An Elementary Treatise on Hydrodynamics and Sound

Front Cover
Deighton, Bell, 1890 - Hydrodynamics - 187 pages
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Contents

The equations of motion
9
1516 Another proof of the equations of motion
10
Pressure is a function of the density
13
Stokes proof that a velocity potential always exists if it exists at any particular instant
14
Physical distinction between rotational and irrotational motion
15
Integration of the equations of motion when a velocity potential exists
16
Impulsive motion
17
Flow and circulation
18
Velocity potential due to a source P 27 do due to a doublet
21
do due to a doublet in two dimensions
22
Image of a source in a plane
23
Motion of a liquid surrounding a sphere which is suddenly annihilated
25
Torrieellis theorem
27
The vena contracta
28
Giffards injector
30
Examples
31
B H
33
CHAPTER II
35
Boundary conditions for a cylinder moving in a liquid
36
Velocity potential and current function due to the motion of a circular cylinder in an infinite liquid
38
Motion of a cylinder in a liquid which is bounded by a concentric cylin drical envelop
40
Current function due to the motion of a cylinder whose cross section is a lemniscate of Bernoulli
42
do do an elliptic cylinder
43
Current function due to the motion of an elliptic cylinder
44
Failure of solution when the elliptic cylinder degenerates into a lamina Discontinuous motion
45
Motion of a sphere under the action of gravity
47
Motion may become unstable owing to the existence of a hollow
49
Effect of viscosity and definition of the coefficient of viscosity
50
Resistance experienced by a ship in moving through water
52
Motion of a spherical pendulum when the liquid is contained within a rigid spherical envelop
53
Examples
55
MOTION OF A SINGLE SOLID IN AN INFINITE LIQUID
59
Values of the components of momentum
66
Motion of an elliptic cylinder under the action of gravity
74
CHAPTER IV
85
Long waves in shallow water
96
do produced by wind
102
CHAPTER V
107
Velocity potential due to a vortex
109
Kircbhoff s elliptic vortex Ill 101 Discussion on the stability of a vortex
112
Motion of two vortices of equal strengths
113
Motion of two vortices of equal and opposite strengths
114
Motion of a vortex inside a circular cylinder
115

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Page 177 - ... every pressure, but the movement is extremely minute. When relieved they return at once to their normal positions, and they exert a 'strong effort of restitution, in virtue of a property which is termed elasticity. We are now dealing with gases, and may define their elasticity as follows : — Def. The elasticity of a gas under any given conditions is the ratio of any small increase of pressure to the cubical compression thereby produced. The term cubical compression denotes the ratio of the...
Page 174 - It is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.
Page 174 - It is impossible for a self-acting machine, unaided by any external agency, to convey heat from one body to another at a higher temperature ; or heat cannot of itself (that is, without compensation) pass from a colder to a warmer body.
Page 169 - The First Law of Thermodynamics. — When work is transformed into heat, or heat into work, the quantity of work is mechanically equivalent to the quantity of heat.
Page 23 - The image in a sphere, of a doublet whose axis passes through the centre of the sphere, can also be found by elementary methods.
Page 120 - ... latter. 8. Prove that three infinitely long straight cylindrical vortices of equal strengths will be in stable steady motion, when situated at the vertices of an equilateral triangle whose sides are large compared with the radii of the sections of the vortices; and that if they are slightly displaced, prove that the time of a small oscillation is the same as that of the time of revolution of the system in its undisturbed state. 9. A straight cylindrical vortex column of uniform rotation f, is...
Page 55 - In the irrotational motion of a liquid, prove that the motion derived from it, by turning the direction of motion at each point in one direction through 90 without changing the velocity, will also be a possible irrotational motion, the conditions at the boundaries being altered so as to suit the new motion. Discuss the motion obtained in this way from the preceding example.
Page 32 - A solid sphere of radius a is surrounded by a mass of liquid whose volume is 47rc3/3, and its centre is a centre of attractive force varying directly as the square of the distance. If the solid sphere be suddenly annihilated, shew that the velocity of the inner surface, when its radius is x, is given by && {(3?
Page 34 - The motion of a liquid is in two dimensions, and there is a constant source at one point A in the liquid and an equal sink at another point B ; find the form of the stream lines, and prove that the velocity at a point P varies as (AP.
Page 15 - Stokes, an important physical distinction in the character of the motion which takes place, according as a velocity potential does or does not exist. Conceive an indefinitely small spherical element of a fluid in motion to become suddenly solidified, and the fluid about it to be suddenly destroyed. By the instantaneous solidification velocities will be suddenly generated or destroyed in the different portions of the element, and a set of mutual impulsive forces will be called into action. Let x,...

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