acm_fluid_121210

MECHANICS OF FLUIDS AMIE

POST BOX NO.77, 2 ND FLOOR, SULTAN TOWERS, ROORKEE – 247667 UTTARANCHAL PH: (01332) 266328 Email : [email protected] 1

TOTAL PAGES: 14

AMIE(I) STUDY CIRCLE(REGD.)

A Focused ApproachDated: 12-12-10

Before taking printout of this chapter, please ensure that this material has not been covered

in the printed course material provided to you (by us).

Additional Course Material

Physical Properties of Fluids

Example

Determine the power required to run a 300 mm dia shaft at 400 rpm in journals with uniform oil thickness of 1 mm. Two bearings of 300 mm width are used to support the shaft. The

dynamic viscosity of oil is 0.03 Pas. (Pas = (N/m2) × s).

Solution

Shear stress on the shaft surface = = (du/dy) = (u/y)

= DN/60 = x 0.3 x400/60 = 6.28 m/s

= 0.03 {(6.28 – 0)/ 0.001} = 188.4 N/m2

Surface area of the two bearings, A = 2DL

Force on shaft surface = x A = 188.4 x(2 xx0.3 x 0.3) = 106.6 N

Torque = 106.6 0.15 = 15.995 Nm

Power required = 2 NT/60 = 2 x × 400 x15.995/60 = 670 W.

P = LR3/450 h = 669.74 W

ASSIGNMENT

Q.1. (AMIE S05, 09, 10 marks): Define a fluid and distinguish between: (i) ideal and real fluids (ii) compressible and incompressible fluids (iii) Newtonian and non Newtonian fluids (iv) surface tension and capillarity

Q.2. (AMIE W06): Define (i) fluid (ii) Newtonian fluid

Q.3. (AMIE S06, 10 marks): Define (i) density (ii) specific gravity (iii) viscosity (iv) specific volume (v) cohesion and adhesion.

Q.4. (AMIE S06, 5 marks): Explain the following (i) surface tension (ii) compressibility

Q.1. (AMIE W07, S09, 10 marks): Define(i) specific gravity, (ii) vapour pressure, (iii) viscosity, (iv) compressibility, and (v) surface tension.

Q.2. (AMIE S08, 10 marks): Define (i) weight density, (ii) specific volume, (iii) capillarity, and (iv) Newton's law of viscosity.

OBJECTIVE QUESTIONS

P.B. NO.77, 2 ND FLOOR, SULTAN TOWERS, ROORKEE – 247667 UTTARANCHAL PH: (01332) 266328 Email : [email protected] 2


A Focused ApproachQ.3. (AMIE W08, 10 marks): Define (i) Newtonian fluid, (ii)ideal plastic fluid,(iii) kinematic viscosity, (iv) specific weight, and (v) specific gravity.

Q.4. (AMIE S10, 6 marks): What are the characteristic fluid properties of which the following phenomena are attributable: (i) Rise of sap in a tree, (ii) Cavitation, and (iii) water hammer.

Q.5. (AMIE S10, 6 marks): What is viscosity? Discuss its role in fluid flow and the factors on which shear stress rate depends in a flowing fluid. Give examples of different kinds of fluids from day-to-day life under different categories.

Q.6. (AMIE W09, S10, 6 marks): Slate Newton's law of viscosity. What is the effect of temperature on viscosity of water and that of air ?

Q.5. (AMIE S06, 5 marks): What is capillarity? Derive expression for height of capillary rise.

Q.6. (AMIE W2000): A piston 7.95 cm diameter and 30 cm long works in a cylinder 8 cm diameter. The annular space is filled with an oil viscosity 2 poise. If an axial load of 10 N is applied on the piston, find the speed of piston.

Answer: 0.1668 m/s

Hint: Load = x d x l and = x (U/dr) where dr is radial clearance.

Q.7. (AMIE S2001, 8 marks): A uniform film of oil 0.13 mm thick separates two discs, each of 200 mm diameter, mounted coaxially. Neglecting edge effects, calculate the torque necessary to rotate one disc relative to other at a speed of 420 rpm, if the oil has a viscosity of 0.14 Pas. Prove any formula you use.

Answer: 7.44 Nm

Q.1. (AMIE W2001, 10 marks): Calculate the power absorbed by fluid friction in a thrust bearing consisting of a flat disc of 100 mm diameter placed at the lower end of a vertical shaft. The oil film is 0.25 mm thick and the viscosity of the oil is 1.3 poise. The shaft rotates at 2000 rpm. Neglect end effects. Prove any formula you use.

Answer: T = 1.0692 N-m, Power = T = 223.93 Watts

Q.2. (AMIE S2001, 8 marks): An oil of viscosity 5 poise is used for lubrication between a shaft and sleeve. The diameter of the shaft is 0.5 m and it rotates at 200 rpm. Calculate the power lost in oil for a sleeve length of 100 mm. The thickness of oil film is 1.0 mm.

Answer: 2.153 kW

Fluid Statics

Revised topics

Conditions of Equilibrium of a floating Body

There are three possible situations for a body when immersed in a fluid.

If the weight of the body is greater than the weight of the liquid of equal volume then the body will sink into the liquid (To keep it floating additional upward force is

required).

If the weight of the body equals the weight of equal volume of liquid, then the

body will submerge and may stay at any location below the surface.

If the weight of the body is less than the weight of equal volume of liquid, then the body will be partly submerged and will float in the liquid.

OBJECTIVE QUESTIONS



A Focused ApproachComparison of densities cannot be used directly to determine whether the body will float or sink unless the body is solid over the full volume like a lump of iron. However the apparent density calculated by the ratio of weight to total volume can be used to check whether a body will float or sink. If apparent density is higher than that of the liquid, the body will sink. If these are equal, the body will stay afloat at any location. If it is less, the body will float with

part above the surface.

A submarine or ship though made of denser material floats because, the weight/volume of the ship will be less than the density of water. In the case of submarine its weight should equal the weight of water displaced for it to lay submerged.

Stability of a body

A ship or a boat should not overturn due to small disturbances but should be stable and return, to its original position. Equilibrium of a body exists when there is no resultant force or moment on the body. A body can stay in three states of equilibrium.

Stable equilibrium: Small disturbances will create a correcting couple and the

body will go back to its original position prior to the disturbance.

Neutral equilibrium Small disturbances do not create any additional force and so the body remains in the disturbed position. No further change in position occurs in

this case.

Unstable equilibrium: A small disturbance creates a couple which acts to increase the disturbance and the body may tilt over completely.

Under equilibrium conditions, two forces of equal magnitude acting along the same line of action, but in the opposite directions exist on a floating/submerged body. These are the gravitational force on the body (weight) acting downward along the centroid of the body and buoyant force acting upward along the centroid of the displaced liquid. Whether floating or submerged, under equilibrium conditions these two forces are equal and opposite and act

along the same line.

When the position of the body is disturbed or rocked by external forces (like wind on a ship), the position of the centre of gravity of the body (with respect to the body) remains at the same position. But the shape of the displaced volume of liquid changes and so its centre of gravity shifts to a new location. Now these two forces constitute a couple which may correct the original tilt or add to the original tilt. If the couple opposes the movement, then the body will regain or go back to the original position. If the couple acts to increase the tilt then the

body becomes unstable. These conditions are illustrated in Fig.

OBJECTIVE QUESTIONS



A Focused Approach

Figure (i) and (ii) shows bodies under equilibrium condition. Point C is the centre of gravity. Point B is the centre of buoyancy. It can be seen that the gravity and buoyant forces are equal

and act along the same line but in the opposite directions.

Figure (iii) shows the body under neutral equilibrium. The centre of gravity and the centre of

buoyancy coincide.

Figures (iv) and (v) shows the objects in Figures (i) and (ii) in a slightly disturbed condition. Under such a condition a couple is found to form by the two forces, because the point of application of these forces are moved to new positions. In the case of Figure (iv) the couple formed is opposed to the direction of disturbance and tends to return the body to the original position. This body is in a state of stable equilibrium. The couple is called righting couple. In the case of Figure (ii) the couple formed is in the same direction as the disturbance and hence tends to increase the disturbance. This body is in unstable equilibrium. In the case of figure (iii) no couple is formed due to disturbance as both forces act at the same point. Hence the

body will remain in the disturbed position.

In the case of top heavy body (Figure (ii)) the couple created by a small disturbance tends to

further increase the tilt and so the body is unstable.

It is essential that the stability of ships and boats are well established. The equations and calculations are more involved for the actual shapes. Equations will be derived for simple shapes and for small disturbances. (Note: For practical cases, the calculations will be

elaborate and cannot be attempted at this level.)

Conditions for the Stability of Floating Bodies

When the centre of buoyancy is above the centre of gravity of the floating body, the body is always stable under all conditions of disturbance. A righting couple is

always created to bring the body back to the stable condition.

When the centre of buoyancy coincides with the centre of gravity, the two forces act at the same point. A disturbance does not create any couple and so the body just

OBJECTIVE QUESTIONS



A Focused Approachremains in the disturbed position. There is no tendency to tilt further or to correct

the tilt.

When the centre of buoyancy is below the centre of gravity as in the case of ships,

additional analysis is required to establish stable conditions of floating.

This involves the concept of metacentre and metacentric height. When the body is disturbed the centre of gravity still remains on the centroidal line of the body. The shape of the

displaced volume changes and the centre of buoyancy moves from its previous position.

The location M at which the line of action of buoyant force meets the centroidal axis of the body, when disturbed, is defined as metacentre. The distance of this point from the centroid

of the body is called metacentric height. This is illustrated in Figure.

If the metacentre is above the centroid of the body, the floating body will be stable. If it is at the centroid, the floating body will be in neutral equilibrium. If it is below the centroid, the

floating body will be unstable.

When a small disturbance occurs, say clockwise, then the centre of gravity moves to the right of the original centre line. The shape of the liquid displaced also changes and the centre of buoyancy also generally moves to the right. If the distance moved by the centre of buoyancy is larger than the distance moved by the centre of gravity, the resulting couple will act anticlockwise, correcting the disturbance. If the distance moved by the centre of gravity is

larger, the couple will be clockwise and it will tend to increase the disturbance or tilting.

The distance between the metacentre and the centre of gravity is known is metacentric height. The magnitude of the righting couple is directly proportional to the metacentric

height.

Larger the metacentric height, better will be the stability. Referring to following figure, the

centre of gravity G is above the centre of buoyancy B.

OBJECTIVE QUESTIONS



A Focused ApproachAfter a small clockwise tilt, the centre of buoyancy has moved to B'. The line of action of this force is upward and it meets the body centre line at the metacentre M which is above G. In this case metacentric height is positive and the body is stable. It may also be noted that the couple is anticlockwise. If M falls below G, then the couple will be clockwise and the body

will be unstable.

Example

A right circular cylinder of diameter D m and height h m with a relative density of (S < 1) is to float in water in a stable vertical condition. Determine the limit of the ratio D/h for the

required situation.

Solution

For stability, the limiting condition is that the metacentre approach the centre of gravity.

(V – volume displaced),

MB = I/V

MG = (I/V) GB.

Here MG = 0 for the limiting condition.

(I/V) = GB

I = D4/64, V = D2 h S/4,

(I/V) = D2/16hS

Also from basics

GB = (h/2) – (h S/2) = h (1 – S)/2; equating, (D2/16hs) = [h (1 – S)]/2

(D/h) = 2 [2 S (1 – S)]0.5 ... (1)

For example if S = 0.8,

D/h = 1.1314

D > h.

The diameter should be larger than the length. This is the reason why long rods float with length along horizontal. The same expression can be solved for limiting density for a given

D/h ratio. Using equation 1

(D/h)2 = 8 S (1 – S) or 8 S2

– 8S + (D/h)2 = 0

S = {1 ± [1 – (4/8) (D/h)2]0.5 }/2,

say if (D/h) = 1.2, then S = 0.7646 or 0.2354

ASSIGNMENT Q.3. (AMIE W07, 4 marks): Explain the terms (i) total pressure (ii) centre of pressure.

OBJECTIVE QUESTIONS



A Focused ApproachQ.4. (AMIE S09, 7 marks): What is centre of pressure? Obtain an expression for the depth of centre of pressure when the lamina is immersed in a liquid and is at angle with the horizontal.

Q.5. (AMIE S06, 5 marks): Define Buoyancy and centre of Buoyancy.

Q.6. (AMIE W08, 4 marks): Explain the terms (i) buoyant force (ii) centre of buoyancy.

Q.7. (AMIE W08, 2 marks): What is the magnitude of buoyant force and where does the line of action of buoyant force pass?

Q.8. (AMIE W08, 2 marks): What is the necessary condition for a body to float in stable equilibrium?

Q.9. (AMIE W07, 4 marks): Explain the terms (i) metacentre (ii) metacentric height.

Q.10. (AMIE S08, W09, 7 marks): With neat sketch, explain the condition of equilibrium of submerged bodies.

Q.11. (AMIE W07, 6 marks): A circular plate, 2.50 m in diameter, is immersed in water. Its greatest and least depth below the free surface being 3 m and 1 m, respectively. Find the (i) total pressure on one face of the plate, and (ii) position of centre of pressure.

Answer: 96310 N, 2.125 m

Q.12. (AMIE W2001, 10 marks): A vertical circular lamina of radius R is kept immersed in a liquid such that its topmost point A is on the free surface. Determine the depth and width of the horizontal chord BC, so that the

thrust due to hydrostatic pressure on the ABC is maximum. Also determine the ratio of total force on ABC in

this case to that on the entire lamina.

Answer: width of BC = 25.R/3, depth = (5/3)R, 0.43936

Q.13. (AMIE S07, 10 marks): A cylindrical gate of 4 m dia and 2 m long has water on its both sides as shown in Fig.. Determine the magnitude and direction of the resultant force exerted by the water on the gate. Also, find the least weight of the cylinder so that it may not be lifted away from the floor.

Answer: 219206 N, = 57031'

Q.14. (AMIE W09, 8 marks): A rectangular plane surface, 2 m wide and 3 m deep, lies in water in such a way that its plane makes an angle of 300 with the free surface of water. Determine the total pressure and position of centre of pressure when the upper edge is 1.5 m below the free-water surface.

Answer: 132435 N, 2.333 m.

Q.15. (AMIE S08, 7 marks): Show that a cylindrical body of 1 m diameter and 2 m height weighing 7.848 kN will not float vertically in sea water of density 1030 kg/m3.

Q.16. (AMIE W08, 6 marks): A piece of wood (specific gravity = 0.6) of 10cm2 in cross-section and 2.5 m long floats in water. How much lead (specific gravity = 12) need to be fastened at the water end of the stick so that it floats upright with 0.5 m length out of water.

Answer: 53.5 N

OBJECTIVE QUESTIONS



A Focused ApproachQ.17. (AMIE W07, 4 marks): A wooden cylinder of circular section, uniform density, and specific gravity 0.6, is required to float in oil of specific gravity 0.8. If the diameter of the cylinder is d and its length is l, show that l cannot exceed 0.817d for cylinder to float with its longitudinal axis vertical.

Q.18. (AMIE W08, 7 marks): A wooden cylinder of specific gravity 0.6 and circular in cross-section is required to float in oil (specific gravity = 0.90). Find the L/D ratio for the cylinder to float with its longitudinal axis vertical on oil, where L is the height of cylinder and D its diameter.

Answer: < 0.75

Q.19. (AMIE S05, 10 marks): Find the density of a metallic body which floats at the interface of mercury of specific gravity 13.6 and water such that 40% of its volume is submerged in mercury and 60% in water.

Answer: 6040 kg/m3

Q.20. (AMIE W05, 5 marks): A slab of wood 2m x 2m x 1 m depth has specific gravity of 0.5 floats in water with 12000 N load on it. Determine the depth of submergence of slab in water.

Answer: 0.80 m

Q.21. (AMIE W06, 5 marks): The specific gravity of the block shown in figure is 1.6. Find the specific gravity of the unknown fluid.

Answer: 1.9

Q.22. (AMIE W06, 2 marks): Define metacentric height.

Q.23. (AMIE S06, 10 marks): A wooden block of specific gravity 0.75 floats in water. If the size of the block is 1 m x 0.5 m x 0.4 m, find its metacentric height.

Answer: 0.01944 m

Q.24. (AMIE W2001, 8 marks): Show that for small angle of tilt, the time period of oscillation of a ship floating in stable equilibrium in water is given by

2 k

Tg.MG

where k is radius of gyration about the axis of rotation and MG is the metacentric height.

Q.25. (AMIE S2000, 10 marks): A closed vertical cylinder 0.4 m in diameter and 0.4 m in height is completely filled with oil of specific gravity 0.8. If the cylinder is rotated about its vertical axis at 200 rpm, calculate the thrust of oil on top and bottom covers of the cylinder. Derive any formula you use.

Answer: 440.98 N (top), 835.46 N (bottom)

Fluid Kinematics

Laminar and Turbulent Flow

If the flow is smooth and if the layers in the flow do not mix macroscopically then the flow is called laminar flow. For example a dye injected at a point in laminar flow will travel along a continuous smooth line without generally mixing with the main body of the fluid.

OBJECTIVE QUESTIONS



A Focused ApproachMomentum, heat and mass transfer between layers will be at molecular level of pure

diffusion. In laminar flow layers will glide over each other without mixing.

In turbulent flow fluid layers mix macroscopically and the velocity/temperature/mass

concentration at any point is found to vary with reference to a mean value over a time period.

For example 'u u u where u is the velocity at an instant at a location and u is the average

velocity over a period of time at that location and u′ is the fluctuating component. This causes higher rate of momentum/heat/mass transfer. A dye injected into such a flow will not flow

along a smooth line but will mix with the main stream within a short distance.

The difference between the flows can be distinguished by observing the smoke coming out of an incense stick. The smoke in still air will be found to rise along a vertical line without mixing. This is the laminar region. At a distance which will depend on flow conditions the smoke will be found to mix with the air as the flow becomes turbulent. Laminar flow will prevail when viscous forces are larger than inertia forces. Turbulence will begin where inertia

forces begin to increase and become higher than viscous forces.

Path Line and Streak Line

Path line is the trace of the path of a single particle over a period of time. Path line shows the direction of the velocity of a particle at successive instants of time. In steady flow path lines

and stream lines will be identical.

Streak lines provide an instantaneous picture of the particles, which have passed through a given point like the injection point of a dye in a flow. In steady flow these lines will also

coincide with stream lines

Particles P1, P2, P3, P4, starting from point P at successive times pass along path lines shown. At the instant of time considered the positions of the particles are at 1, 2, 3 and 4. A line

joining these points is the streak line.

SOURCE FLOW

A source flow consists of a symmetrical flow field with radial stream lines directed outwards from a common point, the origin from where fluid is supplied at a constant rate q. As the area increases along the outward direction, the velocity will decrease and the stream lines will spread out as the fluid moves outwards. The velocity at all points at a given radial distance

will be the same.

OBJECTIVE QUESTIONS



A Focused ApproachSINK SOURCE

Sink is the opposite of source and the radial streamlines are directed inwards to a common point, origin, where the fluid is absorbed at a constant rate. The velocity increases as the fluid moves inwards or as the radius decreases, the velocity will increase. In this case also the velocity at all points at a given radial distance from the origin will be the same. The origin is

a singular point. The circulation around any closed curve is zero.

FLOWNET

The plot of stream lines and potential flow lines for a flow in such a way that these form curvilinear squares is known as flow net. The idea that stream lines and potential lines are

orthogonal is used in arriving at the plot.

Such a plot is useful for flow visualisation as well as calculation of flow rates at various locations and the pressure along the flow. The lines can be drawn by trial or electrical or

magnetic analogue can also be used.

An example is shown in figure for flow through a well rounded orifice in a large tank. The flow rate along each channel formed by the stream lines will be equal. The pressure drop

between adjacent potential lines will also be equal.

With the advent of computer softwares for flow analysis, the mechanical labour in the

plotting of such flow net has been removed. However the basic idea of flow net is useful.

Example

Explain how the potential function can be obtained if the stream function for the flow is

specified.

Solution

(1) Irrotational nature of the flow should be checked first. Stream function may exist, but if the flow is rotational potential function will not be valid.

(2) The values of u and v are obtained from the stream function as

uy

vx

(3) From the knowledge of u and v, can be determined using the same procedure as per the

determination of stream function

OBJECTIVE QUESTIONS



A Focused Approach

( )

ux

udx f y

where f(y) is a function of y only

/ y is determined and equated to – v

Comparing f '(y) is found and then f(y) is determined and substituted in equation A

( )udx f y C

Example

For the following stream functions, determine the potential function

(i) 2 2(3 / 2)( )x y

(ii) 8xy

(iii) x y

Solution

(i) Step1: We have

3u yy

3 3v x v xx

To check for irrotationality

v u

x y

here, both are -3, so checks

Step 2: 3u y , also ux

3 ( )ydx f y

3 ( )xy f y (A)

Diff. eq. (A) wrt y and equating to v,

3 ' 3x f y v xy

'( ) 0f y and so f(y) = constant.

Substituting in A,

OBJECTIVE QUESTIONS



A Focused Approach 3 tanxy cons t

Check 2 2 2

2 2 20, 3 , 0y

x y y x

So also 2

20

y

So checks

3 , 3u y v xx y

also checks.

(ii) Do yourself. 2 24 4x y

(iii) Do yourself. x y

FORCES ON MOVING BLADES

This topic has been deleted.

MOMENT OF MOMENTUM EQUATION

This topic has been deleted.

ASSIGNMENT Q.26. (AMIE W05, 07, S08, 7 marks): Define and distinguish between (i) stream line (ii) streak line and (iii) path line. Are these same in steady flow?

Q.27. (AMIE W07, 3 marks): Define (i) rotational and irrotational flow (ii) uniform and non uniform flow.

Q.28. (AMIE S08, 09, 6 marks): Define (i) steady and non steady flow (ii) one, two and three dimensional flow (iii) laminar and turbulent flow.

Q.29. (AMIE W08, 6 marks): Define (i) steady and unsteady flow (ii) compressible and incompressible flow.

Q.30. (AMIE W09, 6 marks): Define (i) Compressible and incompressible flow (ii) one, two and three dimensional flow.

Q.31. (AMIE W09, 5 marks): Define (i) source flow (ii) sink flow (iii) free vortex flow (iv) superimposed flow.

Q.32. (AMIE S10, 6 marks): Explain what is meant by a point source and a sink source.

Q.33. (AMIE S06, 5 marks): Write short note on (i) momentum theorem. (ii) Reynolds transport theorem

Q.34. (AMIE S06, 5 marks): Derive the expressions of continuity and momentum equations.

Q.35. (AMIE S08, 10 marks): Derive an expression for continuity equation for a three dimensional flow.

Q.36. (AMIE W05, 4 marks): What is potential flow and how does it differ from viscous flow.

Q.37. (AMIE S06, W08, 09, 10 marks): Explain briefly the following : (i) velocity potential (ii) stream function. Describe relation these two.

Q.38. (AMIE W08, 10 marks): What are the properties of stream function and what do you mean by equipotential line and a line of constant stream function.

Q.39. (AMIE W07, 4 marks): describe the use and limitations of flow nets.

OBJECTIVE QUESTIONS



A Focused ApproachQ.40. (AMIE W07, 10 marks): Derive, from first principles, the condition for irrotational flow. Prove that, for potential flow, both the stream function and velocity function satisfy the Laplace equation.

Q.41. (AMIE W05, 8 marks): Does the velocity potential function 2 22(x 2y y ) describe the

possible flow of an incompressible fluid? If so, find out the equation for the velocity vector V. Also determine the equation of streamline.

Answer: Yes, V iu jv 4x i (4y 4) j

, dx dy dx dy

or4x 4y 4 x y 1

Q.42. (AMIE W06, 5 marks): Determine whether the following flow is irrotational or not? Also determine its velocity potential.

2 2u xy ; v x y

Answer: Flow is irrotational. 2 2x y

cons tan t2

Q.43. (AMIE W07, 4 marks): A stream function is given by 2 33x y . Determine the magnitude of

velocity components at point (2,1).

Answer: u = 3 units/s, v = 12 units/s

Q.44. (AMIE W08, 4 marks): The velocity potential function is given by 2 25( )x y . Calculate the

velocity component at (4, 5).

Answer: u = 40 units/s, v = -50 units/s

Q.45. (AMIE S08, 6 marks): A fluid flow field is given by

2 2 2(2 )V x yi y zj xyz yz k

Prove that it is a case of possible steady incompressible fluid flow. Calculate the velocity at the point (2, 1, 3).

Answer: 21.58

Q.46. (AMIE S09, 10 marks): For a doublet of strength 20 m2/s, calculate the velocity at point P(1, 2) and the value of stream function passing through it.

Answer: 0.63 m/s

Hint: For doublet sin2 r

in Cartesian coordinates

2 2

.2

y

x y

Q.47. (AMIE W09, 10 marks): A point P(0.5, 1) is situated in the flow field of a doublet of strength 5 m2/s. Calculate the velocity at this point and also the value of the stream function.

Answer: 0.6366 m/s, stream function = 0.6366 m3/s

Q.48. (AMIE S10, 8 marks): If u = 2x and v = -2y are respectively x and y components of possible fluid flow, determine stream function.

Answer: 2xy C

OBJECTIVE QUESTIONS



A Focused ApproachQ.49. (AMIE W05, 10 marks): What force components are required to hold the black box, shown in the figure, stationary? Assume no mass accumulation in the box.

Answer: Fx = 423.6 N, Fy = 210.85 N

Hint: Q = 0 to find unknown Q. Now find components of forces.

Q.50. (AMIE S2000, 12 marks): A 0.4 m x 0.3 m, 900 vertical reducing bend carries 0.5 m3/s of oil of specific gravity 0.85 with a pressure of 118 kN/m2 at inlet to the bend. The volume of the bend is 0.1 m3. Find the magnitude and direction of the force on the bend. Neglect friction and assume both inlet and outlet sections to be at same horizontal level. Also assume that water enters the bend at 450 to the horizontal.

Answer: Resultant force on bend = 18665.6 N, 283.580 from x direction (anticlockwise)

Q.51. (AMIE S2007, 10 marks): Determine the hydrodynamic force on a uniform 90° pipe elbow of 15 cm diameter through which water flows at a constant velocity of 8.4 m/s and constant pressure of 116kPa (gauge). Assume the elbow to be in horizontal plane.

Answer: 4657 N, = 450

Bernoulli's Equation and its Applications

Example (AMIE Summer 2007, 10 marks)

A liquid of specific gravity 1.52 is discharged from a tank through a siphon whose summit point is 1.2 m above the liquid level in the tank. The siphon has a uniform diameter of 10 cm and it discharges the liquid into atmosphere whose pressure is 101 kPa. If the vapour pressure of the liquid is 28 kPa (abs), how far below the liquid level in the tank can the outlet be safely located? What is the maximum discharge? Neglect all losses of head.

Solution

Referring to figure, apply Bernoulli's equation on 2 and 3

22

3 32 22 32 2

p Vp VZ Z

g g g g

OBJECTIVE QUESTIONS



A Focused Approach

where Z3 = 0, V2 = V3 = V, p3 = 101 kPa, Z2 = 1.2 + h

2 228 1000 101 1000

(1.2 ) 0(1.52 1000) 9.81 2 (1.52 1000) 9.81 2

x V x Vh

x x g x x g

Solving h = 3.7 m

This is the safe limit of h.

Applying Bernoulli's equation between points 1 and 3

2101 1000 101

0 3.7 0(1.52 1000) 9.81 (1.52 1000) 9.81 2

x V

x x x x g

Solving V = 8.52 m/s

2(0.1) 8.52 0.06694

Q AV x

m3/s

ASSIGNMENT Q.52. (AMIE S06, 09, 10, 10 marks): State and prove Bernoulli’s equation. Mention its limitations.

Q.53. (AMIE S05, 10 marks): State Bernoulli’s theorem for steady flow of an incompressible fluid. Derive expression for Bernoulli’s equation from first principle and state the assumptions made for such a derivation.

Q.54. (AMIE W07, 6 marks): Derive Bernoulli's equation from Euler's equation of motion.

Q.55. (AMIE W08, 09, 6 marks): Derive Euler's equation of motion stating the assumptions.

Q.56. (AMIE S05, 10 marks): State the different devices that can be used to measure the discharge through a pipe also through an open channel. Describe one of such devices with a neat sketch and explain how one can obtain the actual discharge with its help.

Q.57. (AMIE S06, 10 marks): Describe how the pitot tube is used to determine the mass flow weighed mean value of the velocity in a large duct.

Q.58. (AMIE W09, 5 marks): How will you determine the velocity of flow at any point with the help of pitot tube?

Q.59. (AMIE S06, 5 marks): What are the relative advantages and disadvantages of the orifice meter, venturimeter and the flow nozzle for measuring flow rates of gases?

Q.60. (AMIE W07, 10 marks): Write short notes on (i) Orifice meter (ii) Pitot tube

Q.61. (AMIE W08, 20 marks): Write short notes on (a) Venturimeter (b) Bernoulli's equation (c) Euler's equation (d) Pitot tube.

Q.62. (AMIE S08, 10 marks): Define an orificemeter. prove that the discharge through an orifice meter is given by

OBJECTIVE QUESTIONS



A Focused Approach

2 20 1 1 02 /dQ C a a gh a a

where a1 is area of pipe; a0 is the area of orifice, and Cd is the coeff. of discharge.

Q.63. (AMIE W09, S09, 10 marks): What is a venturimeter? Draw a neat sketch of the venturimeter showing the arrangement of the manometer. Derive an expression for the rate of flow of fluid through it.

Q.64. (AMIE S10, 8 marks): Define orifice, mouth piece, notch and weirs.

Q.65. (AMIE S09, 6 marks): A horizontal water pipe of diameter 15 cm converges to 7.5 cm diameter. If the pressures at two sections arc 400 kPa and 150 kPa, respectively, calculate the flow rate of water.

Answer: 0.102 m3/s

Q.66. (AMIE W06, 5 marks): Water flows in a circular pipe. At one section, the diameter is 0.2 m, the static pressure is 250 kPa gauge, the velocity is 3 m/s and the elevation is 10 m above ground level. At a down stream section, the pipe diameter is 0.15 m and elevation is 0 m. Find the gauge pressure at the downstream section. Mention the assumptions.

Answer: 338.4 kPa, gauge

Q.67. (AMIE W08, 6 marks): Water flows through a 100 mm pipe at the rate of 0.027 m3/s and then through a nozzle attached to the end of the pipe. The nozzle tip is 50 mm in diameter, and the coefficients of velocity and contraction for the nozzle are 0.950 and 0.930, respectively. What pressure head must be maintained at the base of the nozzle if atmospheric pressure surrounds the jet?

Answer: 11.424 m

Q.68. (AMIE W06, 10 marks): Water flows through the horizontal Y branch shown in figure. For steady flow and neglecting losses, determine the force components required to hold the body Y in place.

The gauge pressure at section (1) is 30 kPa with volumetric inflow 15 x 10-3 m3/s, volumetric outflow from section (2) equals to 10-2 m3/s and water density is 1000 kg/m3.

Answer: Fx = -47.2 N (in –x direction), Fy = 1.172 N

Q.69. (AMIE W07, 6 marks): A venturi meter, having a diameter of 75 mm at the throat and 150 mm diameter at the enlarged end, is installed in a horizontal pipeline 150 mm in diameter carrying an oil of specific gravity 0.9. The difference in pressure head between the enlarged end and the throat recorded by on U tube is 175 mm of mercury. Determine the discharge through the pipe. Assume the coefficient of discharge of the meter as 0.97.

Answer: 0.0308 m3/s

Q.70. (AMIE W07, 6 marks): A venturimeter of 30 cm inlet diameter and 15 cm throat diameter is provided in a vertical pipeline carrying oil of specific gravity 0.9, the flow being upward. The difference in elevation of the throat section and entrance section of the venturimeter is 30 cm. The differential U-tube mercury manometer shows a gauge deflection of 25 cm. Calculate the (i) discharge of oil, (ii) pressure difference between the entrance section and the throat section. Take coefficient of meter as 0.98 and specific gravity of mercury as 13.6.

Answer: 148.795 l/s, 3.379 N/cm2

OBJECTIVE QUESTIONS



A Focused ApproachQ.71. (AMIE S10, 12 marks): A venturi meter is to be fitted in a pipe of 0.25 m diameter, where the pressure head is 7.6 m of following liquid and the maximum flow is 8.1 m3/min. Find the least diameter of the throat to ensure that the pressure head does not become negative. Take Cd = 0.96.

Answer: 11.94 cm

Q.72. (AMIE S08, 6 marks): An orifice meter with diameter 15 cm is inserted in a pipe of 30 cm diameter. The pressure difference measured by a mercury oil differential manometer on two sides of the orifice meter gives a reading of 50 cm of mercury. Find the rate of flow of oil of specific gravity 0.9 when the coefficient of discharge of the meter is 0.64.

Answer: 137.42 l/s

Q.73. (AMIE S08, 6 marks): An orifice meter, with orifice diameter 10 cm, is inserted in a pipe of 20 cm diameter. The pressure gauges fitted upstream and downstream of the orifice meter given readings of 19.62 N/cm2 and 9.81 N/cm2, respectively. Coefficient of discharge for the meter is given as 0.6. Find the discharge of water through pipe.

Answer: 68.21 l/s

Q.74. (AMIE S09, 10 marks): A submarine moves horizontally in sea with its axis much below the surface of water. A pitot tube, properly placed just in front of the submarine and along its axis, is connected to two limbs of a U tube containing mercury. The difference of mercury level is found to be 17 cm. Find the speed of the submarine knowing that density of mercury is 13.6 and that of sea water is 1.026 with respect to freshwater.

Answer: 22.556 km/h

Q.75. (AMIE W05, 8 marks): Derive an expression for the discharge through a triangular notch. When would you recommend to use it.

Flow Through Pipes

Loss of Head Due to Friction

Proof:

Consider following figure.

Applying Bernoulli's theorem at 1-1 and 2-2

2 2

1 1 2 21 22 2 f

p V p Vz z h

g g g g

where hf is frictional loss.

But z1 = z2 as pipe is horizontal; V1 = V2 as same diameter

OBJECTIVE QUESTIONS



A Focused Approach

1 2f

p ph

g g (1)

Frictional resistance = frictional resistance per unit wetted area per unit velocity x wetted area x (velocity)2

or 21 '( )F f dL V

The forces acting on the fluid between sections 1-1 and 2-2 are

1. Pressure force at section 1-1 = 21 4

p x d

2. Pressure force at 2-2 = 22 4

p d

Resolving all forces

2 2 21 2 '( ) 0

4 4p x d p x d f dL V

(2)

or 2

1 2

' 4( )

f Lx Vp p

d

But from (1)

1 2( ) fp p gh

2' 4

f

f Lx Vh g

d

or 24 '

f

f LVh

gd

Putting f'/ = f/2 where f is coeff. of friction, the above equation reduces to

24

2f

flVh

gd

ASSIGNMENT Q.76. (AMIE W2001, 6 marks): Starting the assumptions, deduce an expression for the head loss due to sudden expansion of streamlines in a pipe.

Q.77. (AMIE S08, 10 marks): Derive the expression for the loss of head due to friction in pipe

24 / 2fh flV gd

where hf is the loss of head due to friction; L, the length of the pipes; f, the coeff. of friction; V, the velocity, and d, the diameter of pipe.

Q.78. (AMIE W05, 10 marks): A pipeline of 0.6 m dia. is 1.5 km long. In order to increase the discharge, another parallel line of the same diameter introduced in the second half of the length. Neglecting minor losses, find the increase in discharge if f = 0.004. The head at the inlet is 30 m over that at the outlet.

OBJECTIVE QUESTIONS



A Focused ApproachAnswer: 26.5% increase

Q.79. (AMIE W06, 6 marks): Water flow through a pipe with a flow rate Q and head H1. The pipe later divides into two pipes A and B with diameter of DA and DB and lengths LA and LB as shown in figure. Find an expression for QA and QB, the flow rates through pipes A and B. Pipes A and B have friction factor of fA and fB.

Answer: B 1/ 25B B A

5A A B

QQ

f L D1 .

f L D

; A 1/ 25B B A

5A A B

QQ

f L D1 .

f L D

Q.80. (AMIE S07, 12 marks): Two reservoirs, whose water level elevations differ by 12m, are connected to the following horizontal compound pipes starting from the higher level reservoir: L1 = 200m; D1 = 0.2 m, f1 = 0.008 and L2 = 500 m, D2 = 0.3 m, f2 = 0.006. Considering all head losses and assuming that all changes of section are abrupt, compute the discharge through the system. Determine the equivalent length of a 0.25 m diameter pipe, if minor losses are neglected and friction factors are assumed to be same.

Answer: 0.0754 m3/s, 811.3 m

Laminar Flow

Example (AMIE S2009, 6 marks)

Determine the (i) kinetic energy correction factor (ii) momentum correction factor for laminar flow in a round pipe.

Solution

(i) 2 2( )u K R r

where 1

4

dpK

dx

for horizontal pipes and ( /

.4

d p ZK

ds

for

inclined pipes.

Average velocity 2 0

1.2 .

RV u r dr

R

= 2 32 0

2)

RKR r r dr

R

= 4 4

22

2

2 4 2

K R R KR

R

Kinetic energy correction factor

OBJECTIVE QUESTIONS



A Focused Approach

33

1u dA

V A

= 3 2 2 3

02 2

1( ) 2

2

RK R r rdr

KR R

= 2 2 38 0

16( )

RR r rdr

R

= 6 7 4 3 2 58 0

16[ 3 3 ]

RR r r R r R r dr

R

= 1 1 3 3

16 22 8 4 6

2

(ii) 22

1

av

u dAAU

= 2

2 22 2

12 1 2av

av

rU rdr

R U R

= 2 4

2 0

81 2

R r rrdr

R R R

= 3 5

2 2 40

8 2R r rr dr

R R R

= 2 4 6

2 2 4

0

8 2 4

2 4 6 3

Rr r r

R R R

Example

Oil with a kinematic viscosity of 241 × 10-6 m2/s and density of 945 kg/m3 flows through a pipe of 5 cm dia. and 300 m length with a velocity of 2 m/s. Determine the pump power,

assuming an overall pump efficiency of 45%, to overcome friction.

Solution

Re = uD/ = 2 × 0.05/241 × 10-6 = 415. So the flow is laminar.

hf = (64/415) × [(22 × 300)/(2 × 9.81 × 0.05)] = 188.67 m head of oil

Mass flow = (π × 0.052/4) 2 × 945 kg/s = 3.711 kg/s

Power required = mg H/η = 3.711 × 9.81 × 188.67/0.45 W = 15,263 W or 15.263 kW.

OBJECTIVE QUESTIONS



A Focused ApproachASSIGNMENT

Q.8. (AMIE S05, 10 marks): Prove that the velocity distribution for viscous flow between two parallel plates, when both plates are fixed across section is parabolic in nature. Also, prove that the maximum velocity is equal to one and a half times the average velocity.

Q.9. (AMIE W06, 10 marks): Prove that for flow between two stationary parallel plates, the pressure drop is given by

avg

2

12 udp

dx h

where h is the distance between the plates and uavg is the average flow velocity.

Q.10. (AMIE W05, 10 marks): Derive the velocity distribution formula for a plane Poiseuille flow and prove that the average velocity is 2/3 of its centre line velocity.

Hint: See flow between two parallel plates

Q.11. (AMIE S08, W08, 09, 10 marks): Derive an expression for the velocity distribution for viscous flow through a circular pipe. Also, sketch the velocity distribution and shear stress distribution across the section of pipe.

Q.12. (AMIE S10, 6 marks): Derive the expression for Hagen Poiseuille flow.

Q.13. (AMIE S06, 10 marks): Explain briefly (i) Couette flow (ii) Poiseuille flow

Q.1. (AMIE S07, 12 marks): Starting from continuity and momentum equations, derive the governing equation for Couette flow. Determine velocity profile when lower plate is stationary and upper plate moves with velocity U for different pressure gradient situation.

Q.2. (AMIE W07, 6 marks): Show that for laminar flow in circular pipes, the friction factor is inversely proportional to Reynolds number.

Q.14. (AMIE S05, 10 marks): Two parallel plates kept 0.01 m apart have a laminar flow of oil between them. Taking dynamic viscosity of oil to be 0.8 poise, determine the velocity distribution, discharge and shear stress on the upper plate that moves horizontally at relative velocity 1 m/s with respect to the lower plate which is stationary. Further, the pressure drops in the flow direction from 180 kPa to 100 kPa over a distance of 80 m.

Answer: Flow is Couette flow. u = y(162.5 – 6250y); 0.00604 m3/s; = 3 Ns/m2

Q.15. (AMIE S10, 6 marks): Air ( = 1.2 kg/m3, = 1.81 x 10-5 Ns/m2) is forced at 25 m/s through a = 0.3 m2

steel duct, 148 m long. Calculate the head loss and power if f = 0.015.

Q.1. (AMIE S08, 10 marks): A laminar flow is taking place in a pipe of diameter 200 mm. The maximum velocity is 1.5 m/s. Find the mean velocity and the radius at which this occurs. Also, calculate the velocity at 4 cm from the wall of the pipe.

Answer: 0.75 m/s, 7.07 cm, 0.96 m/s

Q.2. (AMIE S09, 6 marks): A fluid of viscosity 0.7 Ns/m2 and specific gravity 1.3 is flowing through a circular pipe of diameter 10 cm. The maximum shear stress at the pipe wall is given as 196.2 N/m2. Find (i) the pressure gradient, (ii) average velocity, and (iii) Reynold number of the flow.

Answer: - 7848 N/m2, 3.5 m/s, 650

OBJECTIVE QUESTIONS



A Focused ApproachQ.16. (AMIE S05, W2000, 10 marks): A crude oil of viscosity 0.97 Poise and relative density 0.9 is flowing through a horizontal circular pipe of diameter 10 cm and of length 10 m. Calculate the difference of pressure at the two ends of the pipe, if 100 kg of oil is collected in a tank in 30 sec.

Answer: 1463.8 N/m2

Q.1. (AMIE S09, 10 marks): A straight stretch of horizontal pipe of 5 cm diameter was used in the laboratory to measure the viscosity of a crude oil (specific weight 9000 N/m3). During the test run, a pressure differential of 18000 N/m2 was recorded from two pressure gauges located 6 m apart on the pipe. The oil was allowed to discharge into a weighing tank and 5000 N of oil was collected in 3 min duration. Work out dynamic viscosity of the oil.

Answer: 0.1491 Ns/m2

Turbulent Flow

Example

A pipeline 30 cm in diameter carries 300 L/s of petrol ( = 680 kg/m3, = 2.9 x 10-4 Pa.s).

Calculate the (i) friction factor (ii) shear stress at the boundary (iii) shear stress and velocity at a radial distance of 5 cm from the pipe axis (iv) maximum velocity (v) thickness of laminar

sublayer. Assume smooth pipe.

Solution

Mean velocity

2

0.3004.244 /

( / 4)(0.30)

QV m s

A

Reynolds number

64

4.244 0.30 680Re 2.985 10

2.9 10

VD x xx

x

(i) By using the explicit formula for f

6 0.237

0.2210.0032 0.009654

(2.985 10 )f

x

(ii) * 0 / / 8u V f

2 2

0

680(4.244) (0.009654)14.78

8 8

V fPa

14.78

* 0.14743 /680

u m s

(iii) At r = 5 cm, y = r0 - r = 15 - 5 = 10 cm

00

514.78 4.927

15

rx Pa

r

OBJECTIVE QUESTIONS



A Focused Approach The velocity at y = 0.1 m is given by the smooth pipe formula

*

*

5.75log 5.5u yu

u

Noting that /

4

*

0.14743 0.1 6805.75log 5.5 31.597

2.9 10

u x x

u x

*(31.597) 0.14743 31.597 4.658 /u u x m s

(iv) The maximum velocity um is related to V as

*

3.75mu V

u

*3.75 4.244 3.75 0.14743 4.797 /mu V u x m s

(v) Laminar sublayer thickness

4

5

*

11.6 2.9 10 1' 11.6 3.356 10 0.0336

680 0.14743

xx m mm

u

ASSIGNMENT Q.2. (AMIE S05, 5 marks): Write a short note on turbulent flow.

Q.3. (AMIE W07, 8 marks): A smooth brass pipeline, 75 mm in diameter and 900 m long, carries water at the rate of 7 litres/sec If the kinematic viscosity of the water is 0.0195 stokes, calculate the loss of head, wall shearing stress, centre line velocity and thickness of the laminar sublayer. Take Q = 1000 kg/m3.

Answer: 30.922m, 6.3198 N/m2, 1.91 m/s, 0.02829 cm

Q.4. (AMIE S08, 10 marks): A smooth pipe of diameter 400 mm and length 800 m carries water at the rate of 0.04 m3/s. Determine the head lost due to friction, wall shear stress, centre-line velocity and thickness of laminar sub-layer. Take the kinematic viscosity of water as 0.018 stokes.

Answer: 0.2 m, 0.3789 m/s, 0.133 cm

Q.5. (AMIE S08, 10 marks): A smooth pipe of diameter 80 mm and 800 m long carries water at the rate of 0.480 m3/min. Calculate the loss of head, wall shearing stress, centre line velocity, velocity and shear stress at 30 mm from pipe wall. Also, calculate the thickness of laminar sub-layer. Take kinematic viscosity of water as 0.015 stokes. Take the value of coefficient of friction f from the relation f = 0.0791/Re1/4 where Re = Reynold number.

Answer: 23.93 m, 5.869 N/m2, 1.88 m/s,

Boundary layer

Characteristics of Boundary Layer

increases as distance from leading edge x increases.

decreases as U increases.

OBJECTIVE QUESTIONS



A Focused Approach increases as kinematic viscosity increases.

0 decreases as x increases. However, when boundary layer becomes turbulent (see

below), it shows a sudden increase and then decreases with increasing x.

If U increases in the downstream direction, i.e. /p x is -ve, boundary layer growth

is reduced.

If U decreases in the downstream direction, i.e. /p x is +ve, flow near the boundary

is further retarded, boundary layer growth is faster and boundary layer is susceptible to separation.

All characteristics of the boundary layer on flat plate, such as variation of , 0 or

force F, are governed by inertial and viscous forces; hence they are functions of either

Ux/ or UL/.

When Ux/ is less than 5 x 105, boundary layer is laminar and velocity distribution in

boundary layer is parabolic. When Ux/ > 5 x 105 the boundary layer on that portion

is turbulent.

Critical value of Ux/ at which boundary layer changes from laminar to turbulent

depends on turbulence in ambient flow, surface roughness, pressure gradient, plate curvature and temperature difference between fluid and boundary.

Velocity distribution in laminar boundary layer follows parabolic law while that in turbulent boundary layer follows logarithmic law or power law.

In turbulent boundary layer a thin layer known as laminar sub-layer ' exists near the

boundary, if it is smooth.

ASSIGNMENT Q.6. (AMIE S08, 5 marks): Explain with necessary sketch the following

(i) Laminar boundary layer

(ii) Turbulent boundary layer

(iii) Laminar sub-layer

(iv) Boundary layer thickness

Q.7. (AMIE W07, 08, 09, 6 marks): Explain the characteristics of laminar and turbulent layer.

Q.8. (AMIE W08, 8 marks): Discuss the effect of pressure gradient on boundary layer separation.

Q.9. (AMIE S10, 8 marks): Explain the phenomena of boundary layer growth over a flat plate. Explain phenomenon of boundary layer separation with a neat sketch.

Q.10. (AMIE W06, 5 marks): What are boundary layer equations? How do they differ from the Navier Stokes equations?

Q.11. (AMIE S99, W06, 08, 5 marks): Define displacement thickness * and momentum thickness . Write

their formula.

OBJECTIVE QUESTIONS



A Focused ApproachQ.12. (AMIE W08, S10, 8 marks): The velocity distribution in boundary layer is given by u/v = y/, where u

is the velocity at distance y from plate and u = v at y = , being boundary layer thickness. Find the (i)

displacement thickness, (ii) momentum thickness, (iii) energy thickness, and (iv) value of */ .

Answer: /2, /6, /4, 3

Q.13. (AMIE S06, 10 marks): Discuss laminar boundary layer, boundary layer thickness and boundary layer control.

Q.14. (AMIE S99, 2000, W05, 10 marks): Explain the phenomenon of separation of boundary layer and formation of wake. Give a list of various methods of boundary layer control.

Q.15. (AMIE S05, 10 marks): What do you mean by separation of boundary layer? What is the effect of pressure gradient on boundary layer separation? How will you determine whether a boundary layer flow is attached flow, detached flow or on the verge of separation?

Q.16. (AMIE W06, 10 marks): Air is blowing past a flat plate. Assume that air velocity profile is given by expression

3 4u 2y y y

2U

Show that the boundary layer thickness is given by

x

x5.836

Re

Q.17. (AMIE W07, 12 marks): If a laminar boundary layer at zero pressure, gradient over a flat plate is described by the velocity profile.

30/ (3 / 2) / 2V V

in which = (y/). Show that the boundary layer thickness, , wall shear stress 0 and coefficient of drag CD are

given by

2

00

0.3224.65; ; 1.328 Re

Re ReD x

x x

VxC

Q.18. (AMIE S09, 8 marks): Assuming one of the standard velocity distributions for laminar boundary flow, obtain expression for the drag coefficient, boundary shear stress, and thickness of the boundary layer.

Hint: Standard velocity distribution is

30/ (3 / 2) / 2V V

in which = (y/).

Answer: 2

00

0.3224.65; ; 1.328 Re

Re ReD x

x x

VxC

Hint: energy thickness =

2

0

u u1 dy

U U

and energy loss per metre length of spillway will be

(1/2)eU3.

OBJECTIVE QUESTIONS



A Focused ApproachQ.19. (AMIE W05, S10, 10 marks): Water at 150C flows over a flat plate at a speed of 1.2 m/s. The plate is 0.3 m long and 2 m wide. The boundary layer on each side is laminar. Assume the velocity profile is approximately

a linear for which ex/ x 3.46 / R . Determine the drag force on the plate. For water: = 1.1 x 10-6 m2/s,

= 1000 kg/m3.

Answer: 0.873 N/m2

Q.20. (AMIE S2001, 12 marks): A smooth flat plate of length 5 m and width 2 m is moving with a velocity of 4 m/s in stationary air of density 1.2 kg/m3 and kinematic viscosity 1.5 x 10-5 m2/s. If the boundary layer flow changes from laminar to turbulent at Reynolds number of 5 x 105, determine (a) the total drag on one side of the plate and (b) boundary layer thickness at the transition point, if it exists.

Answer: 0.30 N, 13.258 mm

Q.21. (AMIE W06, 5 marks): Air is flowing over a flat plate with a free stream velocity of 10 m/s. At a

distance of 1 m/s from the leading edge, calculate , the boundary layer thickness. Assume turbulent flow over

for a 1/7th power law velocity profile. Assume = 1.23 kg/m3 and = 1 x 10-5 m2/s.

Answer: 25.386 mm Hint: For 1/7th power law, 1/5

x

0.371

x (Re )

Q.22. (AMIE W08, 12 marks): Water is flowing over a thin smooth plate of length 4 m and width 2 m at a velocity of 1.0 m/s. If the boundary layer flow changes from laminar to turbulent at a Reynold number 5 x 105, find the (i) distance from leading edge up to which boundary layer is laminar, (ii) thickness of the boundary

layer at the transition point, and {iii) drag force on one side of the plate. Take velocity of water = 9.8 x 10-4

Ns/m2

Answer: 4.9 cm, 0.344 cm, 12 N

Q.23. (AMIE S09, 10 marks): Atmospheric air at 250 C flows parallel to a flat plate al a velocity of 3 m/s. Use the exact Blasius solution to estimate the boundary layer thickness and the local skin friction coefficient at x = l m from the leading edge of the plate. How these values would compare with the corresponding values obtained from the approximate von-Karman integral technique? Assume cubic velocity profile for air at 250 C, n = 15.53 x 10-6 m2/s.

Answer: Blasius eq.: = 5x/Rex = 1.1376 cm, Cf = 0.664/Rex = 1.51075 x 10-3; Van Karman; = 4.64x/Rex

= 0.010557 m, Cf = 0.646/Rex = 1.4698 x 10-3

Q.24. (AMIE W97, 12 marks): Water is flowing at a velocity of 1 m/s over a smooth flat plate of length 1 m and width 2 m. Taking viscosity of water as 9.81 x 10-4 N-s/m2, determine the drag force on one side of the plate. Assume Blasius solution for determination of CD for laminar flow and CD = 0.072/(Re)0.2 for turbulent flow, where CD is drag coefficient and NR is Reynolds number.

Answer: 12.08 N on one side of plate

Hint: Flow is combined i.e. laminar upto certain distance and turbulent after that.

Hence FD = (FD)laminar + (FD)turbulent

Q.25. (AMIE S10, 12 marks): Air( = 1.23 kg/m3 and = 1.5x 10-5 m2/s) is flowing over a flat plate. The free

stream speed is 15 m/s. At a distance of 1m from the scaling edge (leading edge), calculate and w for (i)

completely laminar flow, and (ii) completely turbulent flow for a one-seventh power low velocity profile.

Answer: (i) = 4.96x/Re = 4.96 mm, x = (1/2)Cf = 0.092 N/m2 where Cf = 0.644/Re (ii) = 0.371x/Re0.2 =

23.4 mm, 0.503 N/m2, Cf = 0.0576/Re0.2

OBJECTIVE QUESTIONS



A Focused Approach

Compressible Flow

SOUND WAVE IN COMPRESSIBLE FLOW

With reference to following figure in which a sound wave os flowing with velocity c.

Applying continuity equation between (1) and (2)

( )( )cA d c dV A

Neglecting product of small quantities,

c c dV cd

or dV cd (1)

Applying momentum equation

Force in right direction = mass x change in velocity from (1) and (2)

or ( ) {( ) }A p dp A cA c dv c

or d cdV

or dp

dVc

Substituting for dV in equation (1)

dp

cdc

or 2 dpc

d

or dp

cd

This is required expression.

Example (AMIE S08, W08, S10, 8 marks)

Derive an expression for change of area and velocity for compressible fluid in a nozzle

2( 1)dA dV

MA V

OBJECTIVE QUESTIONS



A Focused ApproachSolution

From continuity eq.

.AV

m AV const

here is specific volume.

Taking logarithms, and then differentials

0dA dV d

A V

or 1dA d dV dV V d

A V V dV

(1)

From steady flow energy equation

2

.2

Vh const here h is specific enthalpy.

or 0dh VdV dh VdV (2)

From first law of thermodynamics

0Tds dh dp for isentropic flow.

or dh dp (3)

From (2) and (3)

VdV dp (4)

The isentropic relation for an ideal gas gives

pv const

Taking log, and then differentials

0dp d

p

or pd dp (5)

From (4) and (5)

VdV pdv

or d V

dV p

Substituting for /d dV in relation (1)

1dA dV V V

A V p

OBJECTIVE QUESTIONS



A Focused Approach

= 2

1dV V

V RT

as p = RT.

= 2

21

dV V

V a

as 2a RT = sonic velocity2

= 2 1dV

MV

as V/a = M, the Mach number.

Example (AMIE Summer 2010, 6 marks)

Prove that

/ 1

20 11

2

y yP y

MP

where P0 = stagnation pressure; y = isentropic index; M = Mach number.

Solution

For compressible flow

0 0

p

p

and 1/

0 0

T p

T p

etc.

The stagnation enthalpy h0 is related to static enthalpy h by the steady flow energy equation

2

0 2

Vh h

For an ideal gas, h = CpT and h0 = CpT0

2

0 0 2p

VC T C T

or 2

0 12 p

T V

T C T

But 1p

RC

2 2

0 11 1 .

22

1

T V V

T RTR

Now, 2RT a , a = sonic velocity and V/a = mach number = M.

OBJECTIVE QUESTIONS



A Focused ApproachHence, the above relation becomes,

2

202

1 11 . 1

2 2

T VM

T a

From isentropic relation,

/ 1 / 1

20 0 11

2

p TM

p T

Hence Proved.

ASSIGNMENT Q.26. (AMIE W07, 4 marks): Define Mach number and explain its significance in compressible fluid flow.

Q.27. (AMIE W07, 09, 8 marks): Obtain an expression for the sound wave in a compressible fluid in terms of change of pressure and change of density.

Q.28. (AMIE S05, 3 marks): What is a normal shock and how is it obtained?

Q.29. (AMIE W05, 10 marks): What is shock wave and under what conditions would it occur? Derive the Prandtl Mayer relation.

Q.30. (AMIE S06, 10 marks): How is a shock wave produced in a compressible fluid? Explain normal and oblique shocks.

Q.31. (AMIE W08, 4 marks): Define the terms (i) subsonic flow (ii) supersonic flow.

Q.32. (AMIE W09, 4 marks): Define (i) Mach number (ii) subsonic flow (iii) sonic flow (iv) supersonic flow.

Q.33. (AMIE S05, 3 marks): Write the basic equations i.e. continuity, momentum and energy equation for a control volume having normal shocks.

Q.34. (AMIE S05, 4 marks): How the velocity, temperature, density and entropy change across a normal shock wave.

Q.35. (AMIE W05, 10 marks): The conditions of a gas in a combustor at entry are: p1 = 0.343 bar, T1 = 310 K and C1 = 60 m/s. Determine the mach number, pressure, temperature and velocity at the exit if the increase in

stagnation enthalpy of the gas between entry and exit is 1172; 5 kJ/kg. Take Cp = 1.005 kJ/kg and = 1.4. The

question is to be solved using appropriate tables.

Answer: 0.45, 0.277 bar, 1423.2K, 340.3 m/s

Q.36. (AMIE S05, 10 marks): A tank contains air at a pressure 135 kPa and temperature 270C. The local barometric pressure is 100 kPa. Air discharges out of the tank and into atmosphere through a convergent nozzle. Determine the output flow velocity and the mass flow rate of air. The cross sectional area at the nozzle outlet is 500 m2.

Answer: 222.7 m/s, 0.14089 kg/s

Q.37. (AMIE W06, 10 marks): A convergent divergent nozzle with supersonic flow at exit has a throat area of 500 mm2 and exit area of 1000 mm2. Air enters the nozzle with a stagnation temperature of 360 K and stagnation pressure of 1 MPa. Determine the maximum flow rate that the nozzle can pass.

Answer: 1.0639 kg/s

Q.38. (AMIE W07, 08, 8 marks): Find the velocity of air flowing at the outlet of a nozzle, fitted to a large vessel which contains air at a pressure of 29.43 bar (abs) and at a temperature of 20 °C. The pressure at the outlet of the nozzle is 20.6 bar (abs). Take K= 1.4 and R = 287 J/kg K.

Answer: 238.8 m/s

OBJECTIVE QUESTIONS



A Focused ApproachQ.39. (AMIE S08, 8 marks): Determine the exit velocity and mass flow rate for isentropic flow of air through a nozzle from inlet stagnation conditions of 7 bar and 3200C to an exit pressure of 1.05 bar and the exit area of the

nozzle is 6.25 cm2. Also, determine the throat area of the nozzle. Assume = 1.4.

Answer: 706.2 m/s, 0.4683 kg/s, 4.026 cm2

Q.40. (AMIE W09, 8 marks): A nozzle is required to expand the air from 4.5 bar and 750°C to 1.1 bar. Find the throat area, outlet area and outlet temperature for a mass flow rate of 0.5 kg/s. Take the nozzle efficiency as 85% and assume the following properties for air:

Ratio of specific heat, = 1.4

Gas constant, R = 0.287 kJ/kg °K

Constant pressure specific heat, Cp = 1.005 kJ/kg. K.

Answer: 8.79 cm2, 12.49 cm2, 461.870C

Q.41. (AMIE W09, 8 marks): Air at an absolute pressure 60.0 kPa and 27°C enters a passage at 486 m/sec. The cross-sectional area at the entrance is 0.02 m2. At section 2, further downstream the pressure is 78.8 kPa (absolute). Assuming isentropic flow, calculate Mach number at section 2. Also, identify type of the nozzle.

Given = 1.4.

Answer: 1.2

Objective Questions

Set 2

1. One SI unit of viscosity is equal to

(a) 10 poise

(b) 981 poise

(e) 0.1 poise

(d) None of these

2. Pressure head of a fluid is the ratio of pressure to

(a) fluid height

(b) specific weight

(e) density

(d) specific gravity

3. In a differential manometer, the flowing fluid is water and the gauge fluid is water and the manometer reading is 10 cm, the differential head in m of water is

(a) 13.6

(b) 1.36

{c) 1.47

(d) 1.26

4. Along a stream line,

OBJECTIVE QUESTIONS



A Focused Approach(a) velocity is constant

(b) is zero

(c) is zero

(d) is constant

5. Doublet is a combination of

(a) source and vortex

(b) source and sink

(c) source and uniform flow

(d) None of these

6. Flow in a whirlpool in a river is an example of

(a) free vortex

(b) spiral vortex

(e) forced vortex

(d) radial vortex

7. The coefficient of contraction of a sharp-edged small orifice under normal conditions is

(a) 0.985

(b) 0.82

(e) 0.707

(d) 0.62

8. An error of 1.5mm is committed in the measurement of head over a triangular notch. The bead over the notch is 0.5 m. The percent error in computing discharge is

(a) 0.5

(b) 0.75

(e) 1.5

(d) 3.0

9. In series pipes, the parameter which is same in each pipe is

(a) hf

(b) Q

(e)f

(d) None of these

10. The following is not a minor loss

OBJECTIVE QUESTIONS



A Focused Approach(a) friction loss

(b) bend loss

(e) enlargement loss

(d) contraction loss

11. In laminar flow between two fixed parallel plates, the ratio of average to maximum velocities is

(a) 1/3

(b) 1/2

(e) 2/1

(d) 2/3

12. In which of the following types of flows, the shear stress is uniform across the cross-section

(a) Simple Couette flow

(b) Generalised Couette flow

(e) Poiseuille flow

(d) None of these

13. The thickness of a laminar boundary layer on a flat plate varies

(a) x

(b) x1/2

(e) x-1/2

(d) x1/7

14. The average drag coefficient of a laminar layer is

(a) 1.328

Re

(b) 0.664

Re

(c) 1/5

0.0587

Re

(d) 1/5

0.0735

Re

15. Reynolds turbulent shear stress is given by

(a) lu

(b) .u v

OBJECTIVE QUESTIONS



A Focused Approach

(c) du

dy

(d) du

dy

16. In a given rough pipe, the losses depend on

(a) Q

(b) Re

(c) ,

(d) f, D, V

17. Kinematic eddy viscosity has units

(a) poise

(b) Pascal

(e) N/s

(d) m2/s

18. Effect of compressibility of fluid can be neglected if Mach number.

(a) 0.3 to 1.0

(b) < 0.3

(c) > 1

(d) none of these

19. In a diverging passage, the velocity of supersonic flow

(a) decreases linearly

(b) decrease exponentially

(c) increases

(d) remains constant

20. Across a normal shock in compressible fluid, there is

(a) increase in p and decrease in M

(b) increase in p and s, decrease in M

(c) increase in p and M, no change in s

(d) increase in p, M and T

21. Poise is the unit of

(a) mass density

(b) kinematic viscosity

OBJECTIVE QUESTIONS



A Focused Approach (c) viscosity

(d) velocity potential

22. The point of action of hydrostatic force is known as

(a) centre of gravity

(b) centre of buoyancy

(c) centre of pressure

(d) metacentre

23. Bernoulli's theorem deals with law of conservation of

(a)mass

(b) momentum

(c) energy

(d) None of the above

24. The coefficient of friction for laminar flow through a circular pipe is given by

(a) 1/40.791/ Ref

(b) 16 / Ref

(c) 64 / Re

(d) 32 / Ref

25. The boundary layer separation takes place if

(a) pressure gradient is zero.

(b) pressure gradient is positive

(c) pressure gradient is negative

(d) pressure gradient is constant

26. Compressibility is equal to

(a) ( / ) /dV dP

(b) / ( / )dP dV

(c) /dP d

(d) /dP d

27. When the fluid is at rest the shear stress is

(a) maximum

(b) zero

(c) minimum

OBJECTIVE QUESTIONS



A Focused Approach(d) None of the above

28. For a floating body, if the metacentre coincides with the centre of gravity, the equilibrium is called

(a) stable

(b) unstable

(c) neutral

(d) None of the above

29. An oil of specific gravity 0.7 and pressure 0.14 kgf/cm2 will have the height of oil as

(a) 70 cm of oil

(b) 2 m of oil

(c) 20 cm of oil

(d) 14 cm of oil

30. The velocity distribution across a section of a circular pipe having viscous flow is given by

(a) 2max[(1 ( / ) ]u u r R

(b) 2 2max[ ]u u R r

(c) 2max[1 / ]u u r R

(d) None of the above.

31. The buoyant force for the floating body passes through the

(a) centre of gravity of the body

(b) meta-centre of the body

(c) centroid of the displaced volume

32. The pressure difference between inside and outside of a droplet of water is given by

(a) 2 /d

(b) 4 /d

(c) 8 /d


33. An ideal fluid is one which

(a) is frictionless and incompressible

(b) is viscous

(c) obeys Newton's law of viscosity

(d) All of the above

OBJECTIVE QUESTIONS



A Focused Approach34. The position of centre of pressure of a plane surface immersed in a static fluid is

(a) at the centroid of the submerged surface

(b) always above centroid

(c) always below centroid


35. Stream lines and path line always coincide in

(a) steady flow

(b) uniform flow

(c) non-uniform flow

(d) laminar flow.

36. Navier Stokes equations are useful in the analysis of

(a) turbulent flows

(b) vortex flows

(c) viscous flows

(d) rotational flows

37. The velocity distribution at any section of a pipe for steady laminar flow is

(a) linear

(b) exponential

(c) parabolic

(d) hyperbolic

38. The speed of pressure wave depends upon

(a) initial velocity of fluid

(b) viscosity of flowing fluid

(c) diameter of pipe

(d) density of flowing fluid

39. The velocity of sound is largest in

(a) air

(b) kerosene

(c) water

(d) steel

40. The range of coefficient of discharge for a venturimeter is

(a) 0.6 - 0.7

OBJECTIVE QUESTIONS



A Focused Approach(b) 0·7 - 0.85

(c) 0.92 - 0.98

(d) 0.85 - 0.92

41. Correct unit of kinematic viscosity is

(a) m2/s

(b) Ns/m2

(c) m/kg-s

(d) kg/m2-s

42. All liquid surfaces tend to stretch. This phenomenon is called

(a) cohesion

(b) adhesion

(c) surface tension

(d) cavitation

43. A metallic piece weighs 78.5 N in air and 58.8 N in water. The relative density of the metal would be

(a) 8

(b) 4

(c) 6

(d) 3

44. In uniform flow, the velocities of fluid particles are:

(a) equal at all sections

(b) always dependent on time

(c) mutually perpendicular to each other

(d) the fluid particles move in well defined paths.

45. Pressure loss for laminar flow through pipeline is dependent

(a) directly on square of flow velocity

(b) directly on square of pipe radius

(c) directly as length of pipe

(d) inversely on viscosity of flowering medium.

46. The velocity potential function in a two-dimensional flow field is given by 2 2x y . The magnitude of velocity at point P(1, 1) is

(a) zero

OBJECTIVE QUESTIONS



A Focused Approach(b) 2

(c) 22

(d) 8

47. Identify the Bernoulli's equation, where each term represents energy per unit mass:

(a) 2

.2

P Vy const

w g

(b) 2

.2

P Vgy const

(c) 2

2

VP wy const

(d) none of these

48. Discharge is measured by

(a) current meter

(b) pitot tube

(c) venturimeter

(d) hot wire anemometer

49. The relation 2 2

2 20

x y

for an irrotational flow is referred to as

(a) Euler's equation

(b) Laplace equation

(c) Reynold's equation

(d) Cauchy-Riemann's equation

50. Fluid is a substance which offers no resistance to change of

(a) pressure

(b) flow

(c) shape

(d) volume

(e) temperature

51. A fluid is said to be ideal, if it is

(a) incompressible

(b) inviscous

(c) viscous and incompressible

OBJECTIVE QUESTIONS



A Focused Approach(d) viscous and compressible.

(e) inviscous and incompressible

52. The unit of viscosity is

(a) m2/s

(b) kg/m-s

(c) Ns/m2

(d) Ns2/m

(e) None of the above

53. A one-dimensional flow is one which

(a) is uniform flow

(b) is steady uniform flow

(c) takes place in straight lines

(d) involves zero transverse component of flow

(e) takes place in one dimension.

54. The line of action of the buoyant force acts through the

(a) centroid of the, volume of fluid vertically above the body

(b) centre of the volume of floating body

(c) centre of gravity of any submerged body

(d) centroid of the displaced volume of fluid

(e) None of the above

55. Differential manometer is used to measure

(a) pressure in pipes, channels, etc.

(b) atmospheric pressure

(c) very low pressure

(d) difference of pressure between two points

(e) velocity in pipes

56. An ideal flow of any fluid must satisfy

(a) Pascal law

(b) Newton's law of viscosity

(c) boundary layer theory

(d) continuity equation

(e) Bernoulli's theorem

OBJECTIVE QUESTIONS



A Focused Approach57. A stream line is defined as the line

(a) parallel to central axis flow

(b) parallel to outer surface of pipe

(c) a tangent to it at the point gives the direction of velocity.

(d) along which the pressure drop is uniform.

(e) which occurs in all flows.

58. Separation of flow occurs when pressure gradient

(a) tends to approach zero.

(b) becomes negative

(c) changes abruptly

(d) reduces to value when vapour formation starts

59. For a laminar flow

(a) flow .occurs in zig-zag way.

(b) Reynolds number lies between 2000 and 3000 for pipes.

(c) Newton's law of viscosity is of importance.

(d) pipe losses are major considerations

(e) velocity of flow is maximum.

60. Orifices are used for

(a) velocity

(b) pressure

(c) rate of flow

(d) none of these

61. For supersonic flow, if the area of flow increases, then

(a) velocity decreases

(b) velocity increases

(c) velocity remains constant

(d) none of these

62. Pressure drag results from

(a) skin friction

(b) deformation drag

(c) principle cause of skin friction

(d) always occur when deformation drag predominates

63. The growth of boundary layer is supported when

OBJECTIVE QUESTIONS



A Focused Approach (a) /p x is positive

(b) /p x is zero

(c) /p x is negative

Key to Set 2

1. a

2. b

3. d

4. d

5. b

6. a

7. d

8. b

9. b

10. a

11. d

12. a

13. b

14. a

15. b

16. d

17. d

18. b

19. c

20. b

21. c

22. c

23. c

24. b

25. b

26. a

27. b

28. c

29. b

30. a

31. c

32. b

33. a

34. c

35. a

36. c

37. c

38. d

39. d

40. c

41. a

42. c

43. b

44. a

45. c

46. c

47. b

48. c

49. b

50. c

51. e

52. c

53. d

54. d

55. d

56. e

57. c

58. b

59. c

60. c

61. b

62. b

63. a

POST BOX NO.77, 2 ND FLOOR, SULTAN TOWERS, ROORKEE – 247667 UTTARANCHAL PH: (01332) 266328 Email : [email protected] 43

TOTAL PAGES:


A Focused Approach