Physics for Geologists, Second edition

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116 Fluids and fluid flow Fluid statics The first equation, Equation 1.1, p = pgz, states that pressure in a fluid is proportional to depth and to the mass density of the fluid (p). We must first satisfy ourselves that this is true independently of the shape of the vessel or container. The inability of fluids to resist shear stress leads to some fundamental propositions. (1) The free surface of a liquid in static equilibrium is horizontal This is a matter of common observation and practical utility in building and survey- ing, and has been for centuries. If the slope of the surface is not horizontal but has a slope 0, the weight ( W) of a small element of liquid at this surface can be resolved into a normal component, W cos 0, and a shear component, W sin 0. The shear stress must be zero in a liquid, so 0 must be zero. (2) Surfaces of equal pressure in static fluids are horizontal This really follows from the first. If the fluid is not homogeneous and p is not a function of the depth, shear stresses will exist that will cause flow until they are elim- inated, and p becomes a function of depth only. The shape of the container does not alter this. (3) The pressure at a point in static equilibrium is equal in all directions Consider a prism within the water that is so small that its weight is insignificant compared to the pressure around it (Figure 12.5), and consider side b to have unit area. The area of side a is then sin a; and of side c, cos a. The force (pressure x area) exerted by the liquid on side a is then pa sina, and on side c it is p, cos a. The force pb on unit side b can be resolved into horizontal and vertical components, pb sin ct and pb cos a, respectively. Since the prism is in static equilibrium, the sum of the forces acting on it must be zero. Hence, Copyright 2002 by Richard E. Chapman pa sin a - pb sina = 0; and pa = pb, p,cosa-pbcosa=O; andp, =pb, A Figure 12.5 The forces acting on a small prism of water.
Fluids and fluid flow 117 and the forces acting on the two parallel sides are clearly equal and opposite from the second proposition. Since pa = pb = p, is independent of the angle a, which may arbitrarily be assigned a value and orientation, the pressure at a point in a static liquid is equal in all directions. Note carefully that the pressure on a submerged object is not equal in all directions. (4) The horizontal force acting on a surface in a static fluid is the product of the pressure and the vertical projection of the area Consider the prism again. The force acting on side b, of unit area, is pb, the horizontal component of which is pb sin a. But sin a is the area of side a, which is the area of the projection of b onto a vertical surface. This proposition can also be shown to be true for curved surfaces by considering very small areas and their tangential slope. The propositions above apply equally to the water filling the pore space of particulate solids, such as sand. If we now fill the containers with sand, it is clear that the water pressure in the pore spaces between the sand grains will follow the same relationship (bearing in mind that the grains will displace water and so raise the free upper surface). What about the pendular water? The short answer is that there is no water that behaves as pendular water in a single-phase liquid. It is the interfacial tension that makes the pendular water immobile, and this does not exist in a single-phase liquid. What pressure does the solid-water mixture exert on the bottom of the container? and what about the solid component? What follows must be regarded as thought experiments because in small containers the pressure of particulate solids on the bottom does not increase once the thickness is greater than twice the diameter of the container. Clearly the weight of the water-saturated sand exerts a pressure on the bottom of the container, and that total force is made up of a force due to the water and a force due to the solids. The fractional porosity f is the ratio of pore-water volume to total volume. The bulk weight is the sum of the weights of the water and the solids, pbgv; and the bulk weight density is the bulk weight divided by the bulk volume, pbg (=yb). The total weight of the water-saturated sediment is made up of the weight of solids and the weight of the water in the pores the sum of which is Buoyancy reduces the effective weight of the solids by the weight of water displaced. Should we not take this into account? No. If we do that we find 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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62 Atomic structure and age-dating
Page 81 and 82: 64 Atomic structure and age-dating
Page 83 and 84: 66 Electromagnetic radiation radiat
Page 85 and 86: 68 Electromagnetic radiation Source
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: Fluids and fluid flow 115 Figure 22
Page 135 and 136: water above a vertical thickness h
Page 137 and 138: Fluids and fluid flow 121 where V i
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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