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CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^1
CIVE1400 – Fluid Mechanics Examples.This handout contains typical examples of basic engineering fluid mechanics. It covers all sectionsof the module. The examples are similar to what you will find in the final examination. You shouldattempt as many of these as possible so that you get accustomed to using the formula we havediscussed in the lectures. The examples marked ‘DIFFIULT’ should be left to the end.Solutions are available on the web page for this module:www.efm.leeds.ac.uk/CIVE/FluidsLevel1Fluid Properties Examples 1. Explain why the viscosity of a liquid decreases while that of a gas increases with a temperature rise.The following is a table of measurement for a fluid at constant temperature.Determine the dynamic viscosity of the fluid. -1 du/dy (s ) 1.^ 0.0^ 1.^ 0.2^ 1.^ 0.4^ 1.^ 0.
1.^ 0.
-2^ 2.^ 0.0^ 2.^ 1.0^ 2.^ 1.9 W (N m)^
2.^ 3.1^ 2.^ 4.
(^3) 2. The density of an oil is 850 kg/m. Find its relative density and Kinematic viscosity if the dynamic-3 (^) viscosity is 5 u 10 kg/ms. 3. The velocity distribution of a viscous liquid (dynamic viscosity
(^2) P = 0.9 Ns/m) flowing over a fixed (^2) plate is given by u = 0.68y - y(u is velocity in m/s and y is the distance from the plate in m).What are the shear stresses at the plate surface and at y=0.34m? (^3) 4. 5.6mof oil weighs 46 800 N. Find its mass density, U^ and relative density,^ J.
5.^ From table of fluid properties the viscosity of water is given as 0.01008 poises.^2 What is this value in Ns/mand Pa s units?6.^ In a fluid the velocity measured at a distance of 75mm from the boundary is 1.125m/s. The fluid hasabsolute viscosity 0.048 Pa s and relative density 0.913. What is the velocity gradient and shear stressat the boundary assuming a linear velocity distribution.
CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^2 Pressure and Manometers 1.1What will be the (a) the gauge pressure and (b) the absolute pressure of water at depth 12m below the^3 surface?^ U= 1000 kg/m, and p^ water^ atmosphere
(^2) = 101kN/m. (^2 2) [117.72 kN/m, 218.72 kN/m] 1.2At what depth below the surface of oil, relative density 0.8, will produce a pressure of 120 kN/m
2? What depth of water is this equivalent to?[15.3m, 12.2m]1.3^2 What would the pressure in kN/mbe if the equivalent head is measured as 400mm of (a) mercury
J=13.
(^3) (b) water ( c) oil specific weight 7.9 kN/m 3 (d) a liquid of density 520 kg/m? (^2 2 2) [53.4 kN/m, 3.92 kN/m, 3.16 kN/m, 2.04 kN/m
2 ]
1.4A manometer connected to a pipe indicates a negative gauge pressure of 50mm of mercury. What is theabsolute pressure in the pipe in Newtons per square metre if the atmospheric pressure is 1 bar?^2 [93.3 kN/m] 1.5What height would a water barometer need to be to measure atmospheric pressure of 1 bar?[>10.19m]1.6An inclined manometer is required to measure an air pressure of 3mm of water to an accuracy of +/- 3%.The inclined arm is 8mm in diameter and the larger arm has a diameter of 24mm. The manometric fluid^3 has density 740 kg/mand the scale may be read to +/- 0.5mm.What is the angle required to ensure the desired accuracy may be achieved?[7.6q] 1.7Determine the resultant force due to the water acting on the 1m by 2m rectangular area AB shown in thediagram below.[43 560 N, 2.37m from O 1.8Determine the resultant force due to the water acting on the 1.25m by 2.0m triangular area CD shown inthe figure above. The apex of the triangle is at C.^3 [23.8u^10 N, 2.821m from P]
CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^3 Forces on submerged surfaces 2.1Obtain an expression for the depth of the centre of pressure of a plane surface wholly submerged in afluid and inclined at an angle to the free surface of the liquid.A horizontal circular pipe, 1.25m diameter, is closed by a butterfly disk which rotates about a horizontalaxis through its centre. Determine the torque which would have to be applied to the disk spindle to keepthe disk closed in a vertical position when there is a 3m head of fresh water above the axis.[1176 Nm]2.2 (DIFFUCULT)A dock gate is to be reinforced with three horizontal beams. If the water acts on one side only, to a depthof 6m, find the positions of the beams measured from the water surface so that each will carry an equalload. Give the load per meter.[58 860 N/m, 2.31m, 4.22m, 5.47m]2.3The profile of a masonry dam is an arc of a circle, the arc having a radius of 30m and subtending an angleof 60q^ at the centre of curvature which lies in the water surface. Determine (a) the load on the dam in N/mlength, (b) the position of the line of action to this pressure.^6 [4.28^ u^10 N/m length at depth 19.0m]2.4 (DIFFUCULT)The arch of a bridge over a stream is in the form of a semi-circle of radius 2m. the bridge width is 4m.Due to a flood the water level is now 1.25m above the crest of the arch. Calculate (a) the upward force onthe underside of the arch, (b) the horizontal thrust on one half of the arch.[263.6 kN, 176.6 kN]2.5The face of a dam is vertical to a depth of 7.5m below the water surface then slopes at 30
q^ to the vertical. If the depth of water is 17m what is the resultant force per metre acting on the whole face?[1563.29 kN]2.6A tank with vertical sides is square in plan with 3m long sides. The tank contains oil of relative density0.9 to a depth of 2.0m which is floating on water a depth of 1.5m. Calculate the force on the walls and theheight of the centre of pressure from the bottom of the tank.[165.54 kN, 1.15m] Application of the Bernoulli Equation 3.1In a vertical pipe carrying water, pressure gauges are inserted at points A and B where the pipe diametersare 0.15m and 0.075m respectively. The point B is 2.5m below A and when the flow rate down the pipe is0.02 cumecs, the pressure at B is 14715 N/m
2 greater than that at A. Assuming the losses in the pipe between A and B can be expressed as
(^2) v k where^ v^ is the velocity at A, g 2 find the value of^ k. If the gauges at A and B are replaced by tubes filled with water and connected to a U-tube containingmercury of relative density 13.6, give a sketch showing how the levels in the two limbs of the U-tubediffer and calculate the value of this difference in metres.[ k^ = 0.319, 0.0794m]
CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^4 3.2A Venturimeter with an entrance diameter of 0.3m and a throat diameter of 0.2m is used to measure thevolume of gas flowing through a pipe. The discharge coefficient of the meter is 0.96.Assuming the specific weight of the gas to be constant at 19.62 N/m
3 , calculate the volume flowing when the pressure difference between the entrance and the throat is measured as 0.06m on a water U-tubemanometer.^3 [0.816 m/s]3.3A Venturimeter is used for measuring flow of water along a pipe. The diameter of the Venturi throat istwo fifths the diameter of the pipe. The inlet and throat are connected by water filled tubes to a mercuryU-tube manometer. The velocity of flow along the pipe is found to be
2 5.^ H^ m/s, where^ H^ is the manometer reading in metres of mercury. Determine the loss of head between inlet and throat of theVenturi when^ H^ is 0.49m. (Relative density of mercury is 13.6).[0.23m of water]3.4 (DIFFUCULT)Water is discharging from a tank through a convergent-divergent mouthpiece. The exit from the tank isrounded so that losses there may be neglected and the minimum diameter is 0.05m.If the head in the tank above the centre-line of the mouthpiece is 1.83m. a) What is the discharge?b) What must be the diameter at the exit if the absolute pressure at the minimum area is to be 2.44m ofwater? c) What would the discharge be if the divergent part of the mouth piece were removed. (Assumeatmospheric pressure is 10m of water).^3 3 [0.0752m, 0.0266m/s, 0.0118m/s] 3.5 (DIFFUCULT)A closed tank has an orifice 0.025m diameter in one of its vertical sides. The tank contains oil to a depthof 0.61m above the centre of the orifice and the pressure in the air space above the oil is maintained at^2 13780 N/mabove atmospheric. Determine the discharge from the orifice.(Coefficient of discharge of the orifice is 0.61, relative density of oil is 0.9).^3 [0.00195 m/s] 3.6 (DIFFUCULT)The discharge coefficient of a Venturimeter was found to be constant for rates of flow exceeding a certainvalue. Show that for this condition the loss of head due to friction in the convergent parts of the meter can^2 be expressed as^ KQm^ where^ K^ is a constant and
Q^ is the rate of flow in cumecs. Obtain the value of^ K^ if the inlet and throat diameter of the Venturimeter are 0.102m and 0.05mrespectively and the discharge coefficient is 0.96.[ K =1060]3.7A Venturimeter is to fitted in a horizontal pipe of 0.15m diameter to measure a flow of water which may^3 be anything up to 240m/hour. The pressure head at the inlet for this flow is 18m above atmospheric andthe pressure head at the throat must not be lower than 7m below atmospheric. Between the inlet and thethroat there is an estimated frictional loss of 10% of the difference in pressure head between these points.Calculate the minimum allowable diameter for the throat.[0.063m]3.8A Venturimeter of throat diameter 0.076m is fitted in a 0.152m diameter vertical pipe in which liquid ofrelative density 0.8 flows downwards. Pressure gauges are fitted to the inlet and to the throat sections.The throat being 0.914m below the inlet. Taking the coefficient of the meter as 0.97 find the dischargea) when the pressure gauges read the same b)when the inlet gauge reads 15170 N/m
2 higher than the throat gauge.^3 3 [0.0192m/s, 0.034m/s]
CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^7 Application of the Momentum Equation 6.1The figure below shows a smooth curved vane attached to a rigid foundation. The jet of water,rectangular in section, 75mm wide and 25mm thick, strike the vane with a velocity of 25m/s. Calculatethe vertical and horizontal components of the force exerted on the vane and indicate in which directionthese components act.[Horizontal 233.4 N acting from right to left. Vertical 1324.6 N acting downwards]^45 q
25 q 6.2A 600mm diameter pipeline carries water under a head of 30m with a velocity of 3m/s. This water main isfitted with a horizontal bend which turns the axis of the pipeline through 75
q^ (i.e. the internal angle at the bend is 105q). Calculate the resultant force on the bend and its angle to the horizontal.[104.044 kN, 52q^ 29’]6.3^3 2 A horizontal jet of water 2u^10 mmcross-section and flowing at a velocity of 15 m/s hits a flat plate at 60 q^ to the axis (of the jet) and to the horizontal. The jet is such that there is no side spread. If the plate isstationary, calculate a) the force exerted on the plate in the direction of the jet and b) the ratio between thequantity of fluid that is deflected upwards and that downwards. (Assume that there is no friction andtherefore no shear force.)[338N, 3:1]6.4A 75mm diameter jet of water having a velocity of 25m/s strikes a flat plate, the normal of which isinclined at 30q^ to the jet. Find the force normal to the surface of the plate.[2.39kN]6.5The outlet pipe from a pump is a bend of 45
q^ rising in the vertical plane (i.e. and internal angle of 135
q). The bend is 150mm diameter at its inlet and 300mm diameter at its outlet. The pipe axis at the inlet ishorizontal and at the outlet it is 1m higher. By neglecting friction, calculate the force and its direction if^2 the inlet pressure is 100kN/mand the flow of water through the pipe is 0.3m
3 /s. The volume of the pipe (^3) is 0.075m. [13.94kN at 67q^ 40’ to the horizontal]6.6The force exerted by a 25mm diameter jet against a flat plate normal to the axis of the jet is 650N. What^3 is the flow in m/s? (^3) [0.018 m/s] 6.7A curved plate deflects a 75mm diameter jet through an angle of 45
q. For a velocity in the jet of 40m/s to the right, compute the components of the force developed against the curved plate. (Assume no friction).[R=2070N, R=5000N down]x^ y
CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^8 6.8A 45q^ reducing bend, 0.6m diameter upstream, 0.3m diameter downstream, has water flowing through it^3 at the rate of 0.45m/s under a pressure of 1.45 bar. Neglecting any loss is head for friction, calculate theforce exerted by the water on the bend, and its direction of application.[R=34400N to the right and down,^ T^ = 14q
]
Laminar Pipe Flow 7.1The distribution of velocity, u, in metres/sec with radius r in metres in a smooth bore tube of 0.025 m^2 bore follows the law, u = 2.5 - kr. Where k is a constant. The flow is laminar and the velocity at the pipesurface is zero. The fluid has a coefficient of viscosity of 0.00027 kg/m s. Determine (a) the rate of flow^3 in m/s (b) the shearing force between the fluid and the pipe wall per metre length of pipe.-4^3 -3^ [6.14x10m/s, 8.49x10N] 7.2A liquid whose coefficient of viscosity is m flows below the critical velocity for laminar flow in a circularpipe of diameter d and with mean velocity u. Show that the pressure loss in a length of pipe is 32u
2 P/d.
Oil of viscosity 0.05 kg/ms flows through a pipe of diameter 0.1m with a velocity of 0.6m/s. Calculate theloss of pressure in a length of 120m.^2 [11 520 N/m] 7.3 (DIFFUCULT)A plunger of 0.08m diameter and length 0.13m has four small holes of diameter 5/1600 m drilled throughin the direction of its length. The plunger is a close fit inside a cylinder, containing oil, such that no oil isassumed to pass between the plunger and the cylinder. If the plunger is subjected to a vertical downwardforce of 45N (including its own weight) and it is assumed that the upward flow through the four smallholes is laminar, determine the speed of the fall of the plunger. The coefficient of velocity of the oil is 0.2kg/ms.[0.00064 m/s]7.4A vertical cylinder of 0.075 metres diameter is mounted concentrically in a drum of 0.076metres internaldiameter. Oil fills the space between them to a depth of 0.2m. The rotque required to rotate the cylinder inthe drum is 4Nm when the speed of rotation is 7.5 revs/sec. Assuming that the end effects are negligible,calculate the coefficient of viscosity of the oil.[0.638 kg/ms]
CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^9 Dimensional analysis 8.1A stationary sphere in water moving at a velocity of 1.6m/s experiences a drag of 4N. Another sphere oftwice the diameter is placed in a wind tunnel. Find the velocity of the air and the drag which will givedynamically similar conditions. The ratio of kinematic viscosities of air and water is 13, and the density^3 of air 1.28 kg/m. [10.4m/s 0.865N]8.2Explain briefly the use of the Reynolds number in the interpretation of tests on the flow of liquid in pipes.Water flows through a 2cm diameter pipe at 1.6m/s. Calculate the Reynolds number and find also thevelocity required to give the same Reynolds number when the pipe is transporting air. Obtain the ratio ofpressure drops in the same length of pipe for both cases. For the water the kinematic viscosity was-6^2 1.31u^10 m/s and the density was 1000 kg/m
3 -6^. For air those quantities were 15.1u^10 m
2 /s and (^3) 1.19kg/m. [24427, 18.4m/s, 0.157]8.3Show that Reynold number,^ Uud/P, is non-dimensional. If the discharge Q through an orifice is a functionof the diameter d, the pressure difference p, the density
1/2 U, and the viscosity P, show that Q = Cp 2 1/2d/U where C is some function of the non-dimensional group (d
1/2^ 1/2^ Ud/P). 8.4A cylinder 0.16m in diameter is to be mounted in a stream of water in order to estimate the force on a tallchimney of 1m diameter which is subject to wind of 33m/s. Calculate (A) the speed of the streamnecessary to give dynamic similarity between the model and chimney, (b) the ratio of forces.^3 Chimney:^ U^ = 1.12kg/mP^ = 16
-6^ u 10 kg/ms (^3) Model: U = 1000kg/mP^ = 8 -4^ u 10 kg/ms [11.55m/s, 0.057]8.5If the resistance to motion, R, of a sphere through a fluid is a function of the density
U^ and viscosity^ P^ of the fluid, and the radius r and velocity u of the sphere, show that R is given by
2 ur §·PU R f^ ¨^ ¸©¹UP Hence show that if at very low velocities the resistance R is proportional to the velocity u, then R = k
Pru where k is a dimensionless constant.A fine granular material of specific gravity 2.5 is in uniform suspension in still water of depth 3.3m.Regarding the particles as spheres of diameter 0.002cm find how long it will take for the water to clear.Take k=6S^ and^ P=0.0013 kg/ms.[218mins 39.3sec]
CIVE1400: Fluid Mechanics^
All Sections: Examples All Sections: Examples^
CIVE1400: Fluid Mechanics^10
