Mechanics of Fluids

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Example 6.2 Two stationary, parallel, concentric discs of external radius R2 are a distance c apart. Oil is supplied at a gauge pressure p ∗ 1 from a central source of radius R1 in the lower disc. From there it spreads radially outwards between the two discs and escapes to atmosphere. Assume the flow is laminar. (a) Starting from the equation for flow between fixed surfaces u = 1 � dp∗ � (y 2µ dx 2 − cy) show that the pressure distribution gives rise to a vertical force F on the upper disc given by F = πp ∗ � 1 R 2 � � R2 1 + 2 x 1 − loge (x/R1) � � dx loge (R2/R1) R1 (b) Calculate the rate at which oil of viscosity 0.6 kg · m −1 · s −1 must be supplied to maintain a pressure p ∗ 1 of 15 kPa when R2/R1 = 6 and the clearance c is 1 mm. Solution (a) � c � c Q = 2πxu dy = 2πx 0 0 1 � dp∗ � (y 2µ dx 2 − cy)dy =− πxc3 � dp∗ � 6µ dx Hence dp ∗ =− 6µQ πc3 dx x Integrating with respect to x: p ∗ =− 6µQ πc3 loge x + A The boundary conditions are So A = p ∗ 1 p ∗ = p ∗ 1 , x = R1; p ∗ = 0, x = R2 + 6µQ πc 3 log e R1 and A = 6µQ πc 3 log e R2 Combining these expressions we obtain p ∗ − p ∗ 1 =−6µQ πc 3 log e x R1 and p ∗ 1 Steady laminar flow between parallel planes 203 6µQ = πc3 log R2 e R1 Eliminating (6µQ)/(πc 3 ) between these two relations produces p ∗ = p ∗ 1 − p∗ loge (x/R1) 1 loge (R2/R1)
204 Laminar flow between solid boundaries ✷ F = πR 2 1p∗ 1 + � R2 2πxp R1 ∗ dx = πR 2 1p∗ � � R2 1 + 2π = πp ∗ 1 � R 2 1 (b) From the above analysis So Q = = R1 � R2 + 2 x R1 πc 3 p ∗ 1 6µ log e (R2/R1) p ∗ 1 − p∗ 1 � p ∗ 1 =−6µQ πc3 log R2 e R1 � loge (x/R1) xdx loge (R2/R1) 1 − loge (x/R1) � � dx loge (R2/R1) π × 1mm 3 × 15 × 10 3 Pa 6 × 0.6 kg · m −1 · s −1 × 1.792 × (10 3 mm/m) 3 Q = 7.3 × 10 −6 m 3 · s −1 6.5 STEADY LAMINAR FLOW BETWEEN PARALLEL PLANES, ONE OF WHICH IS MOVING We now consider steady laminar flow between plane boundaries that move relative to one another while still remaining parallel and the same distance apart. We may conveniently assume that one of the boundaries is stationary and that the other moves with a velocity V in the direction of flow. Even if both are moving, this assumption still serves: all velocities are considered relative to one of the boundaries which is then supposed at rest. Here we may assume the lower plane in Fig. 6.5 to be the stationary one. Again we may consider the flow from left to right and, for consistency, we consider the velocity of the upper plane as positive from left to right. The analysis proceeds as before, and again we arrive at eqn 6.18. The boundary conditions, however, are now different. The velocity at the lower plane where y = 0 is again zero and hence B = 0. But the fluid at y = c must have the velocity V of the upper plane. Hence, substituting the values in eqn 6.18: � δp∗ � c2 � δp∗ � c µV = µV + Ac and so A = − δx 2 δx 2 c
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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
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2.5 Two small vessels are connected
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2.18 In the vertical end of an oil
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The principles governing fluids in
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planes B and C (Fig. 3.2), δs bein
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an actual fluid is thus often remar
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density are small. Then eqn 3.9 has
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General energy equation for steady
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− Energy loss to friction/time Ma
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a decrease of pressure (provided th
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in the neighbourhood of B results i
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it is difficult to measure the heig
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The diagram illustrates an orifice
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� depth h below the free surface
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Solution (a) We consider first the
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of the same liquid. A vena contract
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To ensure that the pressure measure
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Since �h = p1 − p2 ϱwg the flo
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For consistency with the mathematic
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showing the essential form of the r
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With the same assumptions used in d
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∴ Velocity of approach = 0.0660 m
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of the base of the notch. Assume th
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Our aim now is to derive a relation
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The velocity, in general, varies fr
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the control volume is � ϱ1u1ux1
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Applications of the momentum equati
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that the resultant force on the flu
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The existence of the reaction may b
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our reference axes attached to the
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of increase of x-momentum is � D
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This is a simplified picture of wha
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.
Page 196 and 197: To be tested, the model must theref
Page 198 and 199: it follows that Hence V2 V1 = P2 P1
Page 200 and 201: 5.5 A torpedo-shaped object 900 mm
Page 202 and 203: Laminar flow between solid boundari
Page 204 and 205: the place of y: τ = µ ∂u ∂r A
Page 206 and 207: compensate for the reduction in vel
Page 208 and 209: The expression for the total discha
Page 210 and 211: etween the resisting viscous forces
Page 212 and 213: must be zero so as to meet the requ
Page 216 and 217: Thus for any value of y the velocit
Page 218 and 219: Steady laminar flow between moving
Page 220 and 221: in eqn 6.24 we obtain � D Vp 2
Page 222 and 223: and the passage of the liquid level
Page 224 and 225: this book, it is important to give
Page 226 and 227: 6.6.4 Rotary viscometers A simple m
Page 228 and 229: The torque T on the cylinder of rad
Page 230 and 231: stationary when the outer cylinder
Page 232 and 233: Fundamentals of the theory of hydro
Page 240 and 241: part of the bearing then gives when
Page 242 and 243: e determined by our analysis.) From
Page 248 and 249: where a = 6µ�R h3 dh pc dx Funda
Page 250 and 251: Integrating this between θ = 0 and
Page 252 and 253: Rearrangement of eqn 6.71 and subst
Page 254 and 255: 6.4 A cylindrical drum of length l
Page 256 and 257: 7.1 INTRODUCTION Flow and losses in
Page 258 and 259: a maximum value of 2.0. The line CD
Page 260 and 261: America the friction factor commonl
Page 262 and 263: 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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