# Chapter 1 Fluid and their Properties

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1 June -15 Jan - 16 GTU Paper Analysis (New Syllabus) Chapter 1 Fluid and their Properties Sr. No. Questions Theory 2. Explain the following terms: Relative density 2. Kinematic viscosity 3. Cavitation 4. Vapour pressure 5. Continuum 6. Compressibility. Capillary effect Define or explain following terms Newton s law of viscosity 2. Capillarity Examples 2 2. Calculate the shear stress developed in oil of viscosity 2 poise, used for lubricating the clearance between a shaft of diameter 12 cm and its journal bearing. The shaft rotates at 180 rpm and clearance is 4 mm. A 150 mm diameter vertical cylinder rotates concentrically inside cylinder of diameter 151 mm. both the cylinders are 250 mm high. The space between the cylinders is filled with a liquid whose viscosity is unknown. If a torque of 12 Nm is required to rotate the inner cylinder at 100 rpm, determine the viscosity of the liquid. Fluid Mechanics ( )

2 Dec - 09 Mar -10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 June 15 Chapter 1 Fluid and their Properties Sr. No. Questions Theory Explain the continuum concept used in fluid mechanics Explain the difference between the behavior of solids and fluids under an applied force With neat sketch write about the Newtonian and Non Newtonian fluids 3 4. Derive an expression for capillary rise (depression) between two vertical parallel plates Define the capillarity and derive an expression for capillary rise h=4s/ρgd where, h= rise of water, s=surface tension of liquid, 4 4 ρ=density of liquid and d= diameter of droplet 6. Explain the phenomenon of capillarity. Obtain and expression for capillary rise and capillary fall of a liquid (i) Define capillarity and develop a formula for capillary rise of. a liquid between two concentric glass tubes with usual notations. (ii) State and explain Newton s law of viscosity What is the difference between dynamic viscosity and kinematic viscosity? State their units What is compressibility? Derive an expression for it? Define following (any seven) - Density - Dynamic viscosity 4 4 4

3 - Kinematic viscosity - Capillary - Bulk modulus of elasticity - Surface tension - Vapor presser - Cavitation - Cohesion - Adhesion - Continuum - Non-Newtonian Fluid 1 What is fluid? How fluid is differing from solid? Define viscosity, surface tension, compressibility and vapour pressure. Examples A flat plate 30 cm x 50 cm slides on oil ( µ = 0.8 N-s/m2 ) over a large plane surface. What is the force required to drag the plate at 2m/s if separating oil film is 0.4 mm thick? A disc of 100 mm diameter rotates on a table separated by an oil film of 2mm thickness. Find the torque required to rotate the disc at 60 rpm, if the dynamic viscosity of oil is 0.05 poise. Assume the velocity gradient in oil film to be linear. A plate 0.05 mm distance from a fixed plate moves at 2 m/s and requires a force of 2.2 N/m2 to maintain this speed, find the viscosity of the fluid between the plates. A soap bubble 62.5 mm diameter has an internal pressure in excess of the outside pressure of 20 N/m2.what is the tension in soap film? The capillary rise in the glass tube is not to exceed 0.4 mm of water. Determine its minimum size, given that surface tension for water in contact with air s =0.025 N/m and contact angle θ = 25. 4

4 6 8 9 A rectangular plate, 1m X 0.5m, weighing 980. N slides down a 30º inclined surface at a uniform velocity of 2.0 m/sec. If the 2 mm gap between the plate and the inclined surface is filled with caster oil, determine the viscosity (in Poise) of the castor oil. An oil of viscosity 4 poise is used for lubrication between a shaft and sleeve. The diameter of the shaft is 0.5 m and it rotates at 250 rpm. Calculate the power lost in oil for a sleeve length of 100 mm. The thickness of oil film is 1 mm. A 50 mm diameter shaft rotates with 500 rpm in a 80 mm long journal bearing with 51 mm internal diameter. The annular space between the shaft and bearing is filled with lubricating oil of dynamic viscosity 1 poise. Determine the torque required and power absorbed to overcome friction. Calculate the capillary rise in a glass tube of 3mm diameter inserted in water. Surface tension for water is 0.05 N/m. What will be the percentage increase in capillary height if the diameter of glass tube is 2mm. 4

5 Jun - 15 Jan - 16 Chapter 2 Pressure & Head GTU Paper Analysis (New Syllabus) Sr. No. Questions Theory Enlist the various Mechanical gauges for pressure measurement and describe their working with suitable diagram. 2. State and prove pascal s law with usual notations. 04 Examples Find the depth of point below sea water surface where the pressure intensity is kn/m 2. The specific gravity of sea water is Fluid Mechanics ( )

6 Mar-10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 June 15 Dec 15 Chapter 2 Pressure & Head Sr. No. Questions 2. Theory Define atmospheric pressure, gauge pressure and absolute pressure. State and prove Pascal s law for static fluid. Also mention its application. OR Prove that Intensity of pressure at any point in a fluid at rest is same in all direction What do you mean by gauge pressure, vacuum pressure and absolute pressure? Explain the working principle of U-tube differential manometer with neat sketch State Pascal s law and hydrostatic law Describe vertical single column manometer? How will you measure the fluid pressure with it? OR Define manometer. List different types of manometer. Explain single column manometer with usual notation. State advantages and limitations of manometer Enlist types of manometers. Differentiate between u-tube manometer and u-tube differential manometer. 6. Explain: Bourden tube pressure gauge and Diaphragm gauge 0

7 . What is fluid statics? What are the forces acting on the fluid particle in fluid statics? State Pascal law. Also prove that pressure at same level in static fluids is equal. Examples 0 A U-tube manometer is used to measure the pressure of water in a pipe line, which is in excess of atmospheric pressure. The right limb of the manometer contains mercury and is open to atmosphere. The contact between water and mercury is in the left limb. Determine the pressure of water in the main line, if the difference in level of mercury in the limbs of U-tube is 12cm and the free surface of mercury is in level with the centre of the pipe. If the pressure of water in pipe line is reduced to 9810 N/m2. Calculate the new difference in the level of mercury The left limb of a u-tube manometer containing mercury is connected to a pipe in which a fluid of specific gravity 0.85 is flowing. The right limb is open to atmosphere. Mercury levels in the left and right limbs are 0.12 m and 0.4 m below center of the pipe respectively. Calculate the vacuum pressure in the pipe. Also express this vacuum pressure in terms of: i) absolute pressure and, ii) m of water. Take atmospheric pressure as 103 KN/m2 and specific gravity of mercury as (i)distinguish between absolute pressure and gauge pressure. An open tank contains water up to a depth of 2m and above it, oil of specific gravity 0.9 for 1m depth. Find the pressure at the bottom of the tank (ii)state and discuss: Hydrostatic law of pressure variation. An inverted differential manometer is connected with two pipes A and B in which water is flowing as shown in fig. The manometric fluid is oil of specific gravity 0.8. Refer the figure and find the 0 0 0

8 pressure difference between A and B.

9 Jun - 15 Jan - 16 GTU Paper Analysis (New Syllabus) Chapter 3 Static Forces on Surface and Buoyancy Sr. No. Questions Theory Derive expressions for total pressure and centre of pressure for vertically immersed surface 2. Show that the distance between the meta-centre and centre of buoyancy is given by 3. Explain the conditions of stability for a submerged and floating body with neat diagrams. Examples A rectangular pontoon 8 m long, 6 m wide and 2 m deep, floats in sea water (sp. weight = N/m 3 ). It carries an empty boiler on its upper dock of 4 m diameter. The weight of pontoon and boiler are 600 kn and 200 kn respectively. The center of gravity of each unit coincides with geometric centre of the arrangement and lie on same vertical line. Find the metacentric height of arrangement and check the stability. A block of wood has a horizontal cross section 500 mm X 500 mm and height h.it floats vertically in water. If the specific gravity of wood is 0.6, find the maximum height of block so that it can remain in stable equilibrium. A barge in the shape of a rectangular block 8 m wide, 12.8 m long and 3 m deep floats in water with a draft of 8 m. the centre of gravity of the barge is 0.3 m above the water surface. State whether the barge is in stable equilibrium. Calculate the righting moment when the barge heels by Fluid Mechanics ( )

10 Mar - 10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 June 15 Chapter 3 Static Forces on Surface and Buoyancy Sr. No. Questions Theory Define the following terms: (I) Total pressure (II) Centre of pressure (III) Force of buoyancy (IV) Metacentre Derive the expression for total pressure and centre of pressure for a vertical plate submerged in the liquid with usual notations Explain the condition of stability for a submerged body Show that the distance between the meta-centre and center of buoyancy is given by BM=I/. OR Define metacenter and metacentric height. Explain analytical method for determination of metacentric height. 5. Prove that the centre of pressure for any immersed surface always lies below its centroid. Define buoyant force, center of buoyancy, metacenter and 6. metacentric height. Also describe conditions of equilibrium for floating and submerged bodies.. For inclined immersed surface derive with usual notations, expression for total pressure and center of pressure. Examples A body has the cylindrical upper portion of 4m diameter and 2.4m deep. The lower portion, which is curved, displaces a volume of 800 litres of water and its centre of buoyancy is situated 2.6m below the

11 2. 3. top of the cylinder. The centre of gravity of the whole body is 6m below the top of the cylinder and the total displacement of water is 52 kn. Find the 2etacentric height of the body. A solid cylinder of diameter 4 m has a height of 4 m. Find the Metacentric height of the cylinder if the specific gravity of the material of cylinder is 0. and it is floating in water with its axis vertical. State whether the equilibrium is stable or unstable. A wooden block of specific gravity of 0. and dimensions 18 cm wide, 30 cm deep and 100 cm long floats horizontally on 18 cm wide surface in water. Calculate the 2etacentric height and comment on the stability of the block. If the block is given a tilt of 6o in the clockwise direction. Calculate what should be the mass should be kept at a distance from the centre 5 cm on the opposite side of offset the tilt A square plate of diagonal 2m is immersed in a liquid with its diagonal vertical and upper corner is 0.5m below the free surface of the liquid. The specific gravity of the liquid is 4. Find, (i) The force exerted by liquid on the plate. (ii) The position of its centre of pressure A solid cylinder 2.5m in diameter and 2.5m high is floating in water with its axis vertical. If the specific gravity of cylinder material is 0., Find 2etacentric height. Also state whether the equilibrium is stable or unstable. Determine the total pressure on a circular plate of diameter 5 m which is placed vertically in water in such a way that the centre of the plate is 3 m below the free surface of water. Find the position of centre of pressure also. A pontoon of KN displacement is floating in water a weight of KN is moved through a distance of 8 m across the deck of

12 pontoon which tilts the pontoon through an angle of 4 find the 3etacentric height of the pontoon. An equilateral triangular plate having 2.5 m side is immersed in water with its base coinciding with the free surface. Calculate total force and center of pressure if, i) the plate is vertical and, ii) angle of inclination of the plate with the free surface is 60 o. A wooden block of specific gravity 0.5 floats in water. If the size of the block is 1m X 0.5 m X 0.4 m, find its 3etacentric height. An isosceles triangular plate of base 3 m and altitude 3 m is immersed vertically in an oil of specific gravity 0.8. The base of plate coincides with the free surface of oil. Determine total pressure on the plate. Distinguish between centre of pressure and centre of gravity. A rectangular plate 3m X 5m is immersed vertically in water such that the 3m side is parallel to the water surface. Determine the hydrostatic force and the depth of centre of pressure if the top edge of the plate is 2m below water surface. A rectangular body 2 m long, 1 m wide and 0.8 m deep floats in water. The depth of immersion is 0.6 m. What is the weight of the body? Is the body in stable equilibrium? A cylindrical block weight 22 KN having diameter 2m and height 2.5m is to float in sea water (S=025), show that it does not float vertically. Fig. 2 shows a gate having a quadrant shape of radius of 1 m subjected to water pressure. Find the resultant force and its inclination with the horizontal. Take the length of gate as 2 m

13 A solid cube of sides 0.5 m each is made of a material of relative density 0.5. The cube floats in a liquid of relative density 0.95 with two of its faces horizontal. Examine its stability. A logwood 1meter diameter and 2.5 meter long is floating in water. Calculate the depth of immersion of logwood in water if the specific gravity of logwood is 0.6. The weight of stone is 530N in air and reduces to 200N while submerging it into water find the specific gravity of stone

14 Jun 15 Jan 16 GTU Paper Analysis (New Syllabus) Chapter 4 & 6 Kinematics Sr. No. Questions Theory Explain clearly: stream line; path line and streak line Distinguish clearly between: Rotational and Irrotational flow 2. Laminar and Turbulent flow. 3. Derive the continuity equation in Cartesian co-ordinates 0 Examples 0 Velocity for a two dimensional flow field is given by V=(3+2xy+4t2)i+(xy2+3t)j Find the velocity and acceleration at appoint (1,2) after 2 sec. 05 Fluid Mechanics ( )

15 Mar -10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 Jun 15 Dec 15 Chapter 4 & 6 Kinematics Sr. No. Questions Theory Define: - (i) path line (ii) stream line (iii) stream tube 3 2. State the characteristics of Laminar flow and Turbulent flow Derive an expression of stream function and velocity potential function for vortex flow. Define and explain circulation. What is the importance of concept of circulation? OR Define circulation. Prove that circulation Γ = ξda with usual notations. 5. Explain Flow-net and state the importance of flow net Derive an expression for continuity for 3-D flow and reduce it for steady, incompressible 2-D flow in Cartesian coordinate system. Explain briefly : (I) Steady flow and unsteady flow (II) Uniform flow and non uniform flow (III) Laminar and turbulent flow (IV) Compressible and incompressible flow (V) Rotational and irrotational flow Explain the following in brief : (I) Stream function (II) Velocity potential function (III) Circulation

16 (IV) Flow net 9. Distinguish between forced vortex and free vortex flow What do you mean by equipotential line and a line of constant stream function? Explain the following terms: (i) Laminar and turbulent flow. (ii) Draft tube (iii) Stream tube. (iv) Velocity potential function Velocity component of a fluid flow are given as Examples 4 1 Where, x, y, z are given in meter and time t in sec. Determine velocity vector at point P (4, 1, 2) at time t = 4 sec. Also 5 determine the magnitude of velocity and acceleration of the flow for given location and time. An open circular cylinder of 15 cm diameter and 100 cm long 2. contains water up to a height of 80 cm. Find the maximum speed at which the cylinder is to be rotated about its vertical axis so that no water spills. The stream function for a two dimensional flow is given by ψ = 3xy. 3. Calculate the velocity at point P (2, 4). Also find the velocity potential 4 function, Φ. Explain the terms: Rotational flow, relative equilibrium, buoyant 4. force. A tank, 1 m wide, 6 m long and 2.5 m deep is open at top. It contains water to a depth of 2 m. If the tank is accelerated at 2 m/s 2, calculate the volume of water spilled out of the tank. 5. Differentiate between streamline and equipotential line. For a 2-D flow field, the velocity potential is given as Φ = 2xy x. Determine the stream function Ψ at a point P(2, 2). 6. The velocity component in a two-dimensional flow field for an

17 incompressible fluid are as follows: u = y 3 /3 + 2x - x 2 y and v = xy 2 2y x 3 /3 Obtain an expression for the stream function ψ.. 8. A vessel, cylindrical in shape and closed at the top and bottom, contains water up to a height of 80 cm. The diameter of the vessel is 20 cm and length of vessel 120 cm. The vessel is rotated at a speed of 400 r.p.m. about its vertical axis. Find the height of parabola formed. A stream function in a two-dimensional flow is ψ=2xy. Calculate the velocity at the point (3, 2). Find the corresponding velocity potential φ.

18 June -15 Jan - 16 Chapter 5 The Energy Equation and its Application GTU Paper Analysis (New Syllabus) Sr. No. Questions Theory State assumptions for Bernoulli s equations. Derive Bernoulli s equations 2. Derive euler s equation of motion. 3. Derive the expressions for discharge over (i) Rectangular notch and (ii) Triangular notch. 4. Derive the expression for time required to emptying a tank through an orifice at its bottom. Examples A 300 mm X 150 mm venturimeter is placed vertically with throat 250 mm above the inlet section conveys kerosene of density 820 kg/m3. The flow rate is 140 litre/sec. calculate the pressure difference between inlet and throat section. Take C d = 0.9. A crude oil of viscosity 0.9 poise and relative density 0.9 is flowing through a horizontal circular pipe of 3. diameter 120 mm and length 12 m. calculate the difference of pressure at the two ends of the pipe, if 85 N of the oil is collected in tank in 25 seconds. Fluid Mechanics ( )

19 Dec - 09 Mar -10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 June 15 Chapter 5 The Energy Equation and its Application Sr. No. Questions Theory Derive an expression for the measurement of velocity of flow at any point in a pipe or channel by Pitot tube. 2. With usual notations derive the expression for the discharge through a triangular notch. 3. Derive Euler s equation of motion along a streamline and hence obtain Bernoulli s equation clearly state the assumption made. 4. Explain the construction and working of a Venturimeter and 5. also derive an expression for the discharge through it. Compare a triangular notch with a rectangular notch for measuring discharges and derive an expression for the discharge through a triangular notch. 6. State Bernoulli s theorem. 3. What are the advantages of triangular notch over rectangular 3 notch? 8. Find an expression for the discharge over a rectangular notch Define Kinetic Energy correction factor and momentum 3 3 correction factor. 10. State assumptions for Bernoulli s equations. Derive Bernoulli s equations 1 Explain: Vena-contracta. Discuss the characteristics of flow at 3 vena-contracta, in case of an orifice. 12. State Bernoulli s theorem. What are the assumptions in 3

20 Bernoulli s theorem? Sketch the VM and show all the component parts. Why is the length of diverging cone kept larger than the length of converging cone? What is fluid statics? What are the forces acting on the fluid particle in fluid statics? State Pascal law. Also prove that pressure at same level in static fluids is equal. Examples A sharp edged orifice, 5 cm in diameter, in the vertical side of large tank discharges water under a head of 5 m. If Cc = 0.62 and C = 0.98, determine ( a ) the diameter of jet at the vena contracta, ( b ) the velocity at the vena contracta and ( c ) discharge in m3/s. Explain momentum correction factor required for the flow past a section. A liquid flows through the circular pipe 0.6 m diameter. Measurements of velocity taken at interval along a diameter as under. Distance From wall,m Velocity,m/s The total momentum per unit time is kg. Find the true momentum in context of average velocity and find the momentum correction factor. The pipe AB is of uniform diameter ( Figure) and the pressure at A and B are 150 and 250 kn/m2 respectively. Find the direction of the flow and head loss in meters of liquid if liquid has a specific gravity of

21 Describe the procedure of measurement of velocity with the Pitot tube. Find flow rate of water for venturimeter if mercury manometer reads y = 10 cm for the case where D1 = 20 cm and D = 10 cm and z = 0.45 m ( Refer Figure) What is cipolleti weir? For cipolletti weir, derive the slope ( ¼ : 1 ) of the sides of the trapezoidal notch by setting the reduction in discharge due to contraction equal to the increase in discharge due to triangulation area added. In an experiment on 90 0 Vee notch, the flow is collected in a 0.90 m diameter vertical cylindrical tank. It is found that the depth of water increases by m in 16.8 seconds when the head over

22 the notch is 0.2 m. Determine the coefficient of discharge. If the error in observation of head over the notch is 1mm,what will be the error in discharge? The water is flowing through a taper pipe of length 100 m having diameter 600 mm at the upper end and 300 mm at the lower end, at the rate of 50 litres/sec. The pipe has a slope of 1 in 30. Find the pressure at the lower end if the pressure at the higher level is * 10 4 N /m2& lower end is 10 m above datum. A horizontal venturimeter of 200 mm X 100 mm is used to measure the discharge of an oil of specific gravity A mercury manometer is used for the purpose. If the discharged is 100 litres per second and if the coefficient of discharge of the venturimeter is 0.9, find the difference of mercury level in between two limbs of manometer. The inlet and throat diameters of a vertically mounted venturimeter are30 cm and 10 cm respectively. The throat section is below the inlet section at a discharge of 10 cm. The specific gravity of the liquid is 900Kg/m3. The intensity of pressure at inlet is 140 KPa and the throat pressure is 80 KPa. Calculate the flow rate in Lps.. Assume that 2 % of the differential head is lost between inlet and throat. Take coefficient of discharge 0.9. A horizontal Venturimeter with inlet diameter 20cm and throat diameter 10cm is used to measure the flow of oil of sp.gr 0.8. The discharge of oil through venturimeter is 60 Liters/Second. Find the reading of the oil mercury differential manometer take Cd = The head of water over an orifice of diameter.5 cm is.5 m. The jet of water coming out from the orifice is collected in a tank having cross-sectional area of 1 m x 1 m. The rise of water level in this tank is 0.8 m in 25 seconds. The coordinates of a point

23 on the jet measured from venacontracta are 3.5 m horizontal and 0.5 m vertical. Find the co-efficient of discharge, co-efficient of velocity and coefficient of contraction. A horizontal venturimeter with inlet and throat diameters 0.3 m and 0.15 m respectively is used to measure the flow of water in a pipe. The reading of differential manometer connected to the inlet and the throat is 0.25 m of mercury. Determine the rate of flow, if the coefficient of discharge is 0.9. The water is flowing through a tapering pipe having diameters 300 mm and 150 mm at section 1 and section 2 respectively. The discharge through the pipes is 40 liters / sec. The section 1 is 10 m above datum and section 2 is 6 m above datum. Find the intensity of pressure at section 2 if that at section 1 is 400 kn/m2 The co-ordinates of a point on a free jet issued from an orifice under the head of 8m are (5, 0.4). If the dia. of the orifice is 2cm and origin of co-ordinate system is the centre of the jet at vena-contracta, find coefficient of velocity. A horizontal venturimeter (VM) connected to a pipe of 20 cm dia. has 10 cm dia. throat. The difference of pressure between the inlet and the throat is measured by a differential mercury manometer, which shows the deflection of 30 cm. If the coefficient of discharge of VM is 0.9, calculate the discharge of water passing through the pipe. A pipe line carrying oil of specific gravity 0.9, changes in diameter from 250mm diameter at a position 1 to 450 mm diameter at a position 2 which is 6 meter at a higher level. If the pressure at 1 and 2 are 12 N/cm2 and 6 N/cm2 respectively and the discharge is 250 litre/sec. calculate the loss of head and direction of flow. 1 A rectangular channel 2 m wide has a discharge of 250 liters/sec, which is measured by a right angled V-notch. Find the position of 4 4

24 18 the apex of the notch from the bed of the channel if maximum depth of water is not to exceed 3 m. Take Cd = Find the discharge of water flowing through a pipe 30 cm diameter placed in an inclined position where a venture meter is inserted, having a throat diameter of 15 cm. The difference of pressure between the main and throat is measured by a liquid of specific gravity 0.6 in an inverted U-tube which gives a reading of 30 cm. The loss of head between the main and throat is 0.2 times the kinetic head of the pipe.

25 June -15 Jan - 16 Chapter Dimensional Analysis and Similarity GTU Paper Analysis (New Syllabus) Sr. No. Questions Theory State Buckingham s π-theorem. What do you mean by repeating variables? How the repeating variables are selected in dimensional analysis? 5 Define the following dimensionless numbers: (I) Reynold s No. (II) Froude No. (III) Euler s No. (IV) Mach No. 2. OR State the various dimensionless numbers with their significance in fluid flow situations. 3. Define or explain following terms Dimensional homogeneity, Derived units Examples 2 The frictional torque T of a disc of diameter D rotatin viscosity µ and density ρin a turbulent flow is given by, Prove this by Buckingham s p method Using the method of dimensional analysis obtain an expression for the discharge Q over a rectangular weir. The discharge depends on the head H over the weir, acceleration due to ravity g, length of the weir crest over the channel bottom Z and the kinematic viscosity v of the liquid. Fluid Mechanics ( )

26 Dec - 09 Mar -10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 June 15 GTU Paper Analysis(Old Syllabus) Chapter Dimensional Analysis and Similarity Sr. No. Questions Theory State similarity laws. Where are they used? Explain Froude, Euler and Weber model law with applications. State Buckingham s π-theorem. What do you mean by repeating variables? How the repeating variables are selected in dimensional analysis? Define the following dimensionless numbers: (I) Reynold s No. (II) Froude No. (III) Euler s No. (IV) Mach No. OR State the various dimensionless numbers with their significance in fluid flow situations. Discuss different types of similarities that must exist between a prototype and its model Explain Buckingham s Π theorem for dimensional analysis. 6. Explain euler s, weber and mach model law.. Prove that the scale ratio for discharge for a distorted model is given as Q p /Q m = (Lr) H (Lr) v 5 8. What is similarity? What are the various types of it? Explain. 9. Define or explain following terms Dimensional homogeneity, Derived units

27 10. Define: Reynolds number. Also, explain Reynolds model law. Examples Using Buckingham s π-theorem, show that the lift F L on airfoil can be expressed as, [ ] GTU Paper Analysis(Old Syllabus) Where, ρ = mass density, V = velocity of flow, d = characteristic depth, μ = co-efficient of viscosity, α = angle of incidence. The pressure difference ΔP in a pipe of diameter D and length l due to turbulent flow depends on the velocity v, viscosity μ, density ρ and roughness k. Using Buckingham s π-theorem, obtain an expression for ΔP. Derive on the basis of dimensional analysis suitable parameters to present the thrust developed by propeller. Assume that thrust P depends upon the angular velocity ω, speed of advance V, diameter D, dynamic viscosity μ, mass density ρ, elasticity of the fluid medium which can be denoted by speed of the sound in the medium C. The efficiency η of a fan depends on the density ρ, the dynamic viscosity μ of the fluid, the angular velocity ω, diameter D and discharge Q. Express efficiency η in terms of dimensionless parameters by using Buckingham s-π theorem. A pipe of 4m in diameter is required to transport an oil of specific gravity 0.8 and dynamic viscosity 0.04 poise at the rate of 2500 litres per second. Test were conducted on a150mm diameter pipe using water at 20 C. The viscosity of water at 6

28 C is 0.01 poise. Find the rate of flow in the model. A ship 250 m long moves in sea-water, whose density is 1025 kg/m3. A 1:100 model of this ship is to be tested in a wind tunnel. The velocity of air in the wind tunnel around the model is 25 m/s and the resistance of the model is 50 N. Determine the velocity of ship in sea-water and also resistance of the ship in sea-water. The density of air is 24 kg/m2, the kinematic viscosity of sea-water is stokes and viscosity of air is stokes. The efficiency of fan depend upon diameter of rotor, discharge of fluid, density of fluid, dynamic viscosity of fluid and angular velocity of rotor. Find the expression for efficiency in terms of dimensionless number. The resisting force R of a supersonic plane during flight can be considered as dependent upon the length of the aircraft l, velocity V, viscosity of air µ, air density ρ and bulk modulus of air K. Express the functional relationship between these variables with the resisting force. The lift force F on the air foil depends upon the mass density ρ of the medium, velocity of flow V, characteristic length d, the viscosity µ,and angleof attach a. obtain an expression for the lift force by using Buckingham s pi-theorem. The frictional torque T of a disc of diameter D rotatin viscosity µ and density ρin a turbulent flow is given by, Prove this by Buckingham s p method Using the method of dimensional analysis obtain an expression for the discharge Q over a rectangular weir. The discharge depends on the head H over the weir, acceleration due to GTU Paper Analysis(Old Syllabus)

29 12 ravity g, length of the weir crest over the channel bottom Z and the kinematic viscosity v of the liquid. Show, using Buckingham s p-theorem, that the resistance (F) to the motion of a sphere of diameter (D) moving with a uniform velocity (V) through a real fluid of density (ρ) and viscosity (µ) is given by: GTU Paper Analysis(Old Syllabus)

30 Jun - 15 Jan - 16 Chapter 8 Viscous Flow GTU Paper Analysis (New Syllabus) Sr. No. Questions Theory Derive the Hagen Poiseuille equation for laminar flow in the circular pipe. 0 Examples 2. A crude oil of viscosity 0.9 poise and relative density 0.9 is flowing through a horizontal circular pipe of diameter 120 mm and length 12 m. calculate the difference of pressure at the two ends of the pipe, if 85 N of the oil is collected in tank in 25 seconds. Calculate the shear stress developed in oil of viscosity 2 poise, used for lubricating the clearance between a shaft of diameter 12 cm and its journal bearing. The shaft rotates at 180 rpm and clearance is 4 mm. 0 0 Fluid Mechanics ( )

31 Mar -10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 Jun 15 Dec 15 Chapter 8 Viscous Flow Sr. No. Questions Theory Derive an expression for velocity distribution for viscous flow through a circular pipe. Also sketch the velocity distribution and shear stress distribution across the section of a pipe. OR What is Hagen Poiseuille s formula? Derive an expression for Hagen Poiseuille s formula. 2. Write short note on Say bolt viscometer Derive an expression for power absorbed in overcoming viscous resistance in case of a journal bearing. Show that the value of the co-efficient of friction for viscous flow through a circular pipe is given by f =- 16/Re where Re = Reynolds number. 5. Explain Dash pot mechanism and its utility Derive an expression for power absorbed in foot step bearing.. Explain Reynolds experiment with neat sketch. Examples 4 Two parallel plates 80 mm apart have laminar flow of oil between them with maximum velocity of flow is 5 m/s. Calculate : (I) Discharge per meter width (II) Shear stress at the plate (III) The difference in the pressure between two points 20 meter apart. (IV) Velocity gradient at the plates. (V) Velocity at 20 mm from the plate.

32 Assume viscosity of oil 24.5 poise. A laminar flow is taking place in a pipe of diameter of 200 mm. The maximum velocity is 5 m/sec. Find the mean velocity and the radius at which this occurs. Also calculate the velocity at 4 cm from the wall of the pipe. A pipe of 25 cm diameter and 900 m long is carrying an oil of mass density 940 kg/m3 and dynamic viscosity 1Ns/ m3. The oil flow rate is 0.15 m3/sec. Due to increase in atmospheric temperature the viscosity of oil changes by a factor of 8. If the same quantity of oil is to be conveyed compare the cost of pumping. A 50 mm diameter shaft rotates with 500 rpm in a 80 mm long journal bearing with 51 mm internal diameter. The annular space between the shaft and bearing is filled with lubricating oil of dynamic viscosity 1 poise. Determine the torque required and power absorbed to overcome friction. An oil of viscosity 4 poise is used for lubrication between a shaft and sleeve. The diameter of the shaft is 0.5 m and it rotates at 250 rpm. Calculate the power lost in oil for a sleeve length of 100 mm. The thickness of oil film is 1 mm. 6. Two plates are placed at distance of 0.2 mm apart. Lower plate is fixed while upper plate having surface area one m2 is pulled at 0.2 m/s. Find force and power required to maintain this speed. If the fluid separating them is having viscosity of 2.0 poise.. A crude oil of viscosity 0.95 poise and relative density 0.9 is flowing through a horizontal circular pipe of diameter 100 mm and a length of 5 meter long. Calculate the difference of pressure at the two ends of the pipe, if flow rate is m3/s.

33 Jun - 15 Jan - 16 Chapter 9 Turbulent Flow GTU Paper Analysis (New Syllabus) Sr. No. Questions Theory Derive an expression for loss of head due to friction in pipe flow. 0 0 Fluid Mechanics ( )

34 Mar -10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 Jun 15 Dec 15 Chapter 9 Turbulent Flow Sr. No. Questions Theory Derive Darcy-Weisbach equation for the co-efficient of friction in pipes. OR Derive an expression for the loss of head due to friction in pipes. OR What do you understand by the terms major energy loss and minor energy losses in pipe? Derive Darcy-Weisbach equation with usual notations. What is velocity defect? Derive an expression for the velocity defect for turbulent flow in pipes. Derive an expression for shear stress on the basis of Prandtl mixing length theory Write a short note on moody diagram for calculating the head loss due to friction. 6 3

35 June -15 Chapter 11 Flow Through Pipe GTU Paper Analysis (New Syllabus) Sr. No. Questions Theory Derive an expression for the loss of head due to friction in pipes. 5 Examples A pipe AB branches into two pipes BC and BD. The pipe has diameter of 30 cm at A, 20 cm at B, 15 cm at C and 10 cm at D. Determine the discharge at A if flow velocity at A is 2.5 m/s. Also find the velocity at B and D, if the velocity at C is 4.2 m/s. Fluid Mechanics ( )

36 June -15 Jan - 16 GTU Paper Analysis (New Syllabus) Chapter 11 Compressible Fluid Flow Sr. No. Questions Theory Prove that the velocity of sound wave in compressible fluid is given by C = kρ 0 Examples A supersonic aircraft flies at an altitude of 8 km where temperature is 4o C. Determine the speed of the aircraft if its sound is heard 4 seconds after its passage over the head of an observer. Take R = 28 J/kg K and = 4. 0 Fluid Mechanics ( )

37 Dec - 09 Mar -10 Dec - 10 May - 11 Dec 11 May - 12 Jan - 13 May - 13 Nov - 13 May - 14 Dec 14 June 15 Chapter 11 Compressible Fluid Flow Sr. No. Questions Theory Explain Zone of action, Zone of silence, Mach angle, Mach number and Mach-cone with the help of diagram. 2. With the help of diagram, explain relationship between mass flow rate and pressure ratio for compressible flow. Prove that velocity of sound wave in a compressible fluid is 3. given by C= (k/ρ) Where k= Bulk modulus of fluid and ρ = Density of fluid. Derive an expression for the velocity of sound wave in a 4. compressible fluid in terms of change of pressure and change of density. Derive and expression for area velocity relationship for a 5. compressible fluid in the form with usual notation da/a= (dv/v)[m³-1] 6. Define: Sub-sonic flow, Super-sonic flow, Mach angle and Mach cone. Explain: Mach cone, Zone of silence, stagnation pressure,. adiabatic process. Derive the energy equation for compressible flow in an adiabatic process Explain the propagation of pressure waves when the speed of the object is more than the speed of sound. Also, derive 8. the relation between Mach number and Mach angle. OR Explain the propagation of pressure waves with neat sketch

38 2. Examples Calculate the stagnation pressure temp. and density on the stagnation point on the nose of the plane which is flying at 800 km/hr through still air having a pressure 8 N/square cm (abs) and temp. 10 º c. Take R =28 J/Kg K, γ = 4. Calculate the stagnation pressure and temperature on the stagnation point on the nose of a plane, which is flying at 900 km/hr through still air having an absolute pressure 9.0 N/cm 2 and temperature -10 o C. Take R = 28 J/Kg K and k = A projectile is travelling in air having pressure and temperature as 0.1 N/mm 2 and 0 0 C. The mach angle is Calculate the velocity of the projectile. Assume R=0.28 kj/kg k. At what speed the shock wave propels in the flow in the air at 150 kn/m2,absolute, is moving at 150 m/s in the high pressure wind tunnel at 40 C 0.Take R=28. State whether the flow super sonic or not. A supersonic aircraft flies at an altitude of 8 km where temperature is 4o C. Determine the speed of the aircraft if its sound is heard 4 seconds after its passage over the head of an observer. Take R = 28 J/kg K and =

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