Appendix
FLUID MECHANICS Approximate physical properties of selected fluids All properties are given at pressure 101. kn/m and temperature 15 C. Liquids Density (kg/m ) Dynamic viscosity (N s/m ) Surface tension with air (Njm) Acetone 79 0.41 x 10-.7 X 10- Benzene 878 0.760 x 10-.89 X 10- Carbon tetrachloride I 590 1.050 x 10-.69 X 10- Glycerin 1 60 1.50 6. x 10- n-octane 70 0.58 x 10- n-pentane 68 0.44 x 10- Mercury 1 590 1.600 x 10-4.66 X 10-1 Sea-water I 00 1.0 x 10-7.0 X 10- Water 999 1.14 x 10-7.0 X 10- Gases Density (kg/m'') Dynamic viscosity (N s/m ) Air Carbon dioxide Carbon monoxide Hydrogen Methane Nitrogen Oxygen Propane 1. 1.89 1.19 0.085 0.670 1.19 1.6 1.9 18.0 X 10-6 14.4 X 10-6 17. X 10-6 8.7 X 10-6 10.8 X 10-6 17. X 10-6 0. X 10-6 7.9 X 10-6 400
APPENDIX Properties of geometric shapes Circle Diameter d 0_-1- -1-_ 0 Area: 7rd /4 About 0-0, Ie = 7rd 4j64 Rectangle b Area: bh About 0-0, Ie = bh j1 Triangle 1 h O------..'--...:::;:>;E---\--0 Area: bhj Ii =h/ About 0-0, Ie = bh /6 About X-X, Ix = bh j1 xl~----""-----~ --X l b J Parallel axis theorem Or-f-- o --x Second moment of area about axis 0-0 through the centroid C is t., Second moment about the parallel axis X - X is Ix. Distance between axes is d. Then lx = Ie + Ad 401
FLUID MECHANICS Answers to test exercises and further problems Programme 1 Test exercises Further problems 1 6.8 kn/m 1 0.1 MN/m 65 N/m 0.0 N s/m 5.19 kn/m 1.0 kn per m; 1.9 kn 4.7 N/m 4 15. g; 4.95 mm 5 1100 kg/m! 5 0.1 mm (negligibly small) 6 8.6mm Programme 1 5.0 kn 1 1.19 m 8.4 kn 199 kn (a).6 kn 559 N at 7.1 0 to horizontal (b) 1.675 m 4 09 kn (c) 1.4 kn 5 740 N at 1 mm from C 4.1 kn to left 5.0 kn upwards Programme 1 0.8 m 1 0.65 m 1.07 m 0.188 m 0.49 m; stable b/fi 4 Reduced by 0.57 m; 4 1.6 m now unstable 0.914 m 5 (a) 0. m (b) 104 mm lower relative to ship (c) 0.175 m (d) 5 s Programme 4 8.6 m/s; 1.65 m; I.5 m/s 0.75 m 0.10 m lower 59 N/m 894 m/s (a) 9.6 m/s 4 790 kn/m (b) 456 kn/m 40
APPENDIX Programme 5 1 0.448 mjs (a). mjs; (b) 0.17 kgjs (a) 18.7 kn jm (b) 18 W 1 4.1 mjs 0.70 mjs 8691/min 44 kn/m ; 691 I/min Programme 6 18.4 N, at 1 to the vertical.0 kn, at 0 to the horizontal 10 N to the right 1 4 5 16.7 N, at 9 above horizontal 56 m/s A(p + pu ) 6.4 kn to the right (a) 7 mjs (b) 8.1 N downwards; 140 N to the right Programme 7 (i) k~j; (ii) [~] (i) consistent (ii) not consistent (first and second terms [ML jt ] ; third term [MLjT ]) Programme 8 1 (6.pfJpQ); or some power of this ratio, or its reciprocal (a) (FjputP) and (P/pu ), for example. Other forms, such as the product of these, are possible, and are equally correct. Programme 9 (a) 66.1 x 10 ; turbulent (b) 0.0058 approx. (c) 57 knjm 100 kn/m 861jmin I 4 5 0.0149 Njm 7. knjm per metre length 1.1 (a) 0.61 m (b) 1.4 m (c) 54 I/min 7.8 Ijs; 109 kw; 177 m; 17 mm diameter 40
INDEX aerofoil 15, 45 lift force on 45 aircraft, vertical take-off force on 79 air lock 195 angle of contact 48 angle of incidence 45 Archimedes' principle 9, 110 argument, dimensions of 99 barometer 6 Bernoulli's equation 174 derivation 175-6 for a pipe with losses 8 and venturi meter 11 bluff body 4 bucket, Pelton wheel 57,61,64 Buckingham's theorem 1 buoyancy 9,110 centre of 11-16 definition 11 carburettor 18 car ferries, stability of 14 centre of buoyancy 11-16 definition 11 centre of pressure 66, 77-90 of circular gate 80 definition 66 of rectangular gate 81 of T-shaped plate 88 centroid of a plane area 59-64 chord of aerofoil 45 closed-circuit wind tunnel 6 coefficient of contraction coefficient of discharge for orifice for venturi meter 17 coefficient of drag 4 coefficient of lift 4 coefficient of velocity compressibility consistent set of units 91 continuity principle 156 and mean velocity 158 contraction coefficient of in a pipe 159 control surface 16, 5 conversion of units 84 dependent dimensionless ratios dependent variables 1 derivative, dimensions of 0-5 derived unit 91 development of flow pattern in a pipe 6, 65, 66 diffuser 160,11,4 angle 4 efficiency 4 dimensional formula 95 dimensionless quantities 98 dimensionless ratios dependent independent presentation of data in terms of 14-19 dimensions 94-06 discharge, coefficient of 157 for orifice for venturi meter 17 displacement 141 drag coefficient 4 drag force on sphere 0 dynamic pressure dynamic similarity 5, 41 dynamic viscosity 6 ears, popping 19 efficiency, diffuser 4 elevation 189, 86 energy 168 loss in orifice meter loss in venturi meter 5 entrance to a pipe, development of flow pattern in 6, 65, 66 equations, consistency of 8 experimental results, presentation in terms of dimensionless ratios 14-19 fabric, cutting with water jet 00 fittings, head loss in 90 flaps 47 405
INDEX flow laminar 5 turbulent 5 flow rate over sharp-edged weir 9 over v-notch weir 1 frame of reference, moving 196 free jet 5 friction factor 69 European version 69 in laminar flow 71 USA version 69 friction loss in non-circular pipes 9 in a pipe 74 fully developed flow in a pipe 64, 65, 66 gauge I gauge pressure 18 negative geometric similarity 41 grade line, hydraulic 19 Hagen 60 Hagen-Poiseuille equation 59, 64 head 18, 189 pressure 18, 86 total 189 velocity 189, 86 head loss in a pipe 74, 85 hydraulic grade line 19,87, 91 hydraulic mean depth 9 hydrometer 4 hydrostatic force 65 on a circular window in a tank 7 on a curved surface 9-105 on a dished end 9 on the end of a tank 66, 68 on a hemispherical tank end 96 on an inclined plane area, expression for 71 on a pipe elbow 94 on a plane area 65-74 on a quadrant tank end 98 on a radial sluice gate 10 on rectangular sluice gate 7, 74 incidence, angle of 45 inclined manometer 4 incompressibility independent dimensionless ratios independent variables 1 injector 180 inverted V-tube 1 jet free 5 submerged 5 jet engine thrust 48 jet of water, force of 46, 6 on moving Pelton bucket 64 on moving plate 6 kinematic viscosity 79 laminar flow 5 friction factor in 71 pressure drop in a pipe 57 velocity profile in a circular pipe 59 leather, cutting with water jet 00 lift 45 coefficient of 45 liquid cargo, effect on stability 19 locomotive picking up water 197 manometer -5 two-fluid 7 inclined 4 mass flow rate 157 mean velocity and continuity principle 158 meniscus 49 for mercury in glass 50 metacentre 14 metacentric height 1-8 definition 16 experimental determination 17-0 metacentric radius 1 calculation from geometry 18 model testing 4 momentum 44 rate ofchange of 44-77 Moody chart 7 moving frame of reference 196 newton 91 Newtonian fluid 5 Newton's laws ofmotion 4,4,45,46 non-eircular pipes, friction loss in 9 numerical data, presentation in terms of dimensionless ratios 10-19 406
INDEX orifice coefficient of contraction coefficient of discharge coefficient of velocity flow through 1 orifice meter 8 advantages and disadvantages energy loss in in inclined pipe 1 parallel axis theorem 85 Pelton wheel 56 period of rolling of ship 144 piezometer, 188, 0 pipe bend force on 180 0 68 force on 90 0 71 pipe elbow, force on 71 pipe fittings, head loss in 90 pitching motion 19 pitot-static tube 09 pitot tube 0, 04 plan area (of aerofoil) 45 Poiseuille 60 Poiseuille equation 59, 64 power series, dimensions of terms 99 pressure 14-1 definition 14 gauge 18 dynamic 0 stagnation 0 static 187, 09 total 0 pressure drop across orifice across venturi 5 pressure head 189, 86 pressure loss in a pipe 74 measurement variation in a venturi 178 variation with depth 16 relative density 4 Reynolds number 4, 5, 4, 5, 67, 69 Reynolds, Osborne 4, 5 ripples, speed of motion 8 road sign, wind force on 9 road vehicle, drag force on 4 rocket, force on a stationary 4, 44 rolling motion 19 roll-on, roll-off ferries, stability of 14 roughness, internal, of pipes 71 sea waves, speed of 5 second moment of area 79 sensitivity 9 shear stress at wall of a pipe 55 similarity 5, 41 dynamic 5, 41 geometric 41 siphon 19 specific gravity 4 stability of a floating object 1-8 effect of liquid cargo on 19 of a submerged body 117-19 condition for 118 stagnation point 0 stagnation pressure 0 static pressure 187, 09 static tapping 188 streamline 15 streamtube 154 submerged jet 5 surface tension 45-50, 8 definition 45 tapered pipe, force on 7 testing models 4 throat (of venturi etc.) 11 total head line 87, 91 total pressure 0 turbine, Pelton 56 turbulent flow 5 in a pipe, velocity profile 66 units consistency of 8 consistent set of 91 conversion of 84 V-tube inverted 1 vane, force on 49, 54 vapour lock 195 variables dependent 1 independent 1 velocity coefficient of 407
INDEX gradient 10, 59 head 189, 86 of liquid falling from a pipe 177 vena contracta 1 venturi meter 11 advantages and disadvantages coefficient of discharge 17 inclined 14 venturi, pressure variation in 178 vertical take-off aircraft, forces on 79 viscometer 59 viscosity 4-1, 0, 57 coefficient of 5, 11 dynamic 79 kinematic 79 v-notch weir, flow rate over 1 volume flow rate 157 water jet cutting 00 waves, speed of on the surface of the sea 5 weir, sharp-edged 9 windmill, power 0 wind tunnel, closed-circuit 6 work 168 408