.

ON THE MOTION OF TWO CYLINDERS.

207. Expression for kinetic energy......... 219

208. Motion of a cylinder in a liquid bounded by a fixed plane ... 219

209. Ratio of initial to terminal velocity when the cylinder is in contact

with the plane and is projected from it...... 221

210. Conditions that the cylinder may or may not strike the plane .. 221

211-212. Motion of one cylinder when the other is fixed .... 222

213. Motion of a cylinder in a liquid bounded by a fixed plane, when there is circulation........... 223

214. Steady motion of a cylinder when the liquid is bounded by a horizontal plane............ 224

215-216. Deduction of results from general reasoning..... 226

Examples............ 227

^ CHAPTER XI.

ON THE MOTION OF TWO SPHERES.

217. Kinetic energy consists of nine terms only...... 229

218. Motion along the line of centres........ 230

219. Calculation of the coefficients of the velocities in the expression for the

kinetic energy, by the method of images...... 231

220-221. Complete values of the coefficients....... 232

222. Approximate values of the coefficients, in terms of series of powers of

the reciprocal of the distance between the spheres .... 236

223. Motion perpendicular to the line of centres...... 237

224-225. Calculation of the coefficients by means of images .... 237

226. Calculation of the images μ1, v1, μ'1....... 239

227. Transference theorem in spherical harmonics...... 240

228. Kinetic energy depends solely upon harmonics of the first degree .. 242

229. Calculation of the velocity potential, and of the approximate values of

the coefficients........... 243

230. Motion of a sphere in a liquid bounded by a fixed plane ... 244

231. Expressions for the components of the pressure upon the sphere .. 245

232. Small oscillations of two spheres........ 246

233. Oscillations of the second order. Thomson's theorem .... 247

234. Pulsations of two spheres......... 248

235. Velocity potential due to pulsations, is equivalent to that due to an

infinite system of images.........249

236. Formula for determining the pressure upon the spheres ... 249

237-238. Determination of that portion of the pressure which does not

depend upon the square of the velocity...... 250

239-240. Approximate values of the velocity potential, and of the portion of

the pressure which depends on the square of the velocity .. 252

241. The spheres attract one another when their phases differ by less than a

quarter of a period.......... 255

Examples............ 256

APPENDIX.

I. Proof of the equation p = kpγ......... 259

II. Value of a q-series in terms of elliptic functions..... 260

III. Determination of the azimuthal motion of a solid of revolution by

means of Weierstrass's functions....... 261



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