Physical Processes in Earth and Environmental Sciences Phần 4

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Chương 3 Hiển thị các đối tượng bị biến dạng đồng nhất phẳng (Hình 3.83b), các cạnh của hình vuông vuông góc với nhau, nhưng nhận thấy rằng cả hai đường chéo của hình vuông (a và b trong hình 3,83.) Ban đầu ở 90 đã biến dạng kinh nghiệm bằng cách cắt căng thẳng di chuyển đến vị trí a và b trong các đối tượng bị biến dạng.

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Nội dung Text: Physical Processes in Earth and Environmental Sciences Phần 4

LEED-Ch-03.qxd 11/27/05 4:26 Page 88 88 Chapter 3 simple shear into a rhomboid, both the sides and the diag- (a) onals of the square have experienced shear strain. a b 90° 3.14.5 Pure shear and simple shear Pure shear and simple shear are examples of homogeneous strain where a distortion is produced while maintaining Angular shear γ the original area (2D) or volume (3D) of the object. Both (3) g = tan c types of strain give parallelograms from original cubes. g = tan c = tan –32° = –0.62 Pure shear or homogeneous ﬂattening is a distortion which a converts an original reference square object into a rectan- –32° gle when pressed from two opposite sides. The shortening (b) produced is compensated by a perpendicular lengthening c b a 90° (Fig. 3.84a; see also Figs 3.81 and 3.83b). Any line in the object orientated in the ﬂattening direction or normal to it does not suffer angular shear strain, whereas any pair of c perpendicular lines in the object inclined respect to these directions suffer shear strain (like the diagonals or the rec- (c) tangle in Fig. 3.83b or the two normal to each other radii in the circle in Fig. 3.84a). Simple shear is another kind of distortion that trans- 30° forms the initial shape of a square object into a rhomboid, so that all the displacement vectors are parallel to each other and also to two of the mutually parallel sides of the c rhomboid. All vectors will be pointing in one direction, g = tan 30° = 0.57 known as shear direction. All discrete surfaces which slide with respect to each other in the shear direction are named shear planes, as will happen in a deck of cards lying on a Fig. 3.83 Examples of measuring the angular shear in a square object table when the upper card is pushed with the hand (a) deformed into a rectangle by pure shear (b) and a rhomboid (c) (Fig. 3.84c). The two sides of the rhomboid normal to the by simple shear. displacement vectors will suffer a rotation deﬁning an angular shear and will also suffer extension, whereas the sides parallel to the shear planes will not rotate and will remain unaltered in length as the cards do when we dis- shows the object deformed by homogeneous ﬂattening place them parallel to the table. Note the difference with (Fig. 3.83b), the sides of the square remain perpendicular respect to the rectangle formed by pure shear whose sides to each other, but notice that both diagonals of the square do not suffer shear strain. Note also that any circle repre- (a and b in Fig. 3.83) initially at 90 have experienced sented inside the square is transformed into an ellipse in deformation by shear strain moving to the positions a and both simple and pure shear. To measure strain, fossils or b in the deformed objects. To determine the angular other objects of regular shape and size can be used. If the shear, the original perpendicular situation of both lines has original proportions and lengths of different parts in the to be reconstructed and then the angle can be measured. body of a particular species are known (Fig. 3.85a), it is In this case the line a has suffered a negative shear with possible to determine linear strain for the rocks in which respect to b. The line a perpendicular to b has been plot- they are contained. Figure 3.85 shows an example of ted and the angle between a and a deﬁnes the angular homogeneous deformation in trilobites (fossile arthropods) shear. The shear strain is calculated by the tangent of the deformed by simple shear (Fig. 3.85b) and pure shear angle . The same procedure can be followed to calculate (Fig. 3.85c). Note how two originally perpendicular lines the strain angle between both lines plotting a line normal in the specimen, in this case the cephalon (head) and the to a . Note that in this case the shear will be positive as the bilateral symmetry axis of the body, can be used to meas- angle between b and a is smaller than 90 . In the second ure the shear angle and to calculate shear strain. example (Fig. 3.83c) the square has been deformed by
LEED-Ch-03.qxd 11/27/05 4:27 Page 89 Forces and dynamics 89 Flattening direction (a) y (c) x (b) y y c x x Fig. 3.84 Pure shear (a) and simple shear (b) are two examples of homogeneous strain. Both consist of distortions (no area or volume changes are produced; (c) Simple shear has been classically compared to the shearing of a new card deck whose cards slide with respect to each other when pushed (or sheared) by hand in one direction. (a) (b) (c) ψ Fig. 3.85 Homogeneous deformation in fossil trilobites: (a) nondeformed specimen; (b) deformed by simple shear. Note how two originally perpendicular lines such as the cephalon base and the bilateral symmetry axis can be used to measure the shear angle and calculate shear strain (c) deformed by pure shear. If the original size and proportions of three species is known, linear strain can be established. 3.14.6 The strain ellipse and ellipsoid known as the principal axis of the strain ellipse and are mutually perpendicular. The strain ellipse records not only the directions of maximum and minimum stretch or exten- We have seen earlier (Figs 3.80 and 3.84) that when homo- sion but also the magnitudes and proportions of both geneous deformation occurs any circle is transformed into parameters in any direction. To understand the values of a perfectly regular ellipse. This ellipse describes the change the axis of the strain ellipse imagine the homogeneous in length for any direction in the object after strain; it is deformation of a circle having a radius of magnitude 1, called the strain ellipse. For instance, the major axis of the which will be the value of l0 (Fig. 3.86a). Now, if we apply ellipse, which is named S1 (or e1), is the direction of maxi- the simple equation of the stretch S (Equation 2; Fig. 3.81) mum lengthening and so the circle is mostly enlarged in whereas for a given direction, the stretch e is the difference this direction. Any other lines having different positions on in length between the radius of the ellipse and the initial the strained objects which are parallel to the major axis of undeformed circle of radius 1 it is easy to see that the major the ellipse suffer the maximum stretch or extension. axis of the ellipse will have the value of S1 and the minor Similarly the minor axis of the ellipse, which is known as S3 axis the value of S3. An important property of the strain (or e3) is the direction where the lines have been shortened axes is that they are mutually perpendicular lines which most, and so the values of the extension e and the stretch S were also perpendicular before strain. Thus the directions are minimum. The axis of the strain ellipse S1 and S3 are
LEED-Ch-03.qxd 11/27/05 4:27 Page 90 90 Chapter 3 of maximum and minimum extension or stretch corre- shear stress is produced). Shear strain can be determined in spond to directions that do not experience (at that point) the ellipse by two originally perpendicular lines, radii R of shear strain (note the analogy with the stress ellipse in the circle and the line tangent to a radius at the perimeter which the principal stress axis are directions in which no (Fig. 3.87). In (a), before deformation, the tangent line to the circle is perpendicular to the radius R. In (b), after deformation, the lines are no longer normal to each other and so an angular shear can be measured and the shear strain calculated, as explained earlier. Before (a) In strain analysis two different kinds of ellipse can be deformation deﬁned, (i) the instantaneous strain ellipse which deﬁnes the homogeneous strain state of an object in a small incre- r=1 ment of deformation and (ii) the ﬁnite strain ellipse which represents the ﬁnal deformation state or the sum of all the phases and increments of instantaneous deformations that the object has gone through. In 3D a regular ellipsoid will Pure shear S3 develop with three principal axes of the strain ellipsoid, (b) namely S1, S2, and S3, being S1 S2 S3 . Now that we have introduced the concept of the strain S1 ellipse we can return to the previous examples of homoge- neous deformation and have a look at the behavior of the strain axes. In the example of Fig. 3.84 the familiar square is depicted again showing an inner circle (Fig. 3.88). Two (c) mutually perpendicular radius of the circle have been Simple shear marked as decoration. Note that a pure shear strain has S3 been produced in four different steps. The circle has become an ellipse that, as the radius of the circle has a value of 1, will represent the strain ellipsoid, with two principal axes S1 and S3. Note that when a pure shear is produced the orientation of the principal strain axis S1 remains the same through all steps in deformation and so it is called coaxial strain (Fig. 3.88). This means that the directions of maximum and minimum extension are pre- Fig. 3.86 The stress ellipse in 2D strain analysis reﬂects the state of strain of an object and represents the homogeneous deformation of served with successive stages of ﬂattening. A very different a circle of radius 1 transformed into an ellipsoid. As I0 is 1, situation happens when simple shear occurs (Fig. 3.89): S1 I1/1 I1 which represents the stretch S of the long axis. the axes of the strain ellipsoid rotate in the shear direction Similarly S3 I1 giving the stretch S of the short axis. (a) (b) +c S3 R R‘ R=1 S1 Fig. 3.87 Shear strain in the strain ellipse. In (a), before deformation, the tangent line to the circle is perpendicular to the radius R. In (b), after deformation, both lines are not normal to each other, the angular shear can be obtained and the shear strain calculated by tracing a normal line to the tangent to the circle at the point where R’ intercepts the circle, and measuring the angle . The shear strain can be calculated as y tan .
LEED-Ch-03.qxd 11/27/05 4:27 Page 91 Forces and dynamics 91 1 r= S1 S1 S3 S3 S3 S1 S1 S3 Fig. 3.88 Pure shear is considered to be a coaxial strain since the orientation of the axes of the strain ellipse S1 and S3 remain with the same orientation through progressively more deformed situations. S3 S3 S1 S1 S1 S1 r=1 S3 S3 Fig. 3.89 Simple shear can be described as a noncoaxial strain as the orientation of the principal strain axis of the strain ellipse S1 and S3 rotates with progressive steps on deformation. and so the strain is noncoaxial. The orientation of the axes After deformation, the circle has suffered strain and devel- is not maintained, which means that the directions of max- oped into a perfect ellipse by homogeneous ﬂattening imum and minimum extension rotate progressively with (Fig. 3.90b). The original radius R of the circle, with time. length l0, has been elongated and will correspond to the radius R of the ellipse of length l1. Comparing both lengths, the extension, e (Equation 1; Fig. 3.81) or the stretch, S (Equation 2; Fig. 3.81), can be easily calculated. 3.14.7 The fundamental strain equations and The reciprocal quadratic elongation can be directly the Mohr circles for strain (l0/l1)2. The angular deformation can be obtained as measured by plotting the tangent to the ellipse at the point For any strained body the shear strain and the stretch can p, where the radius intercepts the ellipse perimeter, then be calculated for any line forming an angle with respect plotting the normal to the tangent, and measuring the to the principal strain axis S1 if the orientation and values angle with respect to the radius R (Fig. 3.90b). of S1 and S3 are known. As in the case of stress analysis The Mohr circle strain diagram is a useful tool to graph- the approach can be taken in 2D or 3D. Although it is ically represent and calculate strain parameters, following a important to remember that the physical meanings of similar procedure that was used to calculate stress compo- strain and stress are completely different, the equations nents. In this case the ratio between the shear strain and the have the same mathematical form (Fig. 3.90) and can be quadratic elongation ( / ) is represented on the vertical derived using a similar approach. The fundamental strain axis and the reciprocal quadratic elongation ( ) on the equations allow the calculation of changes in length of horizontal axis (Fig. 3.90c). The / ratio is an index of lines, deﬁned by means of the reciprocal quadratic elonga- the relative importance of the angular deformation versus tion( 1/ ), of any line forming an angle with the linear elongation. When the ratio is very small, changes respect to the direction of maximum stretch S1. To illus- in length dominate, in fact when the ratio equals zero, trate the use and signiﬁcance of the Mohr circles for strain, there is no shear strain, which coincides with the directions an original circle of radius R can be used (as in Fig. 3.87). of the principal strain axis. In homogeneous strain of pure
LEED-Ch-03.qxd 11/27/05 4:27 Page 92 92 Chapter 3 (a) (c) g/l The fundamental strain equations Considering the quadratic elongation l = (1 + e)2 = S2 (1) g/l l1 + l3 l1– l3 l= cos 2f + (2) 2 2 2f l To define the strain equations, the reverse of the quadratic elongation, or reciprocal quadratic elongation (l’) is used: l l1 l3 l = 1/l (3) l + l3 l –l l‘ = 1 – 3 1 cos 2f (4) 2 2 Equation 5 relates the ratio between the angular (d) strain g and the linear strain l, with the principal strain axis and the angle f. g/l l –l l 3–l 1 g /l = 3 1 sin 2f (5) l 3 – l 1 sin 2u 2 2 2 2f l (b) l +c l1 l3 l3 g = tan c l 3 – l 1 cos 2 p R f f 2 l1 l1+l3 2 Fig. 3.90 (a) The fundamental strain equations. The Mohr circles for strain display graphically the relations between the / ratio and the reciprocal quadratic elongation . The / ratio reﬂects the relative importance of angular deformation versus linear deformation; (b) strained circle into an ellipse; (c) the Mohr circle strain diagram; and (d) Relation between the Mohr circle and the fundamental strain equations. distortion, where there is no change in volume or area, value, at the right end of the circle, corresponds to 3 and there are two directions that suffer no ﬁnite stretch, where the minimum, at the left end, to 1. Once the circle is the value of 1. Finally two directions of maximum shear plotted, it is possible to calculate the values of / and strain are present, corresponding to the lines forming an (Fig. 3.90d) for any line forming an angle respect to the angle 45 with respect to S1. The Mohr circle for strain direction of the major principal strain axis S1. The line is has an obvious relation to the fundamental strain equations plotted from the center of the circle at the angle 2 sub- (Equations 4 and 5; Fig. 3.90a) as shown in Fig. 3.90d. tended from 1 into the upper half of the circle if the To plot the circle in the coordinate axes, the reciprocal angle is positive or into the lower half if it is negative. The values 1 and 3 of 1 and 3 are ﬁrst calculated and rep- coordinates of the point of intersection between the line and resented along the horizontal axis. The circle will have a the circle have the values / , . Through the value of diameter 3 1 and the center will have coordinates and then that of S can be calculated. Knowing it is (1 )/2, 0. Note that as the expressions on the x-axis also possible to calculate and ﬁnally the angle of shear 3 are the reciprocal quadratic elongations, the maximum strain . 3.15 Rheology 3.15.1 Rheological models experiments: the study of strain–stress relations or how the rocks or other materials respond to stress under certain con- The reaction of rock bodies and other materials to applied ditions is the concern of rheology. Different kinds of experi- stresses can only be observed and studied through laboratory ments are possible, generally undertaken on centimeter-scale
LEED-Ch-03.qxd 11/27/05 4:27 Page 93 Forces and dynamics 93 cylindrical rock samples. Both tensional (the sample is gen- 3.15.2 Elastic model erally pulled along the long axis) and compressive (sample is pushed down the long axis) stresses can be applied, both Elastic deformation is characterized by a linear relationship in laterally conﬁned (axial or triaxial tests) or unconﬁned in stress–strain space. This means that the relation between conditions (uniaxial texts). Experiments involving the the applied stress and the strain produced is proportional application of a constant load to a rock sample and observ- (Fig. 3.91a). An instantaneous applied stress is followed ing changes in strain with time are called creep tests. instantly by a certain level of strain. The larger the stress Experimental results are analyzed graphically by plotting the larger the strain, up to a point at which the rock can be stress, ( ), against strain, ( ), or strain rate (d /dt), the distorted no further and it breaks. This limit is called the latter obtained by dividing the strain by time (Fig. 3.91). elastic boundary and represents the maximum stress that Simple mathematical models can be developed for different the rock can suffer before fracturing. If the stress is regimes of rheological behavior. Stress is usually repre- released before reaching the elastic limit such that no frac- sented as the differential stress ( 1 3). Other important tures are produced, elastic deformation disappears. In variables are lithology, temperature, conﬁning pressure, other words, elastic strained bodies recover their original and the presence of ﬂuids in the interstitial pores shape when forces are no longer applied. The classical ana- causing pore ﬂuid pressures. There are three different pure log model is a spring (Fig. 3.92a). The spring at repose rheological behavioral regimes: elastic, plastic, and viscous represents the nondeformed elastic object. When a load is (Fig. 3.91). Elastic and plastic are characteristic of solids whereas viscous behavior is characteristic of ﬂuids. Solids under certain conditions, for example, under the effect of permanent stresses, can behave in a viscous way. Elastic, (a) plastic, and viscous are end members of a more complex suite of behaviors. Several combinations are possible, such as visco-elastic, elastic–plastic, and so on. Elastic Viscous (a) (b) (b) Stress (s) Stress (t) h= t E = s/e de/dt Strain rate de/dt Strain (e) Object is static (c) Plastic (c) s < sy vy ea H sy Stress (s) s > sy Object moves y av Strain (e) He Fig. 3.91 Strain/stress diagrams for different rheological behaviors. (a) Elastic solids show linear relations. The slope of the straight line is the Young’s modulus; (b) viscous behavior is characteristic of ﬂu- Fig. 3.92 Classical analogical models for (a) elastic behavior, is ids. Fluids deform continuously at a constant rate for a certain stress compared to a spring; (b) viscous behavior is compared to a value. The slope of the line is the viscosity ( ); (c) plastics will not hydraulic piston or dashpots; and (c) plastic behavior, like moving deform under a critical stress value or yield stress ( y). a load by a ﬂat surface with an initial resistance to slide.
LEED-Ch-03.qxd 11/27/05 4:27 Page 94 94 Chapter 3 added to the spring in one of the extremes (as a value for the Young’s modulus means that the level of strain dynamometer) or it is pulled by one of the edges, it will produced is small for the amount of stress applied, whereas stretch by the action of the applied force. The bigger the low values indicate higher deformation levels for a certain load, or the more the spring is pulled on the extremes, the amount of stress. Rigid solids produce high Young’s modu- longer it becomes by stretching. When the spring is lus values as they are very reluctant to change shape or vol- released or liberated from the load in one of the extremes ume. Rigid materials experience brittle deformation when the spring returns to the original length. their mechanical resistance is exceeded by the applied stress Elasticity in rocks is deﬁned by several parameters; the level at the elastic boundary. most commonly used being Young’s modulus (E) and When applying uniaxial compressional tests to rock the Poisson coefﬁcient ( ). Young’s modulus is a measure of samples, vertical shortening may be accompanied by some the resistance to elastic deformation which is reﬂected in the horizontal expansion. The Poisson coef ﬁcient ( ) shows linear relation between the stress ( ) and the strain ( ): E the relation between the lateral dilation or barreling of a / (Fig. 3.93a). This linear relation, which was observed rock sample and the longitudinal shortening produced by initially by Hooke in the mid-seventeenth century by apply- loading: thus lateral/ longitudinal and it can be seen that ing tensile stresses to a rod and measuring the extension, is Poisson’s coef ﬁcient is dimensionless (Fig. 3.93b). When commonly known as Hooke’s Law. Considering that all stresses are applied, if there is no volume loss, the sample parameters used to measure strain (stretch, extension, or has to thicken sideways to account for the vertical shorten- quadratic elongation) are dimensionless, the Young’s modu- ing. Typically, the sample should develop a barrel form lus is measured in stress units (N m 2, MPa) and has negative (nonhomogeneous deformation) or increase its surface values of the order of 104 or 105. The reason why the area as it expands laterally. For perfect, incompressible, values are negative is because the applied stress is extensional isotropic, and homogeneous materials which compensate and hence has a negative value, and the strain produced is a the shortening by lateral dilation without volume loss, the lengthening, which is conventionally considered positive. Poisson’s coef ﬁcient is 0.5; although values for natural Not all rocks follow Hooke’s Law; some deviations occur materials are generally smaller (Fig. 3.93b). In very rigid but they are small enough so a characteristic value of E can rock bodies, the lateral expansion may be very limited be deﬁned for most rock types (Fig. 3.93a). A high absolute or not occur at all; in this case there is a volume loss and (a) (b) Uniaxial compression a Final state Original E ( 104 MPa) Rock type ec c s1 − s3 (MPa) b Marble –4.8 Limestone –5.3 ed Granite –5.6 d Shale –6.8 Ea > Eb Quartzite –7.9 Diorite –8.4 n = ed/ec e (%) E = s/e (Hooke’s Law) n Rock type Schist, biotite 0.01 Shale, calcareous 0.02 Diorite 0.05 Granite 0.11 Aplite 0.20 Siltstone 0.25 Dolerite 0.28 Fig. 3.93 Elastic parameters. (a) The Young’s modulus describes the slope of the stress/strain straight line, being a measure of the rock resistance to elastic deformation. Line a has a higher value of Young’s modulus (Ea) being more rigid than line b (Young modulus Eb) (i.e. it is less strained for the same stress values); (b) Poisson’s coef ﬁcient relates the proportion in which the rock deforms laterally when it is compressed vertically. Comparing the original and ﬁnal lengths before and after deformation strain can be calculated and the Poisson’s ratio established.
LEED-Ch-03.qxd 11/28/05 10:01 Page 95 Forces and dynamics 95 t Longitudinal strain g = tan c s c G = t/g mE Lateral G= 2m + 1 strain E=s/e Fig. 3.95 Shear or rigidity modulus (G) and its relation to Young’s modulus (E) and Poisson’s number (m). Dilation Shortening e Fig. 3.94 Longitudinal and lateral strain experienced by a rock (a) Solid elastic body sample when an uniaxial compression is applied. The relation between both strains may not be linear as in this case, and the Poisson’s ratio is not constant, it varies slightly for different stress values. (b) Fluid viscous body elastic stresses have to be accumulated somehow. Rock samples will fragment at the elastic limit after experiencing very little lateral strain when the Poisson’s ratio is very small (close to zero). The reciprocal to the Poisson’s coef- ﬁcient is called the Poisson’s number m 1/ . This num- ber is also constant for any material, and so the relation Fig. 3.96 (a) Solid elastic bodies are strained proportionally to the applied forces. If the intensity of the force is maintained there is not a between the longitudinal and lateral strains have a linear further increase in strain. When the force is released the object relation. Nonetheless, as in the case of Young’s modulus recovers the initial shape; (b) The viscous ﬂuid will be deformed there may be slight variations in the linear trend of when a shear force is exerted, but even when the intensity of the Poisson’s coefﬁcient (Fig. 3.94). It is important to remem- force is maintained, an increment in deformation will occur, deﬁning ber that experiments to establish elasticity relationships a strain rate. That is why strain rate is used in rheological plots instead of strain as in solids. The ﬂuid body will remain deformed under unconﬁned uniaxial stress conditions allow the rock permanently once the force is removed. samples to expand laterally. In the crust, any cube of rock that we can deﬁne is not only subject to a vertical load due to gravity but also due to adjacent cubes of rock in every stresses, even inﬁnitesimal, are applied. One of the chief direction and is not free to expand laterally; in such cases differences between an elastic solid and a viscous ﬂuid is complex stress/strain relations can develop. that when a shear stress is applied to a piece of elastic mate- Other elastic parameters are the rigidity modulus (G) rial it causes an increment of strain proportional to the and the bulk modulus (K). The rigidity modulus or shear stress, if the same level of stress is maintained no further modulus is the ratio between the shear stress ( ) and the deformation is achieved (Fig. 3.96a). In ﬂuids when a shear strain ( ) in a cube of isotropic material subjected to shear stress ( ) is applied the material suffers certain simple shear: G / (Fig. 3.95). G is another measure of amount of strain but the ﬂuid keeps deforming with time the resistance to deformation by shear stress, in a way even when the stress is maintained with the same value equivalent to the viscosity in ﬂuids. The bulk modulus (K) (Fig. 3.96b). In this case a level of stress gives way to a relates the change in hydrostatic pressure (P) in a block of strain rate (d /dt), not a simple increment of strain as in isotropic material and the change in volume (V) that it the elastic solids. Higher stress values will give way to experiences consequently: K dP/dV. The reverse to the higher strain rates, so the ﬂuid will deform at more speed. bulk modulus is the compressibility (1/K). As in elastic materials there is no initial resistance to deformation even when stresses acting are very small, but the deformations are permanent in the viscous ﬂuid case 3.15.3 Viscous model (Fig. 3.97a,b). As we have seen earlier (Sections 3.9 and 3.10) the parame- Viscous deformation occurs in ﬂuids (Sections 3.9 and 3.10); ter relating stress to strain rate is the coefﬁcient of dynamic vis- fluids have no shear strength and will ﬂow when shear cosity or simply viscosity ( ): /(d /dt), which is
LEED-Ch-03.qxd 11/27/05 4:29 Page 96 96 Chapter 3 Increassing stress applied Stress released Newtonian Final state (h constant) (a) Elastic Stress (t) t h= de/dt (b) Viscous non-Newtonian (h variable) Strain rate de/dt (c) Plastic h (Pa) Fluid 0.8 10–3 Water (30º) Oil 0.08 102 Basalt lava 108 Rhyolite lava s < sy s > sy 1016 Salt 1022 Asthenosphere T1 T4 T5 T2 T3 TIME Fig. 3.98 Viscosity is the resistance of a ﬂuid to deform or ﬂow: it is the slope of the curve stress/strain rate. Fluids showing linear Fig. 3.97 Strain of different materials with time (stages T 1 to T 5) relations (constant viscosity) are Newtonian. Fluids with nonlinear applying increasing levels of stress: (a) Elastic solids show discrete relation ( variable) are non-Newtonian. The table shows the values strain increments with increasing stress levels (linear relation); of viscosity ( ) for some viscous materials. strain is reversible once the stress is removed (T 5); (b) Viscous ﬂuids ﬂow faster (higher strain rates) with increasing stress; the deformation is permanent once the stress is released; (c) Plastic level of stress is required to start deformation, as the mate- solids will not deform until a critical threshold or yield stress is overpassed (at T4 in this case). Deformation is nonreversible rial has an initial resistance to deformation. This stress (at T 5). value is called yield stress y (Fig. 3.91c). After the yield stress is reached the body of material will be deformed a measured in Pascals. Fluids that show a linear relation between big deal instantaneously, and the deformation will be per- the stress and the strain rate, and so have a constant viscosity, manent and without a loss of internal coherence. So, two are called Newtonian. Fluids, whose viscosity changes with the important differences with respect to elastic behavior are level of stress are called non-Newtonian (Fig. 3.98). Viscous that the strain is not directly proportional to the stress, as behavior is generally compared to a piston or a dashpot con- there is an initial resistance, and that the strain is not taining some hydraulic ﬂuid (Fig. 3.92b). The ﬂuid is pressed reversible as in elastic behavior (Fig. 3.97). An analogical by the piston (creating a stress or loading) and the ﬂuid moves model for plastic deformation is that of a heavy load rest- up and down a cylinder, producing permanent deformation; ing on the ﬂoor (Fig. 3.92c). If the force used to slide the the quicker the piston moves the more rapid the ﬂuid deforms load along a surface is not big enough, the load will not or ﬂows up and down. The viscosity can be described as the budge. This would depend on the frictional resistance resistance of the ﬂuid to movement. High viscosity ﬂuids are exerted by the surface. Once the frictional resistance, and more difﬁcult to displace by the piston up and down the cylin- so the yield stress, is exceeded, the load will slide easily and der. For non-Newtonian ﬂuids (Fig. 3.98) as the piston is the movement can be maintained indeﬁnitely as long as pushed more and more strongly in equal increments of added the force is sustained at the same level over the critical stress the rate of movement or strain rate rapidly increases in a threshold or yield stress. The load will not go back on its non-linear fashion. own! So the deformation is not reversible (Fig. 3.92c). 3.15.4 Plastic model 3.15.5 Combined rheological models Plastic deformation is characteristic of materials which do Elastic, viscous, and plastic models correspond to simple not deform immediately when a stress is applied. A certain mathematical relationships which apply to materials under
LEED-Ch-03.qxd 11/27/05 4:30 Page 97 Forces and dynamics 97 ideal conditions; they are considered homogeneous (the an initial elastic ﬁeld of behavior where the strain is recov- rock has the same composition in all its volume) and erable, but once a yield stress ( y) value is reached the isotropic (the rock has the same physical properties in all material behaves in a plastic way. The analogical model is a directions). Rocks are rarely completely homogeneous or spring (elastic) attached to a heavy load (plastic) moving isotropic due to their granular/crystalline nature and over a rough surface (Fig. 3.99b). The spring will deform because of the presence of defects and irregularities in the instantly whereas the load remains in place until the yield crystalline structure, as well as layers, foliations, fractures, stress is reached, then the load will move; after releasing and so on. Nevertheless, although such aberrations would the force, the spring will recover the original shape but the be important in small samples, on a large scale, when large longitudinal translation is not recoverable. Elastic–plastic volumes are being considered, rocks can be sometimes materials thus recover part of the strain (initial elastic) but regarded as homogeneous. Usually, however, natural rhe- partly remain under permanent strain (plastic). Remember ological behavior corresponds to a combination of two or that in a pure elastic material, permanent strain does not even three different simple models, such as elastic–plastic, occur and after the elastic limit is reached the rock breaks visco-elastic, visco-plastic, or elastic–visco-plastic. Also (b, Fig. 3.99c; line I) whereas in a Prandtl material there is materials can respond to stress differently depending on a nonreversible strain (c, Fig. 3.99c, line II). Once the plastic the time of application (as in instantaneous loads versus limit is reached, the material can then break but only after long-term loads). suffering some permanent barreling (d, Fig. 3.99c, line II). A well-known example of a combined rheological model Visco-elastic models correspond to solids (called is the elastic–plastic (Prandtl material) (Fig. 3.99); it shows Maxwell materials) which have no initial resistance to (a) Elastic – Plastic (c) Prandtl material b elastic plastic field limit sy b. elastic limit Stress (s1–s3) MPa stress (s) c I elastic field d. plastic limit a II c strain (e) d sy (b) elastic limit a F a time strain (ε) Fig. 3.99 (a) Elastic–plastic material shows an initial elastic ﬁeld characterized by recoverable deformation strain followed by a plastic ﬁeld in which the strain is permanent. The boundary between both ﬁelds is the elastic limit located at the yield stress value ( y); (b) The analog model is a load attached to a spring; (c) Part of the strain is recovered (the length of the spring) and part is not (the displacement of the load).
LEED-Ch-03.qxd 11/27/05 4:30 Page 98 98 Chapter 3 strain as in both elastic and viscous models (Fig. 3.100a). a surface with an initial resistance to movement; once the Part of the strain will recover following an elastic behavior load is in motion it behaves in viscous fashion. but part will remain permanently deformed. Maxwell solids behave elastically when the stresses are short lived, 3.15.6 Ductile and brittle deformation like a ball of silicon putty that bounces elastically on the ﬂoor when thrown with some force; but will accumulate permanent deformations at a constant rate if the stress or From the different rheological models discussed above it load (like the proper weight of the material) is applied for can be concluded that there are several kinds of deforma- a longer time. Visco-elastic models can be represented by tion. First, strain produced when loads are applied can be a spring attached longitudinally to a dashpot (Fig. 3.100b). reversible; this is characteristic of elastic behavior as in the The spring will provide the recoverable strain whereas the elastic curves or elastic–plastic materials (a, Fig. 3.99c) dashpot will supply the nonrecoverable strain when a when small stress increments are applied. Deformations can pulling force is applied parallel to the system. also be nonreversible, which means that once the load is Visco-plastic materials (called Bingham plastics) only released the rock will be deformed permanently. behave like viscous ﬂuids after reaching a yield stress, the Deformation is said to be ductile when rocks or other solids strain rate subsequently being proportional to the stress; are strained permanently without fracturing, which hap- initially the material does not respond to the applied stress pens in plastic or elastic–plastic materials once the elastic as for plastic solids (Fig. 3.100c). The analogy will be in limit or yield strength (stress value which separates the elas- this case a dashpot attached in parallel to a load sliding on tic and plastic ﬁelds) is reached (as c in Fig. 3.99c). (a) (b) Maxwell material Stress (s) F Strain rate d /dt (c) (d) Bingham material Stress (s) F sy Strain rate de/dt Fig. 3.100 (a) Visco-elastic or Maxwell materials have a recoverable strain part belonging to the elastic component and a permanent strain due to the viscous behavior like a spring attached to a dashpot (b); (c) visco-plastic or Bingham materials behave in a viscous way but after reaching a critical stress value or yield stress ( y) like a dashpot linked to a load moving on a rough surface (d).
LEED-Ch-03.qxd 11/27/05 4:30 Page 99 Forces and dynamics 99 Nonetheless, ductile is a general, descriptive term that does 2.0 not involve a speciﬁc rheological behavior or strain mecha- nism. It is not a synonymous term for plastic, which is a very 25°C well-deﬁned and particular rheological behavior. Strains pro- duced during plastic deformations are larger in magnitude 1.5 than those produced in the elastic ﬁeld and are generally 300°C Differential stress (×103 MPa) formed by dislocations of the crystalline lattices and/or dif- fusive processes. Ductile deformations are also called ductile ﬂows as the material deforms or ﬂows in a solid state (as a gla- 1.0 cier sliding downslope does, Section 6.7.5). Examples of 500°C ductile deformation in rocks are the formation of folds and salt diapirs. Rocks have a limited ability to change their shape 700°C or volume, which also depends on such external parameters 0.5 as the temperature, conﬁning pressure, and so on. Brittle deformation happens when the internal strength 800°C of rocks is exceeded by stresses; they bust, so internal cohesion is lost in well-deﬁned surfaces or fractures. Brittle 0 deformation can occur after the elastic limit is exceeded 0 5 10 15 not only in pure elastic bodies (b, Fig. 3.99c) but also Strain, e (%) when the stresses reach the plastic limit after some ductile Fig. 3.101 Effect of temperature in the strain–stress diagram for deformation has taken place. Such samples will be perma- basalts under the same conﬁning pressure (5 kbars). nently deformed and also fractured (d, Fig. 3.99c). plastic ﬁeld can develop. In elastic–plastic materials, tem- 3.15.7 Parameters controlling rock deformation perature lowers the elastic limit, which is thus reached at lower stress levels. Rocks may also behave in a viscous way Lithology (rock type) is a variable which may cause diverse at high temperatures if the applied stresses are long lasting. modes of stress–strain behavior. Different rocks or sub- Conﬁning pressure (lithostatic or hydrostatic pressure stances may need different rheological models with which acting on all sides of a rock volume) can be simulated in to describe their deformation. Competency is a qualitative laboratory experiments by introducing some ﬂuid that term used to describe rocks in terms of their inner strength exerts a certain amount of pressure in the sample (triaxial or capacity for deformation. Rocks which deform easily tests) in addition to that provided by the compressive load, and generally in a ductile way are described as incompetent, and by isolating the sample in a constraining metal jacket such as salts, shale, mudstone, or marble. Strong or compe- to discriminate and separate the effects of the pore pres- tent rocks are those which are more dif ﬁcult to deform, sure in the rock. Experiments carried out on samples of the such as quartzite, granite, quartz sandstones, or fresh same lithology and at the same temperature show that basalts. Competent rocks are stiffer and deform generally higher conﬁning pressures increase the yield strength in a in a brittle way. Nevertheless, competency depends not rock, and also the plastic ﬁeld, so fracturing, if it happens, only on lithology but also on temperature, conﬁning pres- occurs after more intense straining (Fig. 3.102). This sure, pore pressure, strain rate, time of application of the means that rocks became more ductile at higher levels of stress, etc. To compare competencies of different kinds of conﬁning pressure. rocks, experiments must take place at equal temperatures When there is ﬂuid trapped in the rock pores, it exerts an and conﬁning pressures. additional hydrostatic pressure which has the effect of Temperature has particularly important effects in rheo- counteracting the conﬁning pressure by the same value of logical behavior (Fig. 3.101). Comparing several experi- the ﬂuid pressure in the pores. The state of stress is lowered ments on samples of the same lithology under the same and an effective stress tensor can be deﬁned by subtracting conditions of conﬁning pressure, it is possible to compare the values of the ﬂuid stresses from those of the solid stress–strain relations at different temperatures. At higher normal stresses (Fig. 3.103). The Mohr circle moves temperatures, rocks behave in a more ductile way, so com- toward lower values by an amount equal to the pore pressure petence is reduced and fractures are more dif ﬁcult to pro- (pf) sustained by the ﬂuid. Thus, when ﬂuids are present in duce. For rocks that are elastic at low temperatures a the pores the effect is the same as lowering the conﬁning
LEED-Ch-03.qxd 11/27/05 4:30 Page 100 100 Chapter 3 (a) (b) 1 2 140 130 Differential stress (MPa) 80 300 70 60 200 40 3 4 20 100 Limestone 0 0 2 4 6 8 10 12 14 16 Strain, e (%) Fig. 3.102 (a) Strain–stress diagram showing several curves corresponding to limestone samples of the same composition at different conﬁning pressures (in MPa); (b) Differences in conﬁning pressure give way to different fracturing or deformation modes. Conﬁning pressure from samples (from 0.1 to 35 MPa in the fractured samples and 100 MPa for the ductile ﬂow). t Effective stress Applied stress sn E s3 s1 E s1 s2 s3 E s2 0 Pore fluid pressure (Pf) sxx txy txz spf 00 sxx–spf txy txz Es = tyx syy tyx − 0 spf 0 tyx syy-spf τyx Es = tzx tzy szz 0 0 σpf tzx tzy szz–spf Applied stress Hydrostatic stress (rock) (fluid) Fig. 3.103 When there is some pressurized ﬂuid in the rock pores, part of the stress is absorbed. The state of stress is lowered and an effective stress tensor can be deﬁned subtracting the values of the normal stresses from those of the ﬂuid. The Mohr circle moves toward lower values by an amount equal to the pore pressure (Pf) sustained by the ﬂuid. pressure in the rocks, so that ductility decreases and frac- strength decreases when the stresses are applied for long tures are produced more easily. Being hydrostatic in nature, times under small differential stresses (creep experiments). the effectiveness of the normal stresses is lowered but the Also in relation to time, the rates of loading (velocity of shear stresses remain unaltered. The control of pore pres- increased loading in the experiments) also have important sure in the rocks is of key importance in fracture formation implications for the production of strain. In a single exper- and will be discussed in some more detail in Section 4.14. iment, the rate of strain is generally maintained constant Other important factors are the time of application of the but the rates of strain can be changed from one experiment stresses: the instantaneous or long-term application of a to another. When changes in strain are produced rapidly certain level of stress may cause different rheological behav- (high loading rates) the rock samples become ductile and iors, like the case of the silicon putty discussed earlier. Rock break at higher stress levels.
LEED-Ch-03.qxd 11/27/05 4:30 Page 101 Forces and dynamics 101 Further reading P.M. Fishbane et al.’s Physics for Scientists and Engineers: has a broad appeal at intermediate level and is very thor- Extended Version (Prentice-Hall, 1993) is again invaluable. ough. The best introduction to solid stress and strain is in Many good things of oceanographic interest can be found G.H. Davies and S.J. Reynolds’s Structural Geology of Rocks in the exceptionally clear work of S. Pond and G.L. Pickard and Regions (Wiley, 1996); R.J. Twise and E.M. Moores’s – Introductory Dynamical Oceanography (Pergamon, Structural Geology (1992) and J.G. Ramsay and M. Huber’s 1983), while R. McIlveen’s Fundamentals of Weather and The Techniques of Modern Structural Geology, vol. 1: Strain Climate (Stanley Thornes, 1998) is good on the atmos- Analysis (Academic Press, 1993) are classics on structural pheric side. A more advanced text is D.J. Furbish’s Fluid geology for advanced studies on solid stress. W.D. Means’s Physics in Geology (Oxford, 1997). G.V. Middeton and P.R. Stress and Strain (Springer-Verlag, 1976) takes a careful and Wilcox’s Mechanics in the Earth and Environmental Sciences rigorous course through the basics of the subject.
LEED-Ch-04.qxd 11/26/05 13:11 Page 102 Calculated for: rwater = 1,000 kg m–3 rrock = 2,380 kg m–3 4 Flow, deformation, Geostatic gradient Hydrostatic and transport gradient 4.1 The origin of large-scale ﬂuid ﬂow Earth is a busy planet: what are the origins of all this with the ambient medium? If so, at what rate? How does motion? Generally, we know the answer from Newton’s the interaction look physically? First Law that objects will move uniformly or remain sta- 4 What is the origin and role of variation in ﬂow velocity tionary unless some external force is applied. The uniform with time (unsteadiness problem)? It is to be expected that motion of ﬂuids must therefore involve a balance of forces in accelerations will be very much greater in the atmosphere whatever ﬂuid we are dealing with. In order to try to predict than in the oceans and of negligible account in the solid the magnitude of the motion we must solve the equations of earth (discounting volcanic eruptions and earthquakes). motion that we discussed previously (Section 3.12). Bulk Why is this? ﬂow (in the continuum sense, ignoring random molecular movement) involves motion of discrete ﬂuid masses from 4.1.2 Horizontal pressure gradients and ﬂow place to place; the masses must therefore transport energy: mechanical energy as ﬂuid momentum and thermal energy as ﬂuid heat. There will also be energy transfers between the Static pressure at a point in a ﬂuid is equal in all directions two processes, via the principle of the mechanical equivalent (Section 3.5) and equals the local pressure due to the of heat energy and the First Law of Thermodynamics weight of ﬂuid above. Notwithstanding the universal truth (Section 2.2, conservation of energy). For the moment we of Pascal’s law, we saw in Section 3.5.3 that horizontal shall ignore the transport of heat energy (see gradients in ﬂuid pressure occur in both water and air. Sections 4.18–4.20) since radiation and conduction intro- These cause ﬂow at all scales when a suitable gradient duce the very molecular-scale motions that we wish to exists. The simplest case to consider is ﬂow from a ﬂuid ignore for initial simplicity and generality of approach. reservoir from oriﬁces at different levels (Fig. 4.1). Here the ﬂow occurs across the increasingly large pressure gra- dient with depth between hydrostatic reservoir pressure 4.1.1 Very general questions and the adjacent atmosphere. The gradient of pressure in moving water (Fig. 4.1) is 1 How does ﬂuid ﬂow originate on, above, and within the termed the hydraulic gradient, and the ﬂow of subsurface Earth? For example, atmospheric winds and ocean currents water leads to the principle of artesian ﬂow and the basis of originate somewhere and ﬂow from place to place for certain our understanding of groundwater ﬂow through the oper- reasons. This raises the question of “start-up,” or the begin- ation of Darcy’s law (developed from the Bernoulli nings of action and reaction. approach in Sections 4.13 and 6.7). The ﬂow of a liquid 2 If ﬂuid ﬂow occurs from place A to place B, what hap- down a sloping surface channel is also down the hydraulic pens to the ﬂuid that was previously at place A? For exam- gradient. ple, the arrival of an air mass must displace the air mass Similar principles inform our understanding of the slow previously present. This introduces the concept of an ﬂow of water through the upper part of the Earth’s crust. ambient medium within which all ﬂows must occur. Here, pressures may also be hydrostatic, despite the ﬂuid 3 How does moving ﬂuid interact with stationary or mov- held in rock being present in void space between solid rock ing ambient ﬂuid? For example, does the ﬂow mix at all particles and crystals (Fig. 4.2); this occurs when the rocks
LEED-Ch-04.qxd 11/26/05 13:15 Page 103 Flow, deformation, and transport 103 are porous to the extent that all adjacent pores communi- Interlayering of porous and nonporous rock then leads to cate, as is commonly the case in sands or gravels. Severe high local pressure gradients down which subsurface ﬂuids lateral and vertical gradients arise when pores are closed by may move. In passage down an oil or gas exploration well, compaction, as in clayey rock; the hydrostatic condition pressure may jump quickly from a hydrostatic trend toward now changes to the geostatic condition when pore pres- p0 = Atmospheric p0 = Atmospheric sures are greater due to the increased weight of overlying rock compared to a column of pore water (Fig. 4.3). In the hydrostatic condition Escape from a reservoir at a rate p0 = Atmospheric all liquid levels are equal determined by the local difference There is no change to this h0 in pressure between hydrostatic and principle when the fluid atmospheric occupies void space that h1 has continuous connection u1 to the surface h2 u2 Fig. 4.2 The hydrostatic condition is equally valid for liquid in reser- voirs or porous rock. Exit velocity = u = √ 2gh h3 Pressure, kg m–2 u3 100 . 10 5 500 . 10 5 During flow from a reservoir along a Calculated for: Flow in pipe or channel there is energy loss rwater = 1,000 kg m–3 downstream due to friction (drag) rrock = 2,380 kg m–3 1 Depth km Slope of line gives Geostatic hydraulic gradient gradient 2 Hydrostatic gradient Flow out 3 Fig. 4.1 Flows induced by hydrostatic pressure. Fig. 4.3 Hydrostatic and geostatic pressure gradients in the Earth’s crust. Low High atmospheric atmospheric pressure Strong wind causing wind shear pressure and water “set-up” on lee-shore b High Low Subsurface flow down horizontal Sloping isobars hydrostatic pressure gradient (modified by Coriolis force in 3D) B A Fig. 4.4 Barotropic ﬂow due to a horizontal gradient in hydrostatic pressure caused and maintained by atmospheric dynamics. The spatial gradients in atmospheric pressure and wind shear may act together or separately. In both cases hydrostatic pressures above B are greater than hydrostatic pressures at all equivalent heights above A, by a constant gradient given by the water surface slope.
LEED-Ch-04.qxd 11/26/05 13:16 Page 104 104 Chapter 4 lithostatic, causing potentially disastrous consequences for partially molten rocks of the Earth’s upper crust in crustal the drill rig and possible “blowout.” The regional magma chambers (Section 5.1) may also vary between hydraulic gradient drives the direction of migration of hydrostatic and geostatic values, with obvious implications subsurface ﬂuids like water and hydrocarbon. Pressures in for the forces occurring during volcanic eruptions. Neutral stability for all cases Ambient fluid lockbox r1 fluid r2 Sl op open in Mixing by molecular diffusion only Conditions r1 = r2 g bo un da ry Lower horizontal boundary Ocean and lakes (unstratified) Water surface lockbox Descending plume (open ocean Ambient liquid liquid ρ2 r1 cold convection, ice meltout) open Wall jet Conditions r1 < r2 Wall jet descending flow ∆ r = +ve (turbidity flow, (bottom water production, thermohaline flow) turbidity currents) Water surface lockbox Surface jet Ambient liquid liquid r2 r1 open Rising plume Conditions r1 > r2 ∆ r = –ve Ocean ridge hydrothermal plumes Atmosphere (unstratified) lockbox Ambient gas liquid r2 r1 open Thunderstorm Conditions r1 < r2 downdraught ∆ r = +ve Catabatic wind Sea breeze front Cold front lockbox Ambient gas liquid r2 r1 open Conditions r1 > r2 Rising plume (thermal) ∆ r = –ve Fig. 4.5 Buoyancy-driven ﬂows.
LEED-Ch-04.qxd 11/26/05 13:16 Page 105 Flow, deformation, and transport 105 Nomenclature and possible types of density currents. Table 4.1 Gas ve Gas Gas ve Liquid ve (e.g. Liquid Liquid ve (e.g. cooler air) neutral (e.g. warmer air) Cooler/more saline/ neutral ( Warmer/ suspensions of solids) less saline) Ambient gas Sinking plume Neutral stability Rising plume River ﬂow downslope NA NA Bottom-spreading and No ﬂow Interface-spreading undercutting current jet Ambient liquid NA NA Degassing bubbles in Sinking plume Neutral Rising plume magma or lava Bottom-spreading stability Spreading jet and undercutting wall jet 4.1.3 Flow in the atmosphere and oceans Coriolis force (not considered in Fig. 27.4), are quite sufﬁcient to drive the entire average surface oceanic circu- lation (discussed in Sections 6.2 and 6.4). The atmosphere and oceans are in a constant state of ﬂux, both experiencing “weather”; that is, the velocity of the ocean waters and atmosphere is unsteady with respect to 4.1.4 Buoyancy/density ﬂow either magnitude or direction over timescales of minutes to months. Here we brieﬂy note that their longer-term average ﬂow approximates to the geostrophic condition Many ﬂows that take place in, on, and above the solid (see also Section 3.12). This is when pressure gradients are Earth occur because density contrasts, , give rise to balanced by the Coriolis force alone, with no other forces buoyancy forces (Section 3.6). The resulting ﬂows are involved: the ﬂuid is assumed ideal, that is, inviscid. termed density or gravity currents. These may act between In terms of the relevant equations of motion, we have different parts of the same general state of matter (e.g. air, F (pressure) F (Coriolis). water, magma) or between different states of matter (e.g. In the atmosphere, the pressure variations that cause water in and under air, gases in magma). We may illustrate geostrophic ﬂow are up to 6 percent and are caused by lat- the various possibilities for water and air by means of eral variations in air density between regional pressure cells thought experiments with gravity lockboxes (Fig. 4.5). like the Iceland Low or the Azores High in the northern The lockbox is of unit volume with any side that can be hemisphere (see Fig. 3.21). Water density also varies with opened instantaneously so that the contained ﬂuid, air, or depth in the oceans but in the well-mixed surface layers of water, may be smoothly introduced within ambient masses the open oceans this density variation is not so important. of similar or different ﬂuid. In all cases the gas phase has a Regional ocean pressure gradients are set up due to varia- lower density than the liquid phase. For simplicity, we tions in the elevation of the mean sea surface (Fig. 4.4), examine the gravity lock in two dimension only, opening ignoring short-term topography due to storms, waves, and the locks in the top, bottom, or side as appropriate. The tides. The slopes involved are very small, up to 3 m over sketches show the expected ﬂow direction as each box is distances of a thousand kilometers or so, that is, gradients opened; the types of ﬂows possible are summarized in of c.3 · 10 6. These tiny gradients, in conjunction with the Table 4.1. 4.2 Fluid ﬂow types There is something immensely satisfying in discovering the (by Pouisseille, simulating the ﬂow of blood in veins and efforts of pioneering scientists to reduce apparently compli- arteries), that ﬂuid ﬂow could exhibit two basic kinds of cated natural phenomena to simple essentials governed by behavior while in motion and that two ﬂow “laws” must some overall guiding principle. One such contribution that exist to explain the forces involved. In Reynolds’ elegant stands out in the area of ﬂuid ﬂow was that by Reynolds. words, “either the elements of the ﬂuid follow one another Before Reynolds’ contribution was published in 1883, it along lines of motion which lead in the most direct manner was generally recognized from observations in natural to their destination, or they eddy about in sinuous paths rivers, from experiments on ﬂow in pipes (by Darcy), the most indirect possible.” In a series of careful experi- and from work on capillary ﬂow in very narrow tubes ments (Fig. 4.6), Reynolds visualized these ﬂow types by
LEED-Ch-04.qxd 11/26/05 13:20 Page 106 106 Chapter 4 Siphon for introducing dye streak Float and scale for measuring discharge and velocity Glass-sided tank b – Bell-shaped entrance b containing section to glass tube glass test tube ensures smooth intake immersed in of water to minimize water inlet disturbance Lever used to open outlet valve and allow variable throughflow of water Fig. 4.6 Reynolds’ apparatus as presented in his 1883 paper. Injected dye- Flow streak Laminar flow – dye streak passes downflow undeformed. You must imagine many such streaks, all parallel in section view. Note: the effects of molecular diffusion in mixing water and dye molecules is ignored at these flow velocities Flow Turbulent flow – dye streak passes downflow undeformed until a certain point when the dye streak billows a few times and is then intermixed with the water by a system of flow-wide eddy motions Flow Turbulent flow – the dye streak billows shown above here are viewed with the aid of an instantaneous electrical spark, providing a clearer view of the way that the billows spread and mix the dye through the whole flow Fig. 4.7 Details of ﬂow patterns as sketched by Reynolds.
LEED-Ch-04.qxd 11/26/05 13:22 Page 107 Flow, deformation, and transport 107 carefully introducing a dye streak into a steady ﬂow of water call laminar ﬂow. With increased velocity the dye streak was through a transparent tube (Figs 4.7 and 4.8). At low ﬂow dispersed in eddies, eventually coloring the whole ﬂow. velocities the dye streak extended down the tube as a This was Reynolds’ “sinuous” motion, now known as tur- straight line. This was his “direct” ﬂow, what we nowadays bulent ﬂow. It was Reynolds’ great contribution, ﬁrst, to recognize the fundamental difference in the two ﬂow types and, second, to investigate the dynamic signiﬁcance of 1 these. The latter process was not completed until he pub- lished another landmark paper in 1895 on turbulent stresses (see Section 3.11); more on these in Section 4.5. 2 4.2.1 Energy loss and ﬂow type: Reynolds critical experiments Concerning the forces involved, it was previously known 3 that “The resistance is generally proportional to the square of the velocity, and when this is not the case it takes a sim- pler form and is proportional to the velocity.” Reynolds approached the force problem both theoretically (or “philosophically” as he put it) and practically, in best 4 physical tradition. His philosophical analysis was “that the general character of the motion of ﬂuids in contact with solid surfaces depends on the relation between a physical constant of the ﬂuid, and the product of the linear dimen- sions of the space occupied by the ﬂuid, and the velocity.” Fig. 4.8 Photographic records of laminar to turbulent transition in a Designing the apparatus reproduced in Fig. 4.6, he pipe ﬂow. Turbulent Pressure drop, ∆p, per unit length of tube flow 1.75 1 Transition zone Laminar flow flow 1 1 ∆p Mean flow velocity of water Fig. 4.9 The rate of pressure decrease downﬂow increases in a linear fashion until at some critical velocity, the rate of loss markedly increases as about the 1.8 power of velocity.

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