Mechanics of Fluids

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at its centreline be p. Then, provided that the fluid A is of constant density, the pressure at P is p + ϱAgy (from eqn 2.3), where ϱA represents the density of fluid A. If the other side of the U-tube is open to atmosphere the (gauge) pressure at Q is ϱBgx where ϱB represents the density of liquid B. Therefore p + ϱAgy = ϱBgx and if ϱA and ϱB are known and y and x are measured, the value of p may be calculated. If A is a gas, ϱA is negligible compared with ϱB, and then p = ϱBgx. The arrangement is suitable for measuring pressures below atmospheric as illustrated in Fig. 2.7. Application of the same principles then gives p + ϱAgy + ϱBgx = 0. U-tube manometers are also frequently used for measuring the difference between two unknown pressures, say p1 and p2. Figure 2.8 shows such an arrangement for measuring the pressure difference across a restriction in a horizontal pipe. (When fluid is flowing past the connections to a manometer it is very important for the axis of each connecting tube to be perpendicular to the direction of flow and also for the edges of the connections to be smooth, flush with the main surface and free from burrs. To reduce the The measurement of pressure 51 Fig. 2.6 Fig. 2.7
52 Fluid statics Fig. 2.8 risk of errors resulting from imperfect connections several openings round the periphery of the pipe are often used, all connected to the manometer via a common annulus; an average pressure is thus obtained in which the individual, probably random, errors tend to cancel.) Again applying the principle that pressures at P and Q must be equal for equilibrium, we have p1 + (y + x)ϱAg = p2 + ϱAgy + ϱBgx ∴ p1 − p2 = (ϱB − ϱA)gx (2.8) If it is desired to express this pressure difference as a head difference for fluid A, then h1 − h2 = p1 − p2 ϱAg = � � ϱB − 1 x (2.9) ϱA If, for example, fluid A is water and B is mercury, then a difference x in manometer levels corresponds to a difference in head of water = (13.56−1)x. A common error is to use simply ϱB/ϱA instead of {(ϱB/ϱA)−1} in eqn 2.9. It should not be forgotten that the pressure at P includes a contribution made by the column of fluid A above it. More generally, it may be shown that a differential manometer such as this records the difference of piezometric pressure p ∗ = p + ϱgz: only when points (1) and (2) are at the same level, as in Fig. 2.8, does the manometer reading correspond to the difference of actual pressure p. Many modifications of the U-tube manometer have developed for particular purposes. A common modification is to make one limb of the ‘U’ very much greater in cross-section than the other. When a pressure difference is applied across the manometer the movement of the liquid surface in the wider limb is practically negligible compared with that occurring in the narrow limb. If the level of the surface in the wide limb is assumed constant the height of the meniscus in only the narrow limb need be measured, and only this limb need therefore be transparent. For accurate measurements reasonable values of x are desirable. For small pressure differences such values may be obtained by selecting liquid B so that the ratio ϱB/ϱA is close to unity. If fluid A is a gas, however, this is not
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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
Page 12 and 13: Fundamental concepts 1 The aim of C
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 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 70 and 71: where the suffixes C and Oy indicat
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 110 and 111: General energy equation for steady
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General energy equation for steady
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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
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are for waves of small amplitude) w
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when the wave was formed. The lengt
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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
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water is impulsively displaced and
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is relative movement of one layer o
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the channel bed a drop of 150 mm in
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11.1 INTRODUCTION Compressible flow
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energy ÷ mass, so that the dimensi
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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
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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
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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 =
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oundaries, both of which are curved
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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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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
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Pressure transients 577 Multiplying
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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
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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
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determined, multiplication by the r
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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
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Table 13.1 Type of turbine Approxim
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∴ Hydraulic efficiency = Overall
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If either z (the height of the turb
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∴ Hydraulic efficiency = u1vw1/gH
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shows the general effect of change
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an unmixed blessing and, except for
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orbiting space-craft, for example,
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Losses are also possible at the inl
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Those particles next to the forward
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It should be noted that the values
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∴ (β1)A = arctan(va/u) = arctan(
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so much that the aerofoil stalls (a
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Use these data to deduce the pump c
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Manometric efficiency = 0.75 = gH/u
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and CP = P/ϱω 3 D 5 in place of Q
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yields h 1 = h f = 4fl d ≈ (100Q)
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(iii) the power dissipated by pipe
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Large numbers of test data for pump
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educed by rounding the inlet edges
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further: the turbine part then remo
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for reaction turbines: 1 − η = 1
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75 mm wide at outlet. The blades oc
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Losses at valves, etc. are estimate
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APPENDIX Units and conversion 1 fac
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Table A1.4 (contd.) Force 32.17 pdl
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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
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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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