A treatise on hydrodynamics: with numerous examples, Volume 2 (Google eBook)

Front Cover
Deighton, Bell and Co., 1888 - Hydrodynamics
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Contents

Application to the potentials of anchor rings
26
Difference and mixed difference equations satisfied by zonal functions
27
Expression for the current function of a liquid bounded by an anchor ring
28
Expressions for the component velocities of the liquid
29
Value of the cyclic constant
30
Approximate values of the R and T functions and the value of the current function in terms of them
31
Examples
32
CHAPTER XIII
34
Current function due to a single vortex
35
do due to a number of vortices
36
Single vortex surrounded by liquid in which there is an additional circulation
37
Conditions of stability of the preceding case
38
Kirchhoffs elliptic vortex
41
The cross section of number of vortices remains approximately circular
42
Vibrations of two vortices
43
Motion of two vortices of equal strengths
45
Motion of two vortices of equal and opposite strengths
46
Motion of a vortex inside a circular cylinder
47
Motion of a vortex between two parallel planes
49
rath of a vortex in a parabolic cylinder
50
300304 On the method of inversion
52
Examples
55
ART PAGE 305 Definition of a circular vortex
59
306307 Determination of the current function due to a single vortex of indefinitely small cross section
60
Motion of a single vortex
61
309310 Motion of two vortices whose planes are parallel and whose centres lie in the same straight line
62
Image of a vortex in a sphere
63
Velocity potential due to a vortex
65
313314 Vibrations of a circular vortex J J Thomsons investigations
66
Calculation of the velocities due to the disturbed vortex
67
The boundary conditions
69
Value of j2 q cos eoenffd when is nearly equal to unity
70
Approximate values of the velocity of translation of the vortex and the angular velocity of the liquid composing its surface
73
Stability of the steady motion of the vortex and the value of the period of oscillation
74
Linked Vortices
75
Steady motion and small oscillations of two linked vortices of equal strengths
76
The motion consists of a quick vibration and a slow one
78
The slow vibrations will not be excited unless the vortex suffers some external disturbance
79
The steady motion of more than six rectilinear vortices which are situated at the angular points of a regular polygon is unstable
80
Equation of the cross section
82
Calculation of the current function outside the vortex
83
Conditions at the surface of separation Value of ft
85
Solution of the problem of steady motion
87
Fluted vibrations about a state of steady motion
88
Values of the current functions due to the disturbed motion
89
Values of the velocity potentials due to the disturbed motion
90
Determination of the period equation
91
Discussion of results
92
Pulsations
93
ON THE MOTION OF A LIQUID ELLIPSOID UNDER THE INFLUENCE OF ITS OWN ATTRACTION ART PAGE 341 History of the problem
95
Determination of the velocities of the liquid
96
The hydrodynamical equations and the dynamical equations for rotation
97
Determination of the pressure
99
Riemanns equations of motion
100
The three first integrals of the equations of motion and their in terpretation
101
Dirichlets equations for the motion of an ellipsoid of revolution
103
Motion of an ellipsoid of revolution which is initially at rest
104
Irrotational motion of an ellipsoid of revolution which is not initially at rest
105
Steady motion of an ellipsoid Classification of results
108
Solution of the problem when the axis of rotation is a principal axis
110
Maclaurins ellipsoid of revolution
111
Jacobis ellipsoid
113
Dedekinds ellipsoid
114
360362 Riemanns ellipsoid
115
Stability of a liquid ellipsoid
118
The condition of stability is that the energy must be a minimum in steady motion
119
When the disturbance consists of a couple the motion is usually unstable excepting in the case of Dedekinds ellipsoid
120
When the disturbance does not consist of a couple the motion is usually stable excepting in the case of Maclaurins ellipsoid
121
Maclaurins ellipsoid is unstable if the excentricity exceeds a certain value
123
Small oscillationsPoincares equation
124
Examples
126
CHAPTER XVI
130
Potential of a nearly spherical mass of attracting matter
131
ART PAGE 375377 Calculation of the potential of the two masses
134
Value of the angular velocity
138
Explanation of the results
139
Progressive waves moving parallel to a shore
153
Standing waves whose crests are parallel to a shore which slopes at an angle Jt
156
Waves in a cylinder
157
Waves when the initial displacement is given by the equation ij or cos 6
158
Waves in hyperboloids and cones
159
Analytical theory of long waves
160
Stationary waves in flowing water
162
Theory of group velocity
163
Forced waves produced by a surface pressure equal to C+ + b3 + bixi + bi
164
Form of the free surface when the pressure is equal to C + x
166
Deep sea wavesStokes solution
167
The liquid has a slow motion of translation in the direction of the wave
171
410411 The Solitary WaveLord Rayleighs solution
173
Capillary waves
177
Production of ripples by wind
178
Waves in water covered with ice
180
Examples
181
CHAPTER XVIII
188
Special cases of instability of a plane vortex sheet
189
Motion of a jet bounded by parallel planes
190
Motion of a cylindrical jet
191
422426 Stability of steady motion between two parallel planes
194
Steady motion between two concentric cylinders
197
CHAPTER XIX
199
The equilibrium theory Determination of the height of the tide
200
Correction for continents
202
Tides of long period
203
The semidiurnal tides
204
Laplaces theory
205
Form of the equation of continuity
206
The equations of motion
207
Deduction of the equations for determining the tidal oscillations of the ocean
209
Tides of long period Laplaces solution
210
do Prof Darwins solution when the ocean is of uniform depth
211
The diurnal tides
214
Motion of the water when the tide is diurnal
215
The semidiurnal tide in an ocean of uniform depth
216
Laplaces numerical results
218
Free oscillations of an ocean of uniform depth
220
The canal theory of tides
222
Development of the expression for the disturbing force
223
Solution of the problem when the disturbing body lies on the equator The first semidiurnal tide
226
453454 Solution when the canal is a great circle
227
Discussion of results Tides of long period
228
466457 Tides in estuaries
229
CHAPTER XX
232
The stresses due to viscosity
233
Components of stress across any plane
234
462463 The stress experienced by an element consists of sjx components
236
The stress quadric
237
The stresses due to viscosity depend upon the motion of distortion Stokes three assumptions
238
Determination of the stresses in terms of the rates of strain
240
The final equations of motion
242
Definition of the coefficient of viscosity
243
The equations for determining the rates of change of the components of molecular rotation
245
Discussion of the assumptions by which the equations of motion have been obtained
246
Impulsive motion
247
Values of the coefficient of viscosity of certain liquids
248
do in the case of certain gases
250
Dissipation of Energy Interpretation of Stokes third assumption
252
479482 Steady motion
253
Loss of energy under certain conditions is a minimum
256
Examples
258
CHAPTER XXI
260
488489 Motion of a spherical pendulum
263
Calculation of the resistance experienced by the sphere
265
Final equation of motion of the pendulum
267
ABT PAGE
268
Motion of a cylindrical pendulum
275
When a cylinder is moving parallel to its axis steady motion
281
Calculation of the current function
287
Discussion of results
296
CHAPTER XXIII
304
ART PAGE
308
Instability of viscous liquids Beynolds experiments
315
Form of the final result when the viscosity is either very large or very
322

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