Physics for Geologists, Second edition

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120 Fluids and fluid flow The hydraulic radius is perhaps best understood with channel flow. Con- sider unit length of water in laminar flow down a gently inclined channel of width w, the depth of the water normal to the bottom being h. Its volume is wh, and its wetted surface is w + 2h. The forces acting on this volume of water are gravitational (the component of weight down the slope 8) and frictional (the resistance of the wetted surface mainly but also the viscosity of the water). With laminar flow, these are equal and opposite, so whpg sin 8 - tO(w + 2h) = O), (12.5) where t o is the boundary shear stress. So the ratio of boundary shear stress to the component of weight of unit volume down the slope is -- to - wh pg sin 0 (w + 2h) ' This is the volume divided by the wetted surface area - the hydraulic radius. The hydraulic radius is a measure of both size and shape. It also applies to rivers, where it is called the hydraulic depth because the right-hand side of Equation 12.6 approaches h as w becomes large compared to h. Solids settling in static fluid If you place a small pebble at the surface of some static water and let it fall, it will accelerate until the frictional resistance of the water equals the weight of the pebble. Its velocity will then be constant at what is known as its terminal velocity. Likewise, if you jump out of an aeroplane without a parachute, the speed at which you hit the ground will not be the same for all small heights, but once high enough to achieve your terminal velocity (about 200 km h-' if you spread your arms and legs, but nearer 240 km hkl if you go into a dive with your arms like swept-back wings), that will be the speed at which you hit the ground from any greater height (and it will be terminal!). The larger the solid object, the larger the sympathetic flow that will be generated by its passage through the water. The more solid objects there are, the larger will be the sympathetic flow. The shape also affects the speed at which the object falls. During the Second World War, a special bomb was designed to fall faster than the normal bomb so that it would penetrate deeper into the ground before exploding (it was called the earthquake bomb, and the idea was to shake down bridges and buildings: it worked). The other extreme is the parachute. Dimensional analysis can be used to find the form of the equation giving the terminal velocity of a single small solid sphere: Copyright 2002 by Richard E. Chapman
Fluids and fluid flow 121 where V is the terminal velocity of a small sphere of diameter d and mass density p, falling through static fluid of mass density p, and kinematic viscosity v. The dimensionless numbers a and b must be determined experimentally. When a = 1 and b = 18, we have Stokes' Law. Stokes' Law applies only to a single small sphere and then only when its terminal velocity is small enough for terms in v2 to be neglected. It should not be used for multiple grains settling in a fluid. The point here is that when v2 is not negligible, kinetic energy is not negligible and the relationship between V and p is no longer linear. It has been found that Equation 12.7 applies reasonably well to multiple grains, and grains large enough to give significant terms in v2 -that is, it can be used for sediment settling provided the coefficients a and b are determined for similar material. It was said above that resistance to a moving object is proportional to the square of its velocity, independent of shape. If an object is falling in air, it cannot 'know' whether it is falling, or being suspended by a rising wind. Winnowing In many parts of the world, wheat is separated from the chaff by beating the corn with a stick and then tossing the wheat and chaff in a breeze. The grains fall back into the basket: the chaff blows away. The reason for this is that the grains are aerodynamically better shaped than the chaff (like the parachutist and his parachute). Graded beds are commonly the result of turbidity currents and the settling of the larger grains from it faster than the smaller in deep water. They are commonly graded in three dimensions because as the velocity of the current decreases so the ability of the water to suspend sediment decreases. In any one position a bed may be graded from coarse to fine upwards; and in any one bed, the grades may become finer away from the source. In Chapter 2 we considered briefly the energy consequences of volcanic eruptions that throw ash and dust into the air and found that the velocity required to throw any mass to a height of 25 km was so improbable that volcanic dust must have been taken to those heights by thermal convection currents due to the volcano itself. The velocity at which the material falls to Earth can be seen from Equation 12.7 to be proportional to its size and to its density. In general, the larger, denser fragments will fall back to Earth sooner than the smaller, less dense. We must not expect Stokes' law to prevail because (a) some of the terminal velocities may be large enough for the kinetic energy term in v2 to be significant; and (b) there will be interactions between the mass of fragments falling at different speeds. When Krakatoa (NW end of Java) erupted in 1883, a cloud of fine dust spread around the Earth and affected sunrises and sunsets for several months - as happened after the eruption of Pinotubo in the Philippines in 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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Page 87 and 88: 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: water above a vertical thickness h
Page 139 and 140: Fluids and fluid flow 123 ardentes,
Page 141 and 142: Fluids and fluid flow 125 Figure 12
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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