Mechanics of Fluids

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Our aim now is to derive a relation by which force may be related to the fluid within a given space. We begin by applying Newton’s Second Law to a small element in a stream-tube (shown in Fig. 4.1) The flow is steady and so the stream-tube remains stationary with respect to the fixed coordinate axes. The cross-section of this stream-tube is sufficiently small for the velocity to be considered uniform over the plane AB and over the plane CD. After a short interval of time δt the fluid that formerly occupied the space ABCD will have moved forward to occupy the space A ′ B ′ C ′ D ′ . In general, its momentum changes during this short time interval. If ux represents the component of velocity in the x direction then the element (of mass δm) has a component of momentum in the x direction equal to uxδm. The total x-momentum of the fluid in the space ABCD at the beginning of the time interval δt is therefore � ABCD uxδm The same fluid at a time δt later will have a total x-momentum � A ′ B ′ C ′ D ′ The last expression may be expanded as � ABCD uxδm − � ABB ′ A ′ uxδm uxδm + � DCC ′ D ′ The momentum equation for steady flow 135 uxδm The net increase of x-momentum during the time interval δt is therefore � � A ′ B ′ C ′ D ′ � uxδm after δt � � � − uxδm ABCD before δt Fig. 4.1
136 The momentum equation = = � � − ABCD � � ABCD � � DCC ′ D ′ uxδm − � uxδm ABB ′ A ′ � before δt uxδm − � ABB ′ A ′ uxδm + � uxδm � DCC ′ D ′ after δt uxδm � after δt since, as the flow is assumed steady, �� uxδm � is the same after δt as ABCD before δt. Thus, during the time intereval δt, the increase of x-momentum of the batch of fluid considered is equal to the x-momentum leaving the stream-tube in that time minus the x-momentum entering in that time: � � � � � � uxδm − uxδm DCC ′ D ′ ABB ′ A ′ For a very small value of δt the distances AA ′ , BB ′ are very small, so the values of ux, for all the particles in the space ABB ′ A ′ are substantially the same. Similarly, all particles in the space DCC ′ D ′ have substantially the same value of ux, although this may differ considerably from the value for particles in ABB ′ A ′ . The ux terms may consequently be taken outside the summations. Therefore the increase of x-momentum during the interval δt is � � � � δm δm � ux DCC ′ − D ′ � ux ABB ′ A ′ (4.1) Now �� δm � DCC ′ D ′ is the mass of fluid which has crossed the plane CD during the interval δt and so is expressed by ˙mδt, where ˙m denotes the rate of mass flow. Since the flow is steady, �� δm � ABB ′ A ′ also equals ˙mδt. Thus expression 4.1 may be written ˙m(ux2 − ux1 )δt, where suffix 1 refers to the inlet section of the stream-tube, suffix 2 to the outlet section. The rate of increase of x-momentum is obtained by dividing by δt, and the result, by Newton’s Second Law, equals the net force Fx on the fluid in the x direction Fx = ˙m(ux2 − ux1 ) (4.2) The corresponding force in the x direction exerted by the fluid on its surroundings is, by Newton’s Third Law, −Fx. A similar analysis for the relation between force and rate of increase of momentum in the y direction gives Fy = ˙m(uy2 − uy1 ) (4.3) In steady flow ˙m is constant and so ˙m = ϱ1A1u1 = ϱ2A2u2 where ϱ represents the density of the fluid and A the cross-sectional area of the stream-tube (A being perpendicular to u). We have so far considered only a single stream-tube with a cross-sectional area so small that the velocity over each end face (AB, CD) may be considered uniform. Let us now consider a bundle of adjacent stream-tubes, each of cross-sectional area δA, which together carry all the flow being examined.
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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
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Notation, dimensions, units and rel
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is nearer 2 m, rather than 1 m. Her
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Properties of fluids 13 The relativ
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that is, p1 − p3 = 1 2BC ϱax (1.
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that there is an almost perfect vac
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calorically perfect. (Some writers
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As a liquid is compressed its molec
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or τ = µ ∂u ∂y (1.9) where µ
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shared among the occupants of bb, a
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the origin with slope equal to µ (
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The interplay of these various forc
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non-uniformity is always encountere
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z direction and therefore the same
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Although Reynolds used water in his
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cylinder. For a certain range of Re
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The roles of experimentation and th
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A flow model which assumes steady,
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2.1 INTRODUCTION Fluid statics 2 Fl
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3. dp/dz =−ϱg. (Since the pressu
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that is, dp p =−g dz R T Variatio
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atmosphere it is termed vacuum or s
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at its centreline be p. Then, provi
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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
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
Page 120 and 121: in the neighbourhood of B results i
Page 122 and 123: it is difficult to measure the heig
Page 124 and 125: The diagram illustrates an orifice
Page 126 and 127: � depth h below the free surface
Page 128 and 129: Solution (a) We consider first the
Page 130 and 131: of the same liquid. A vena contract
Page 132 and 133: To ensure that the pressure measure
Page 134 and 135: Since �h = p1 − p2 ϱwg the flo
Page 136 and 137: For consistency with the mathematic
Page 138 and 139: showing the essential form of the r
Page 140 and 141: With the same assumptions used in d
Page 142 and 143: ∴ Velocity of approach = 0.0660 m
Page 144 and 145: of the base of the notch. Assume th
Page 148 and 149: The velocity, in general, varies fr
Page 150 and 151: the control volume is � ϱ1u1ux1
Page 152 and 153: Applications of the momentum equati
Page 154 and 155: that the resultant force on the flu
Page 156 and 157: The existence of the reaction may b
Page 158 and 159: our reference axes attached to the
Page 160 and 161: of increase of x-momentum is � D
Page 162 and 163: This is a simplified picture of wha
Page 164 and 165: since u4 −u1 is the velocity of t
Page 166 and 167: (1) u 1 (2) (3) (4) Assume a sea le
Page 168 and 169: and frictional effects to be neglig
Page 170 and 171: Physical similarity and dimensional
Page 172 and 173: A well-known example of kinematic s
Page 174 and 175: polygon. Now a polygon can be compl
Page 176 and 177: The condition for dynamic similarit
Page 178 and 179: equality of the Mach numbers is not
Page 180 and 181: Consider the incompressible flow al
Page 182 and 183: moving in a straight line with cons
Page 184 and 185: of the analysis, and its correct im
Page 186 and 187: Note: a suffix has been attached to
Page 188 and 189: For example, the parameter Fϱ/µ 2
Page 190 and 191: The application of dynamic similari
Page 192 and 193: atmospheric air, the dynamic viscos
Page 194 and 195: 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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