Mechanics of Fluids

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where the suffixes C and Oy indicate the axes used. Therefore (Ak 2 )Oy = (Ak 2 )C + Ax 2 The direction of the y-axis was arbitrary; hence it may be said that the second moment of a plane area about any axis equals the sum of the second moment about a parallel axis through the centroid and the product of the area and the square of the perpendicular distance between the axes. This result is frequently known as the parallel axes theorem. Moreover, by definition, (Ak 2 )Oy + (Ak 2 � )Ox = � = A A x 2 � dA + y A 2 dA (x 2 + y 2 � )dA = Hydrostatic thrusts on submerged surfaces 59 A r 2 dA The last term corresponds to the second moment of the area about an axis perpendicular to the plane of the area at the origin. Since the origin was arbitrarily chosen we have the perpendicular axes theorem: the second moment of a plane area about an axis meeting the plane perpendicularly at any point P equals the sum of the second moments of that area about two axes in the plane that intersect perpendicularly at P. The second moment of mass about a particular axis is ∫ M z 2 dM, where z represents the perpendicular distance of an element from the axis in question. If the mean value of z 2 is represented by k 2 , the second moment of mass may alternatively be written Mk 2 . The second moment of mass is also known as the moment of inertia, and k is termed the radius of gyration. The dimensional formula of moment of inertia is [ML 2 ] and a suitable unit is kg · m 2 . Unfortunately the second moment of area is sometimes referred to wrongly as moment of inertia. Inertia is a property of matter and has nothing to do with area. Second moment of area Ak 2 , a purely geometric quantity, and moment of inertia Mk 2 are fundamentally different. In this book we use Ak 2 as the symbol for second moment of area, a suitable suffix indicating the axis about which the moment is taken. Examples of second moments about centroidal axes are given in Fig. 2.14. 2.5 HYDROSTATIC THRUSTS ON SUBMERGED SURFACES The pressure of a fluid causes a thrust to be exerted on every part of any surface with which the fluid is in contact. The individual forces distributed over the area have a resultant, and determination of the magnitude, direction and position of this resultant force is frequently important. For a plane horizontal surface at which the fluid is in equilibrium the matter is simple: the pressure does not vary over the plane and the total force is given by the product of the pressure and the area. Its direction is perpendicular to the plane – downwards on the upper face, upwards on the lower face – and its position is at the centroid of the plane. But if the surface is not horizontal the
60 Fluid statics Fig. 2.14 Second moments about centroidal axes. Fig. 2.15 pressure varies from one point of the surface to another and the calculation of the total thrust is a little less simple. 2.5.1 Thrust on a plane surface Figure 2.15 shows a plane surface of arbitrary shape, wholly submerged in a liquid in equilibrium. The plane of the surface makes an angle θ with the horizontal, and the intersection of this plane with the plane of the free
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Mechanics of Fluids
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Mechanics of Fluids Eighth edition
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Contents Preface to the eighth edit
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8 Boundary Layers, Wakes and Other
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Preface to the eighth edition In th
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Fundamental concepts 1 The aim of C
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1.1.1 Molecular structure The diffe
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therefore, to have in place systems
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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
Page 66 and 67: The amount of liquid B on each side
Page 68 and 69: In a pressure transducer the pressu
Page 72 and 73: surface (where the pressure is atmo
Page 74 and 75: which has different depths of water
Page 76 and 77: Hydrostatic thrusts on submerged su
Page 78 and 79: plane and may then be combined into
Page 80 and 81: Since all the elemental thrusts are
Page 82 and 83: neglected when a body is weighed on
Page 84 and 85: couple acting on the body in its di
Page 86 and 87: centre of buoyancy on to the transv
Page 88 and 89: move, but also the centre of gravit
Page 90 and 91: that is, W(OG) = (W + F)(OB + BM) =
Page 92 and 93: is given by ∂p ∂x =−ϱax Equi
Page 94 and 95: Then ∂p/∂ξ = 0 by definition o
Page 96 and 97: 2.5 Two small vessels are connected
Page 98 and 99: 2.18 In the vertical end of an oil
Page 100 and 101: The principles governing fluids in
Page 102 and 103: planes B and C (Fig. 3.2), δs bein
Page 104 and 105: an actual fluid is thus often remar
Page 106 and 107: density are small. Then eqn 3.9 has
Page 108 and 109: General energy equation for steady
Page 116 and 117: − Energy loss to friction/time Ma
Page 118 and 119: 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
Page 430 and 431:
where the hs represent vertical dep
Page 432 and 433:
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
Page 480 and 481:
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
Page 488 and 489:
Hence h = 2.65λ 2π = 2.65 × 225
Page 490 and 491:
As the ratio of these velocity comp
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water is impulsively displaced and
Page 494 and 495:
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
Page 504 and 505:
Solution (a) From the equation of s
Page 506 and 507:
whence (c − u)δρ = (ρ + δρ)
Page 508 and 509:
the sonic velocity as supersonic (M
Page 510 and 511:
velocity into the undisturbed fluid
Page 512 and 513:
whence p2 p1 = 1 + γ M2 1 1 + γ M
Page 514 and 515:
of eqns 11.2 and 11.20 T0 T u2 = 1
Page 516 and 517:
the temperature rises greatly (say
Page 518 and 519:
Fig. 11.10 Oblique shock relations
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however, that for a given upstream
Page 522 and 523:
The deflection through the first wa
Page 524 and 525:
As δθ is very small, cos δθ →
Page 526 and 527:
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
Page 532 and 533:
Some general relations for one-dime
Page 534 and 535:
of minimum cross-section the veloci
Page 536 and 537:
downstream of the minimum cross-sec
Page 538 and 539:
‘telegraphed’ upstream and a pr
Page 540 and 541:
the external pressure. This compres
Page 542 and 543:
Compressible flow in pipes of const
Page 544 and 545:
Table 11.2 Changes with distance al
Page 546 and 547:
Page 548 and 549:
Page 550 and 551:
From the Fanno Tables at pc/pB = 0.
Page 552 and 553:
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Therefore and M1 = u1 a 24.5 m · s
Page 556 and 557:
drag. The abrupt pressure rise thro
Page 558 and 559:
Analogy between compressible flow a
Page 560 and 561:
is, on ∂2n ∂y2 + ∂2 � n ∂
Page 562 and 563:
and the mass flow rate in the duct.
Page 564 and 565:
11.18 Air flows isothermally at 15
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12.2 INERTIA PRESSURE Any volume of
Page 568 and 569:
Equation 12.3 indicates that u beco
Page 570 and 571:
would come to rest later. Although
Page 572 and 573:
a pressure wave in a gas; here we r
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value in eqn 12.7 gives that is, wh
Page 576 and 577:
the moment disregarded. An increase
Page 578 and 579:
at this point remains constant for
Page 580 and 581:
In addition to the reflection of wa
Page 582 and 583:
is closed so that the area coeffici
Page 584 and 585:
The pressure diagrams show that for
Page 586 and 587:
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
Page 592 and 593:
For pipe 2, and hence K ′ = c2 =
Page 594 and 595:
(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
Page 606 and 607:
would obviate the changes of pressu
Page 608 and 609:
demand for electric power - for exa
Page 610 and 611:
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.
Page 616 and 617:
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
Page 622 and 623:
The integrals in eqns 13.4 and 13.5
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diagram of Fig. 13.16 we have Simil
Page 626 and 627:
would be sufficient to describe the
Page 628 and 629:
Table 13.1 Type of turbine Approxim
Page 630 and 631:
∴ Hydraulic efficiency = Overall
Page 632 and 633:
If either z (the height of the turb
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∴ Hydraulic efficiency = u1vw1/gH
Page 636 and 637:
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,
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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
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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
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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
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Mach angle 498 intersection 510-1 r
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Slip in fluid couplings 652 in reci
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