Physics for Geologists, Second edition

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124 Fluids and fluid flow There is also thermal energy Et, but we are considering an ideal liquid of zero viscosity. The principle of the conservation of energy requires that the sum of the work done, potential energy and kinetic energy shall be constant: in which each term has the dimensions of energy, ML~T-~. Since pgA1811 = pgA2812, ( ) (3 + z + = constant. This is known as Bernoulli's theorem, and it is strictly valid only for incompressible, frictionless liquids. Each term has the dimension of length. What meaning is to be attached to the velocity, q? The velocity of a particle of water depends on its position. It is slow near the edge of the fluid container, faster near the centre. As before, we take the velocity to be the volumetric rate of flow divided by the cross-sectional area of flow: L3T-l L-~ = LT-l. So what do we really mean by the kinetic energy of water flowing in a pipe or channel? A particle of mass m and velocity V has kinetic energy mv2/2, or, per unit of weight, ~ ~ 1But 2 water ~ . flowing in a pipe or channel has a profile in three dimensions, so what is its total kinetic energy? If we know the profiles accurately, we can obtain the total kinetic energy; but we can rarely do that. So we write where a is a factor that takes the total water flow into account. For laminar flow in circular pipes, it turns out that a = 2. For turbulent flow, a is a little more than 1. The first term in Equation 12.8, plpg, is known as the pressure head (see Figure 12.7), and it is the vertical height of a column of liquid of mass density p that can be supported by a pressure p. The second term, z, is merely the elevation of the element of liquid above some arbitrary horizontal datum (strictly a surface of constant potential energy). It is called the elevation head (less desirably, the potential head). The third term, q2/2g, is called the velocity head. The sum of these is called the total head, and with an ideal, incompressible liquid of zero viscosity, the total head would remain Copyright 2002 by Richard E. Chapman
Fluids and fluid flow 125 Figure 12.7 Heads. The pressure head (plpg), elevation head (2, negative when below the datum), and total head (h), which is the algebraic sum of the other two. constant. In fluid flow in rocks the velocity head is negligibly small and is neglected. The total head is proportional to the energy of the water (lacking only the factor g). Real liquids are not ideal, and energy will be lost to heat, leading to a loss of total head as the real liquid moves. Fluid flow in rocks Henry Darcy (note the spelling, Henry not Henri, no apostrophe in Darcy) was an engineer responsible for water supply in the town of Dijon in France, and he conducted experiments to determine the effect of sand filters on the flow of water through pipes. He published the results in 1856. Darcy filled a cylinder with sand, supported on a fine mesh, in apparatus shown diagrammatically in Figure 12.8. Manometers were placed near the top and bottom of the sand, and water was passed through the sand in the cylinder. The volumetric rate of flow was measured, as were the elevations of the water1 in the manometers (Figure 12.9). Darcy observed that 'for sands of the same nature' the flow was proportional to the difference in elevation of the levels in the manometers, and to the area normal to flow, and inversely proportional to the length of the sand: 1 Darcy actually used mercury manometers, but it is simpler to treat them as water manometers here. Copyright 2002 by Richard E. Chapman
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Physics for Geologists Second editi
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Copyright 2002 by Richard E. Chapma
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... vlii Contents Polarization 47 L
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Figures Copyright 2002 by Richard E
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Tables Copyright 2002 by Richard E.
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xiv Preface waves diminishes with d
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xvi Preface provides. This book is
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xviii Acknowledgements to K. Whaler
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xx Symbols mass refractive index bu
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2 Basic concepts Table 1.1 Dimensio
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4 Basic concepts Table 1.2 Basic SI
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6 Basic concepts pascal (Pa) and si
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8 Basic concepts all the relevant d
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10 Basic concepts relating them by
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2 Force The dimensions of force are
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14 Force The ratio of the Moon's ac
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16 Force Figure 2.1 The sum of the
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18 Force When a motorcycle takes a
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20 Force Figure 2.3 Torque is the p
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22 Force How does momentum differ f
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24 Force the weights - that is, the
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26 Force Figure 2.6 Wall-sticking i
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3 Gravity Universal gravity When di
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30 Gravity are significant to geolo
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32 Gravity Figure 3.2 The tractive,
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34 Gravity Mean sea level It might
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36 Gravity The benefits of space fl
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38 Gravity Figure 3.6 Profile of th
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40 Gravity Figure 3.8 The siphon at
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4 Optics Light is an electromagneti
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44 Optics Figure 4.2 Refraction. (a
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46 Optics The coeficient of reflect
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48 Optics only the light oscillatin
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50 Optics called Iceland Spar. Two
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52 Optics Diffraction If you shine
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54 Optics Figure 4.7 Geometrical op
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56 Optics Figure 4.9 The mirrors in
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5 Atomic structure and age-dating P
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60 Atomic structure and age-dating
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66 Electromagnetic radiation radiat
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68 Electromagnetic radiation Source
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Heat and heat flow The noun heat ha
Page 89 and 90: 72 Heat and heat flow Second Law: A
Page 91 and 92: 8 Electricity and magnetism Electri
Page 93 and 94: Electricity and magnetism 77 Figure
Page 95 and 96: Electricity and magnetism 79 rivett
Page 97 and 98: Electricity and magnetism 8 1 defin
Page 99 and 100: Electricity and magnetism 83 is kno
Page 101 and 102: 9 Stress and strain Pressure, p, is
Page 103 and 104: Stress and strain 87 than some natu
Page 105 and 106: These three elastic constants are r
Page 107 and 108: Stress and strain 91 Figure 9.1 New
Page 109 and 110: Stress and strain 93 Figure 9.2 Com
Page 111 and 112: Stress and strain 95 Figure 9.3 Ide
Page 113 and 114: Fracture Stress and strain 97 We fo
Page 115 and 116: Stress and strain 99 from which we
Page 117 and 118: 10 Sea waves The nature of water wa
Page 119 and 120: Sea waves 103 Figure 10.1 Wave moti
Page 121 and 122: Sea waves 105 refraction by islands
Page 123 and 124: Acoustics: sound and other waves 10
Page 125 and 126: Acoustics: sound and other waves 10
Page 127 and 128: 12 Fluids and fluid flow Introducti
Page 129 and 130: Fluids and fluid flow 1 13 Figure 1
Page 131 and 132: Fluids and fluid flow 115 Figure 22
Page 133 and 134: Fluids and fluid flow 117 and the f
Page 135 and 136: water above a vertical thickness h
Page 137 and 138: Fluids and fluid flow 121 where V i
Page 139: Fluids and fluid flow 123 ardentes,
Page 143 and 144: - 14 $ 12 E 10 K .- 3 8 r cll + 6 0
Page 145 and 146: Fluids and fluid flow 129 the liqui
Page 147 and 148: Some dangers of mathematical statis
Page 155 and 156: Appendix 139 What would happen if y
Page 157 and 158: Notes 141 and since x is very large
Page 159 and 160: References Allum, J. A. E. (1966) P
Page 161 and 162: References 145 -(1950) 'Mechanism o
Page 163: Further reading 147 Trigg, G. L. an
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Physics for Geologists, Second edition

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