Mechanics of Fluids

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Fundamental concepts 1 The aim of Chapter 1 is to provide a broad introduction to fluid mechanics, as a foundation for the more detailed discussion of specific topics contained in Chapters 2–13. We start by considering the characteristics of liquids and gases, and what it is that distinguishes them from solids. The ability to measure and quantify fluid and flow properties is of fundamental importance in engineering, and so at an early stage the related topics of units and dimensions are introduced. We move on to consider the properties of fluids, such as density, pressure, compressibility and viscosity. This is followed by a discussion of the terminology used to describe different flow patterns and types of fluid motion of interest to engineers. The chapter concludes by briefly reviewing the roles of experimentation and mathematical theory in the study of fluid mechanics. 1.1 THE CHARACTERISTICS OF FLUIDS A fluid is defined as a substance that deforms continuously whilst acted upon by any force tangential to the area on which it acts. Such a force is termed a shear force, and the ratio of the shear force to the area on which it acts is known as the shear stress. Hence when a fluid is at rest neither shear forces nor shear stresses exist in it. A solid, on the other hand, can resist a shear force while at rest. In a solid, the shear force may cause some initial displacement of one layer over another, but the material does not continue to move indefinitely and a position of stable equilibrium is reached. In a fluid, however, shear forces are possible only while relative movement between layers is taking place. A fluid is further distinguished from a solid in that a given amount of it owes its shape at any time to that of the vessel containing it, or to forces that in some way restrain its movement. The distinction between solids and fluids is usually clear, but there are some substances not easily classified. Some fluids, for example, do not flow easily: thick tar or pitch may at times appear to behave like a solid. A block of such a substance may be placed on the ground, and, although its flow would take place very slowly, over a period of time – perhaps several days – it would spread over the ground by the action of gravity. On
2 Fundamental concepts Liquid Gas the other hand, certain solids may be made to ‘flow’ when a sufficiently large force is applied; these are known as plastic solids. Nevertheless, these examples are rather exceptional and outside the scope of mainstream fluid mechanics. The essential difference between solids and fluids remains. Any fluid, no matter how thick or viscous it is, flows under the action of a net shear force. A solid, however, no matter how plastic it is, does not flow unless the net shear force on it exceeds a certain value. For forces less than this value the layers of the solid move over one another only by a certain amount. The more the layers are displaced from their original relative positions, the greater are the internal forces within the material that resist the displacement. Thus, if a steady external force is applied, a state will be reached in which the internal forces resisting the movement of one layer over another come into balance with the external applied force and so no further movement occurs. If the applied force is then removed, the resisting forces within the material will tend to restore the solid body to its original shape. In a fluid, however, the forces opposing the movement of one layer over another exist only while the movement is taking place, and so static equilibrium between applied force and resistance to shear never occurs. Deformation of the fluid takes place continuously so long as a shear force is applied. But if this applied force is removed the shearing movement subsides and, as there are then no forces tending to return the particles of fluid to their original relative positions, the fluid keeps its new shape. Fluids may be sub-divided into liquids and gases. A fixed amount of a liquid has a definite volume which varies only slightly with temperature and pressure. If the capacity of the containing vessel is greater than this definite volume, the liquid occupies only part of the container, and it forms an interface separating it from its own vapour, the atmosphere or any other gas present. A fixed amount of a gas, by itself in a closed container, will always expand until its volume equals that of the container. Only then can it be in equilibrium. In the analysis of the behaviour of fluids an important difference between liquids and gases is that, whereas under ordinary conditions liquids are so difficult to compress that they may for most purposes be regarded as incompressible, gases may be compressed much more readily. Where conditions are such that an amount of gas undergoes a negligible change of volume, its behaviour is similar to that of a liquid and it may then be regarded as incompressible. If, however, the change in volume is not negligible, the compressibility of the gas must be taken into account in examining its behaviour. A second important difference between liquids and gases is that liquids have much greater densities than gases. As a consequence, when considering forces and pressures that occur in fluid mechanics, the weight of a liquid has an important role to play. Conversely, effects due to weight can usually be ignored when gases are considered.
Page 2 and 3: Mechanics of Fluids
Page 4 and 5: Mechanics of Fluids Eighth edition
Page 6 and 7: Contents Preface to the eighth edit
Page 8 and 9: 8 Boundary Layers, Wakes and Other
Page 10 and 11: Preface to the eighth edition In th
Page 14 and 15: 1.1.1 Molecular structure The diffe
Page 16 and 17: therefore, to have in place systems
Page 18 and 19: has been a strong movement in favou
Page 20 and 21: Notation, dimensions, units and rel
Page 22 and 23: is nearer 2 m, rather than 1 m. Her
Page 24 and 25: Properties of fluids 13 The relativ
Page 26 and 27: that is, p1 − p3 = 1 2BC ϱax (1.
Page 28 and 29: that there is an almost perfect vac
Page 30 and 31: calorically perfect. (Some writers
Page 32 and 33: As a liquid is compressed its molec
Page 34 and 35: or τ = µ ∂u ∂y (1.9) where µ
Page 36 and 37: shared among the occupants of bb, a
Page 38 and 39: the origin with slope equal to µ (
Page 40 and 41: The interplay of these various forc
Page 42 and 43: non-uniformity is always encountere
Page 44 and 45: z direction and therefore the same
Page 46 and 47: Although Reynolds used water in his
Page 48 and 49: cylinder. For a certain range of Re
Page 50 and 51: The roles of experimentation and th
Page 52 and 53: A flow model which assumes steady,
Page 54 and 55: 2.1 INTRODUCTION Fluid statics 2 Fl
Page 56 and 57: 3. dp/dz =−ϱg. (Since the pressu
Page 58 and 59: that is, dp p =−g dz R T Variatio
Page 60 and 61: atmosphere it is termed vacuum or s
Page 62 and 63:
at its centreline be p. Then, provi
Page 64 and 65:
possible, and a sloping manometer m
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The amount of liquid B on each side
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In a pressure transducer the pressu
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where the suffixes C and Oy indicat
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surface (where the pressure is atmo
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which has different depths of water
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Hydrostatic thrusts on submerged su
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plane and may then be combined into
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Since all the elemental thrusts are
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neglected when a body is weighed on
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couple acting on the body in its di
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centre of buoyancy on to the transv
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move, but also the centre of gravit
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that is, W(OG) = (W + F)(OB + BM) =
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is given by ∂p ∂x =−ϱax Equi
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Then ∂p/∂ξ = 0 by definition o
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2.5 Two small vessels are connected
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2.18 In the vertical end of an oil
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The principles governing fluids in
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planes B and C (Fig. 3.2), δs bein
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an actual fluid is thus often remar
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density are small. Then eqn 3.9 has
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General energy equation for steady
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− Energy loss to friction/time Ma
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a decrease of pressure (provided th
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in the neighbourhood of B results i
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it is difficult to measure the heig
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The diagram illustrates an orifice
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� depth h below the free surface
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Solution (a) We consider first the
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of the same liquid. A vena contract
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To ensure that the pressure measure
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Since �h = p1 − p2 ϱwg the flo
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For consistency with the mathematic
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showing the essential form of the r
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With the same assumptions used in d
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∴ Velocity of approach = 0.0660 m
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of the base of the notch. Assume th
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Our aim now is to derive a relation
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The velocity, in general, varies fr
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the control volume is � ϱ1u1ux1
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Applications of the momentum equati
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that the resultant force on the flu
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The existence of the reaction may b
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our reference axes attached to the
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of increase of x-momentum is � D
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This is a simplified picture of wha
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since u4 −u1 is the velocity of t
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(1) u 1 (2) (3) (4) Assume a sea le
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and frictional effects to be neglig
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Physical similarity and dimensional
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A well-known example of kinematic s
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polygon. Now a polygon can be compl
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The condition for dynamic similarit
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equality of the Mach numbers is not
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Consider the incompressible flow al
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moving in a straight line with cons
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of the analysis, and its correct im
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Note: a suffix has been attached to
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For example, the parameter Fϱ/µ 2
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The application of dynamic similari
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atmospheric air, the dynamic viscos
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gives rise to waves on the surface.
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To be tested, the model must theref
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it follows that Hence V2 V1 = P2 P1
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5.5 A torpedo-shaped object 900 mm
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Laminar flow between solid boundari
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the place of y: τ = µ ∂u ∂r A
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compensate for the reduction in vel
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The expression for the total discha
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etween the resisting viscous forces
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must be zero so as to meet the requ
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Example 6.2 Two stationary, paralle
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Thus for any value of y the velocit
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Steady laminar flow between moving
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in eqn 6.24 we obtain � D Vp 2
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and the passage of the liquid level
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this book, it is important to give
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6.6.4 Rotary viscometers A simple m
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The torque T on the cylinder of rad
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stationary when the outer cylinder
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Fundamentals of the theory of hydro
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part of the bearing then gives when
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e determined by our analysis.) From
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where a = 6µ�R h3 dh pc dx Funda
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Integrating this between θ = 0 and
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Rearrangement of eqn 6.71 and subst
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6.4 A cylindrical drum of length l
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7.1 INTRODUCTION Flow and losses in
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a maximum value of 2.0. The line CD
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America the friction factor commonl
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course, quite different from the ro
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Variation of friction factor 253 Fi
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straightforward manner without iter
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Distribution of shear stress in a c
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7.5 FRICTION IN NON-CIRCULAR CONDUI
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estimated by simple theory. The pip
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the junction, the curvature of the
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particular, the performance of a di
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A pipe bend thus causes a loss of h
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cross-sectional area available in t
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correlating experimental data on th
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to do this. But sub-atmospheric pre
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For systems comprising only short r
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7.8 PIPES IN COMBINATION 7.8.1 Pipe
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2. There can be only one value of h
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So For two pipes in parallel, and s
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7.9 CONDITIONS NEAR THE PIPE ENTRY
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where u denotes the mean velocity i
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Assuming that entry and exit losses
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of one-per-revolution electrical co
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of the piping system, this meter is
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losses, calculate (a) the total pow
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How are the running costs altered i
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main. The depth of water in the tan
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the features of boundary layer flow
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the surface other considerations ar
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and θ = = = � δ u 0 um � δ 0
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The momentum equation applied to th
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1 = y η δ The laminar boundary la
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Table 8.1 Comparison of results for
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and � � ∂f (η) B = ∂η A =
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From equation 8.17, the frictional
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The turbulent boundary layer on a s
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Hence F = 0.037ϱu 2 m l(Re l) −1
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From eqn 8.29 Hence and xt − x0 x
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from it. This breakaway before the
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this expression into eqn 8.31 we ge
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When, for example, a ship moves thr
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separate vortices in an inviscid fl
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8.8.4 Profile drag of two-dimension
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are negligible. With reduction in t
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Fig. 8.14 Drag coefficients of smoo
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sphere is falling through the atmos
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At high Reynolds numbers, the varia
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this does not prevent the continual
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Eddy viscosity and the mixing lengt
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Eddy viscosity and the mixing lengt
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Velocity distribution in turbulent
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then, although at very small values
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as η → 1 and (1 − η) is then
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When f −1/2 − 4 log 10 (d/k) is
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comparable with the velocity of sou
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The x, y and z components of the mo
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y y x i-1, j+1 i, j+1 i+1, j+1 i-1,
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e completely covered by a turbulent
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9.1 INTRODUCTION The flow of an inv
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Now consider another point P ′′
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number of smaller ones of which M a
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distortion, the element is thus not
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Any function φ that satisfies Lapl
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ecome perfect squares - except wher
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fluid, and the flow net indicates t
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9.6.2 Flow from a line source A sou
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higher orders of small magnitudes b
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Combining eqns 9.16 and 9.17 we obt
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9.6.5 Forced (rotational) vortex Th
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For a free vortex qR = constant = K
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source is unable to move to the lef
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Also we note that the source is pos
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as shown in Fig. 9.23, encloses all
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eqn 9.23 becomes ψ =−U � r −
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Thus the magnitude of the velocity
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At a stagnation point, qt = 0 and t
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function of the combined flow is gi
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or, more approximately, that betwee
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Then dw dz =−U =−u + iv Hence u
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An introduction to elementary aerof
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ound the aerofoil, its magnitude be
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and lower layers. Since the vortex
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where Ɣ0 represents the circulatio
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9.7 An open cylindrical vessel, hav
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steadily south-east at 6 m · s −
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10.2 TYPES OF FLOW IN OPEN CHANNELS
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The steady-flow energy equation for
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where the hs represent vertical dep
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to the turbulent rough flow regime
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selection of an appropriate value r
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10.6 OPTIMUM SHAPE OF CROSS-SECTION
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oughness coefficient n is independe
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thrust divided by width of the chan
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which the waves follow one another
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The conditions for the critical dep
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y downstream conditions. In these c
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may be obtained by reducing eqn 10.
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this depth is greater than the crit
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The dissipation of energy is by no
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friction, the steady-flow momentum
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Fig. 10.28 Notice that, for a chann
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calculated, and then H + u2 1 /2g m
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Fig. 10.32 If the depth of flow ove
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(Section 10.11.1). Rapid flow can n
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where 0 ≤ θ ≤ 90 ◦ and sin
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(a) The continuity equation is The
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oundaries. It can occur when the fl
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into eqn 10.39 we obtain whence dh
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Table 10.2 Gradually varied flow 46
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The slope of a given channel may, o
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direction. The effect of any bounda
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This momentum term, however, is pro
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With surface tension effects ignore
Page 482 and 483:
are for waves of small amplitude) w
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when the wave was formed. The lengt
Page 486 and 487:
At a particular instant the crests
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Hence h = 2.65λ 2π = 2.65 × 225
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As the ratio of these velocity comp
Page 492 and 493:
water is impulsively displaced and
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is relative movement of one layer o
Page 496 and 497:
the channel bed a drop of 150 mm in
Page 498 and 499:
11.1 INTRODUCTION Compressible flow
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energy ÷ mass, so that the dimensi
Page 502 and 503:
Energy equation with variable densi
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Solution (a) From the equation of s
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whence (c − u)δρ = (ρ + δρ)
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the sonic velocity as supersonic (M
Page 510 and 511:
velocity into the undisturbed fluid
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whence p2 p1 = 1 + γ M2 1 1 + γ M
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of eqns 11.2 and 11.20 T0 T u2 = 1
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the temperature rises greatly (say
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Fig. 11.10 Oblique shock relations
Page 520 and 521:
however, that for a given upstream
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The deflection through the first wa
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As δθ is very small, cos δθ →
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eqn 11.41. Substituting M 2 − 1 =
Page 528 and 529:
oundaries, both of which are curved
Page 530 and 531:
For supersonic flow eqn 11.45 is no
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Some general relations for one-dime
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of minimum cross-section the veloci
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downstream of the minimum cross-sec
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‘telegraphed’ upstream and a pr
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the external pressure. This compres
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Compressible flow in pipes of const
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Table 11.2 Changes with distance al
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Page 550 and 551:
From the Fanno Tables at pc/pB = 0.
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Therefore and M1 = u1 a 24.5 m · s
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drag. The abrupt pressure rise thro
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Analogy between compressible flow a
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is, on ∂2n ∂y2 + ∂2 � n ∂
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and the mass flow rate in the duct.
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11.18 Air flows isothermally at 15
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12.2 INERTIA PRESSURE Any volume of
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Equation 12.3 indicates that u beco
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would come to rest later. Although
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a pressure wave in a gas; here we r
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value in eqn 12.7 gives that is, wh
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the moment disregarded. An increase
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at this point remains constant for
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In addition to the reflection of wa
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is closed so that the area coeffici
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The pressure diagrams show that for
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12.3.4 The method of characteristic
Page 588 and 589:
Pressure transients 577 Multiplying
Page 590 and 591:
and velocities are known, then for
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For pipe 2, and hence K ′ = c2 =
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(instantaneously, we will assume) a
Page 596 and 597:
The one-dimensional continuity rela
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to accommodate instantaneous comple
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12.2 A turbine which normally opera
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13.1 INTRODUCTION Fluid machines13
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flow. By way of illustration we sha
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would obviate the changes of pressu
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demand for electric power - for exa
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Turbines 599 Fig. 13.7 Runner of Pe
Page 612 and 613:
determined, multiplication by the r
Page 614 and 615:
Substituting for �vw from eqn 13.
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ut the larger its value the more bu
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ate of flow possible is achieved wh
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the turbine. For the high heads nor
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The integrals in eqns 13.4 and 13.5
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diagram of Fig. 13.16 we have Simil
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would be sufficient to describe the
Page 628 and 629:
Table 13.1 Type of turbine Approxim
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∴ Hydraulic efficiency = Overall
Page 632 and 633:
If either z (the height of the turb
Page 634 and 635:
∴ Hydraulic efficiency = u1vw1/gH
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shows the general effect of change
Page 638 and 639:
an unmixed blessing and, except for
Page 640 and 641:
orbiting space-craft, for example,
Page 642 and 643:
Losses are also possible at the inl
Page 644 and 645:
Those particles next to the forward
Page 646 and 647:
It should be noted that the values
Page 648 and 649:
∴ (β1)A = arctan(va/u) = arctan(
Page 650 and 651:
so much that the aerofoil stalls (a
Page 652 and 653:
Use these data to deduce the pump c
Page 654 and 655:
Manometric efficiency = 0.75 = gH/u
Page 656 and 657:
and CP = P/ϱω 3 D 5 in place of Q
Page 658 and 659:
yields h 1 = h f = 4fl d ≈ (100Q)
Page 660 and 661:
(iii) the power dissipated by pipe
Page 662 and 663:
Large numbers of test data for pump
Page 664 and 665:
educed by rounding the inlet edges
Page 666 and 667:
further: the turbine part then remo
Page 668 and 669:
for reaction turbines: 1 − η = 1
Page 670 and 671:
75 mm wide at outlet. The blades oc
Page 672 and 673:
Losses at valves, etc. are estimate
Page 674 and 675:
APPENDIX Units and conversion 1 fac
Page 676 and 677:
Table A1.4 (contd.) Force 32.17 pdl
Page 678 and 679:
APPENDIX Physical constants and 2 p
Page 680 and 681:
Kinematic viscosity mm 2 ·s -1 App
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Range II: 11 000 m ≤ z ≤ 20 000
Page 684 and 685:
Table A3.1 (contd.) M 1 M 2 p 2/p 1
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Table A3.2 Isentropic flow M p/po
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Table A3.3 Adiabatic flow with fric
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APPENDIX 4 Algebraic symbols Table
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Table A4.1 (contd.) Symbol Definiti
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Table A4.1 (contd.) Symbol Definiti
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Answers to problems 1.1 56.2 m 3 1.
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8.14 0.002955, 2.866 m · s −1 ,
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Absolute pressure 46 Absolute visco
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Eccentricity 230 Eccentricity ratio
Page 704 and 705:
Mach angle 498 intersection 510-1 r
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Slip in fluid couplings 652 in reci
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