# The general rules of statics (as applied in solid mechanics) apply to fluids at rest. From earlier we know that:

Size: px
Start display at page:

Download "The general rules of statics (as applied in solid mechanics) apply to fluids at rest. From earlier we know that:"

Transcription

1 ELEMENTARY HYDRAULICS National Certificate in Technology (Civil Engineering) Chapter 2 Pressure This section will study the forces acting on or generated by fluids at rest. Objectives Introduce the concept of pressure; Prove it has a unique value at any particular elevation; Show how it varies with depth according to the hydrostatic equation and Show how pressure can be expressed in terms of head of fluid. This understanding of pressure will then be used to demonstrate methods of pressure measurement that will be useful later with fluid in motion and also to analyze the forces on submerges surface/structures. 1. Fluids statics The general rules of statics (as applied in solid mechanics) apply to fluids at rest. From earlier we know that: a static fluid can have no shearing force acting on it, and that any force between the fluid and the boundary must be acting at right angles to the boundary. Pressure force normal to the boundary Note that this statement is also true for curved surfaces, in this case the force acting at any point is normal to the surface at that point. The statement is also true for any imaginary plane in a static fluid. We use this fact in our analysis by considering elements of fluid bounded by imaginary planes. We also know that: 1

2 For an element of fluid at rest, the element will be in equilibrium - the sum of the components of forces in any direction will be zero. The sum of the moments of forces on the element about any point must also be zero. It is common to test equilibrium by resolving forces along three mutually perpendicular axes and also by taking moments in three mutually perpendicular planes an to equate these to zero. 2. Pressure As mentioned above a fluid will exert a normal force on any boundary it is in contact with. Since these boundaries may be large and the force may differ from place to place it is convenient to work in terms of pressure, p, which is the force per unit area. If the force exerted on each unit area of a boundary is the same, the pressure is said to be uniform. Units: Newton's per square metre,,. (The same unit is also known as a Pascal, Pa, i.e. 1Pa = 1 ) (Also frequently used is the alternative SI unit the bar, where ) Dimensions:. 3. Pascal's Law for Pressure At A Point (Proof that pressure acts equally in all directions.) By considering a small element of fluid in the form of a triangular prism which contains a point P, we can establish a relationship between the three pressures p x in the x direction, p y in the y direction and p s in the direction normal to the sloping face. 2

3 Triangular prismatic element of fluid The fluid is a rest, so we know there are no shearing forces, and we know that all force are acting at right angles to the surfaces.i.e. acts perpendicular to surface ABCD, acts perpendicular to surface ABFE and acts perpendicular to surface FECD. And, as the fluid is at rest, in equilibrium, the sum of the forces in any direction is zero. Summing forces in the x-direction: Force due to, Component of force in the x-direction due to, ( ) Component of force in x-direction due to, To be at rest (in equilibrium) 3

4 Similarly, summing forces in the y-direction. Force due to, Component of force due to, ( ) Component of force due to, Force due to gravity, To be at rest (in equilibrium) The element is small i.e., and are small, and so is very small and considered negligible, hence thus Considering the prismatic element again, is the pressure on a plane at any angle, the x, y and z directions could be any orientation. The element is so small that it can be considered a point so the 4

5 derived expression. indicates that pressure at any point is the same in all directions. (The proof may be extended to include the z axis). Pressure at any point is the same in all directions. This is known as Pascal's Law and applies to fluids at rest. 4. Variation Of Pressure Vertically In A Fluid Under Gravity Vertical elemental cylinder of fluid In the above figure we can see an element of fluid which is a vertical column of constant cross sectional area, A, surrounded by the same fluid of mass density. The pressure at the bottom of the cylinder is at level, and at the top is at level. The fluid is at rest and in equilibrium so all the forces in the vertical direction sum to zero. i.e. we have Taking upward as positive, in equilibrium we have Thus in a fluid under gravity, pressure decreases with increase in height. 5

6 5. Equality Of Pressure At The Same Level In A Static Fluid Consider the horizontal cylindrical element of fluid in the figure below, with cross-sectional area A, in a fluid of density, pressure at the left hand end and pressure at the right hand end. Horizontal elemental cylinder of fluid The fluid is at equilibrium so the sum of the forces acting in the x direction is zero. Pressure in the horizontal direction is constant. This result is the same for any continuous fluid. It is still true for two connected tanks which appear not to have any direct connection, for example consider the tank in the figure below. Two tanks of different cross-section connected by a pipe We have shown above that and from the equation for a vertical pressure change we have and 6

7 so This shows that the pressures at the two equal levels, P and Q are the same. 6. General Equation For Variation Of Pressure In A Static Fluid Here we show how the above observations for vertical and horizontal elements of fluids can be generalised for an element of any orientation. A cylindrical element of fluid at an arbitrary orientation. Consider the cylindrical element of fluid in the figure above, inclined at an angle to the vertical, length, cross-sectional area A in a static fluid of mass density. The pressure at the end with height is and at the end of height is. The forces acting on the element are There are also forces from the surrounding fluid acting normal to these sides of the element. For equilibrium of the element the resultant of forces in any direction is zero. 7

8 Resolving the forces in the direction along the central axis gives Or in the differential form If then s is in the x or y directions, (i.e. horizontal),so Confirming that pressure on any horizontal plane is zero. If then s is in the z direction (vertical) so Confirming the result 7. Pressure and Head In a static fluid of constant density we have the relationship integrated to give, as shown above. This can be In a liquid with a free surface the pressure at any depth z measured from the free surface so that z = - h (see the figure below) 8

9 Fluid head measurement in a tank. This gives the pressure At the surface of fluids we are normally concerned with, the pressure is the atmospheric pressure,. So As we live constantly under the pressure of the atmosphere, and everything else exists under this pressure, it is convenient (and often done) to take atmospheric pressure as the datum. So we quote pressure as above or below atmospheric. Pressure quoted in this way is known as gauge pressure i.e. Gauge pressure is The lower limit of any pressure is zero - that is the pressure in a perfect vacuum. Pressure measured above this datum is known as absolute pressure i.e. Absolute pressure is As g is (approximately) constant, the gauge pressure can be given by stating the vertical height of any fluid of density which is equal to this pressure. This vertical height is known as head of fluid. 9

10 Note: If pressure is quoted in head, the density of the fluid must also be given. Example: We can quote a pressure of in terms of the height of a column of water of density,. Using, And in terms of Mercury with density,. Pressure Measurement By Manometer The relationship between pressure and head is used to measure pressure with a manometer (also know as a liquid gauge). Objective: To demonstrate the analysis and use of various types of manometers for pressure measurement. 1. The Piezometer Tube Manometer The simplest manometer is a tube, open at the top, which is attached to the top of a vessel containing liquid at a pressure (higher than atmospheric) to be measured. An example can be seen in the figure below. This simple device is known as apiezometer tube. As the tube is open to the atmosphere the pressure measured is relative to atmospheric so is gauge pressure. 10

11 A simple piezometer tube manometer This method can only be used for liquids (i.e. not for gases) and only when the liquid height is convenient to measure. It must not be too small or too large and pressure changes must be detectable. 2. The "U"-Tube Manometer Using a "U"-Tube enables the pressure of both liquids and gases to be measured with the same instrument. The "U" is connected as in the figure below and filled with a fluid called the manometric fluid. The fluid whose pressure is being measured should have a mass density less than that of the manometric fluid and the two fluids should not be able to mix readily - that is, they must be immiscible. A "U"-Tube manometer Pressure in a continuous static fluid is the same at any horizontal level so, For the left hand arm For the right hand arm 11

12 As we are measuring gauge pressure we can subtract giving If the fluid being measured is a gas, the density will probably be very low in comparison to the density of the manometric fluid i.e. man >> pressure give by can be neglected, and the gauge 3. Measurement Of Pressure Difference Using a "U"-Tube Manometer. If the "U"-tube manometer is connected to a pressurised vessel at two points the pressure difference between these two points can be measured. Pressure difference measurement by the "U"-Tube manometer If the manometer is arranged as in the figure above, then 12

13 Giving the pressure difference Again, if the fluid whose pressure difference is being measured is a gas and involving can be neglected, so, then the terms 4. Advances to the "U" tube manometer. The "U"-tube manometer has the disadvantage that the change in height of the liquid in both sides must be read. This can be avoided by making the diameter of one side very large compared to the other. In this case the side with the large area moves very little when the small area side move considerably more. Assume the manometer is arranged as above to measure the pressure difference of a gas of (negligible density) and that pressure difference is. If the datum line indicates the level of the manometric fluid when the pressure difference is zero and the height differences when pressure is applied is as shown, the volume of liquid transferred from the left side to the right And the fall in level of the left side is 13

14 We know from the theory of the "U" tube manometer that the height different in the two columns gives the pressure difference so Clearly if D is very much larger than d then (d/d) 2 is very small so So only one reading need be taken to measure the pressure difference. If the pressure to be measured is very small then tilting the arm provides a convenient way of obtaining a larger (more easily read) movement of the manometer. The above arrangement with a tilted arm is shown in the figure below. Tilted manometer. The pressure difference is still given by the height change of the manometric fluid but by placing the scale along the line of the tilted arm and taking this reading large movements will be observed. The pressure difference is then given by 14

15 The sensitivity to pressure change can be increased further by a greater inclination of the manometer arm, alternatively the density of the manometric fluid may be changed. 5. Choice of Manometer Care must be taken when attaching the manometer to vessel, no burrs must be present around this joint. Burrs would alter the flow causing local pressure variations to affect the measurement. Some disadvantages of manometers: Slow response - only really useful for very slowly varying pressures - no use at all for fluctuating pressures; For the "U" tube manometer two measurements must be taken simultaneously to get the h value. This may be avoided by using a tube with a much larger cross-sectional area on one side of the manometer than the other; It is often difficult to measure small variations in pressure - a different manometric fluid may be required - alternatively a sloping manometer may be employed; It cannot be used for very large pressures unless several manometers are connected in series; For very accurate work the temperature and relationship between temperature and must be known; Some advantages of manometers: They are very simple. No calibration is required - the pressure can be calculated from first principles. Forces on Submerged Surfaces in Static Fluids We have seen the following features of statics fluids Hydrostatic vertical pressure distribution Pressures at any equal depths in a continuous fluid are equal Pressure at a point acts equally in all directions (Pascal's law). Forces from a fluid on a boundary acts at right angles to that boundary. Objectives: We will use these to analyse and obtain expressions for the forces on submerged surfaces. In doing this it should also be clear the difference between: Pressure which is a scalar quantity whose value is equal in all directions and, Force, which is a vector quantity having both magnitude and direction. 1. Fluid pressure on a surface Pressure is defined as force per unit area. If a pressure p acts on a small area on that area will be then the force exerted Since the fluid is at rest the force will act at right-angles to the surface. 15

16 General submerged plane Consider the plane surface shown in the figure below. The total area is made up of many elemental areas. The force on each elemental area is always normal to the surface but, in general, each force is of different magnitude as the pressure usually varies. We can find the total or resultant force, R, on the plane by summing up all of the forces on the small elements i.e. This resultant force will act through the centre of pressure, hence we can say If the surface is a plane the force can be represented by one single resultant force, acting at right-angles to the plane through the centre of pressure. Horizontal submerged plane For a horizontal plane submerged in a liquid (or a plane experiencing uniform pressure over its surface), the pressure, p, will be equal at all points of the surface. Thus the resultant force will be given by Curved submerged surface If the surface is curved, each elemental force will be a different magnitude and in different direction but still normal to the surface of that element. The resultant force can be found by resolving all forces into orthogonal co-ordinate directions to obtain its magnitude and direction. This will always be less than the sum of the individual forces,. 2. Resultant Force and Centre of Pressure on a submerged plane surface in a liquid. 16

17 This plane surface is totally submerged in a liquid of density and inclined at an angle of to the horizontal. Taking pressure as zero at the surface and measuring down from the surface, the pressure on an element, submerged a distance z, is given by and therefore the force on the element is The resultant force can be found by summing all of these forces i.e. (assuming and g as constant). The term to i.e. is known as the 1 st Moment of Area of the plane PQ about the free surface. It is equal where A is the area of the plane and is the depth (distance from the free surface) to the centroid, G. This can also be written in terms of distance from point O ( as ) Thus: The resultant force on a plane 17

18 This resultant force acts at right angles to the plane through the centre of pressure, C, at a depth D. The moment of R about any point will be equal to the sum of the moments of the forces on all the elements pressure. of the plane about the same point. We use this to find the position of the centre of It is convenient to take moments about the point where a projection of the plane passes through the surface, point O in the figure. We can calculate the force on each elemental area: And the moment of this force is: are the same for each element, so the total moment is We know the resultant force from above C, so, which acts through the centre of pressure at Equating gives, Thus the position of the centre of pressure along the plane measure from the point O is: It look a rather difficult formula to calculate - particularly the summation term. Fortunately this term is known as the 2 nd Moment of Area,, of the plane about the axis through O and it can be easily calculated for many common shapes. So, we know: 18

19 And as we have also seen that 1 st Moment of area about a line through O, Thus the position of the centre of pressure along the plane measure from the point O is: and depth to the centre of pressure is How do you calculate the 2 nd moment of area? To calculate the 2 nd moment of area of a plane about an axis through O, we use the parallel axis theorem together with values of the 2 nd moment of area about an axis though the centroid of the shape obtained from tables of geometric properties. The parallel axis theorem can be written where is the 2 nd moment of area about an axis though the centroid G of the plane. Using this we get the following expressions for the position of the centre of pressure (In the examination the parallel axis theorem and the will be given) The second moment of area of some common shapes. 19

20 The table blow given some examples of the 2 nd moment of area about a line through the centroid of some common shapes. Shape Rectangle Area A 2 nd moment of area,, about an axis through the centroid Triangle Circle Semicircle Lateral position of Centre of Pressure If the shape is symmetrical the centre of pressure lies on the line of symmetry. But if it is not symmetrical its position must be found by taking moments about the line OG in the same way as we took moments along the line through O, i.e. but we have so 20

21 3. Submerged vertical surface - Pressure diagrams For vertical walls of constant width it is usually much easier to find the resultant force and centre of pressure. This is done graphically by means of a pressure diagram. Consider the tank in the diagram below having vertical walls and holding a liquid of density to a depth of H. To the right can be seen a graphical representation of the (gauge) pressure change with depth on one of the vertical walls. Pressure increases from zero at the surface linearly by, to a maximum at the base of. Pressure diagram for vertical wall. The area of this triangle represents the resultant force per unit width on the vertical wall, using SI units this would have units of Newtons per metre. So Resultant force per unit width The force acts through the centroid of the pressure diagram. For a triangle the centroid is at 2/3 its height, i.e. in the figure above the resultant force acts horizontally through the point. For a vertical plane the depth to the centre of pressure is given by 21

22 This can be checked against the previous method: The resultant force is given by: and the depth to the centre of pressure by: and by the parallel axis theorem (with width of 1) Giving depth to the centre of pressure These two results are identical to the pressure diagram method. The same pressure diagram technique can be used when combinations of liquids are held in tanks (e.g. oil floating on water) with position of action found by taking moments of the individual resultant forces for each fluid. Look at the examples to examine this area further. More complex pressure diagrams can be draw for non-rectangular or non-vertical planes but it is usually far easier to use the moments method. 4. Resultant force on a submerged curved surface As stated above, if the surface is curved the forces on each element of the surface will not be parallel and must be combined using some vectorial method. It is most straightforward to calculate the horizontal and vertical components and combine these to obtain the resultant force and its direction. (This can also be done for all three dimensions, but here we will only look at one vertical plane). 22

23 In the diagram below the liquid is resting on top of a curved base. The element of fluid ABC is equilibrium (as the fluid is at rest). Horizontal forces Considering the horizontal forces, none can act on CB as there are no shear forces in a static fluid so the forces would act on the faces AC and AB as shown below. We can see that the horizontal force on AC, resultant force on the curved surface., must equal and be in the opposite direction to the As AC is the projection of the curved surface AB onto a vertical plane, we can generalise this to say The resultant horizontal force of a fluid above a curved surface is: R H = Resultant force on the projection of the curved surface onto a vertical plane. We know that the force on a vertical plane must act horizontally (as it acts normal to the plane) and that must act through the same point. So we can say R H acts horizontally through the centre of pressure of the projection of the curved surface onto an vertical plane. Thus we can use the pressure diagram method to calculate the position and magnitude of the resultant horizontal force on a two dimensional curved surface. 23

24 Vertical forces The diagram below shows the vertical forces which act on the element of fluid above the curved surface. There are no shear force on the vertical edges, so the vertical component can only be due to the weight of the fluid. So we can say The resultant vertical force of a fluid above a curved surface is: R V = Weight of fluid directly above the curved surface. and it will act vertically downward through the centre of gravity of the mass of fluid. Resultant force The overall resultant force is found by combining the vertical and horizontal components vectorialy, Resultant force And acts through O at an angle of. The angle the resultant force makes to the horizontal is 24

25 The position of O is the point of integration of the horizontal line of action of of action of. and the vertical line What are the forces if the fluid is below the curved surface? This situation may occur or a curved sluice gate for example. The figure below shows a situation where there is a curved surface which is experiencing fluid pressure from below. The calculation of the forces acting from the fluid below is very similar to when the fluid is above. Horizontal force From the figure below we can see the only two horizontal forces on the area of fluid, which is in equilibrium, are the horizontal reaction force which is equal and in the opposite direction to the pressure force on the vertical plane A'B. The resultant horizontal force, R H acts as shown in the diagram. Thus we can say: The resultant horizontal force of a fluid below a curved surface is: Vertical force The vertical force are acting are as shown on the figure below. If the curved surface were removed and the area it were replaced by the fluid, the whole system would be in equilibrium. Thus the force 25

26 required by the curved surface to maintain equilibrium is equal to that force which the fluid above the surface would exert - i.e. the weight of the fluid. Thus we can say: The resultant vertical force of a fluid below a curved surface is: R v =Weight of the imaginary volume of fluid vertically above the curved surface. The resultant force and direction of application are calculated in the same way as for fluids above the surface: Resultant force And acts through O at an angle of. The angle the resultant force makes to the horizontal is Instructor: E.M.Shalika Manoj Ekanayake Student (Civil Eng.) College of Engineering,Sri Lanka E Mail: Web: 26

### CHAPTER 2 Pressure and Head

FLUID MECHANICS Gaza, Sep. 2012 CHAPTER 2 Pressure and Head Dr. Khalil Mahmoud ALASTAL Objectives of this Chapter: Introduce the concept of pressure. Prove it has a unique value at any particular elevation.

### Any of the book listed below are more than adequate for this module.

Notes For the Level 1 Lecture Course in Fluid Mechanics Department of Civil Engineering, University of Tikrit. FLUID MECHANICS Ass. Proff. Yaseen Ali Salih 2016 1. Contents of the module 2. Objectives:

### For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then:

Hydraulic Coefficient & Flow Measurements ELEMENTARY HYDRAULICS National Certificate in Technology (Civil Engineering) Chapter 3 1. Mass flow rate If we want to measure the rate at which water is flowing

### If a stream of uniform velocity flows into a blunt body, the stream lines take a pattern similar to this: Streamlines around a blunt body

Venturimeter & Orificemeter ELEMENTARY HYDRAULICS National Certificate in Technology (Civil Engineering) Chapter 5 Applications of the Bernoulli Equation The Bernoulli equation can be applied to a great

### Fluid Statics. Pressure. Pressure

Pressure Fluid Statics Variation of Pressure with Position in a Fluid Measurement of Pressure Hydrostatic Thrusts on Submerged Surfaces Plane Surfaces Curved Surfaces ddendum First and Second Moment of

### Introduction to Marine Hydrodynamics

1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering Shanghai Jiao Tong University

### Fluid Mechanics. Forces on Fluid Elements. Fluid Elements - Definition:

Fluid Mechanics Chapter 2: Fluid Statics Lecture 3 Forces on Fluid Elements Fluid Elements - Definition: Fluid element can be defined as an infinitesimal region of the fluid continuum in isolation from

### Static Forces on Surfaces-Buoyancy. Fluid Mechanics. There are two cases: Case I: if the fluid is above the curved surface:

Force on a Curved Surface due to Hydrostatic Pressure If the surface is curved, the forces on each element of the surface will not be parallel (normal to the surface at each point) and must be combined

### GATE PSU. Chemical Engineering. Fluid Mechanics. For. The Gate Coach 28, Jia Sarai, Near IIT Hauzkhas, New Delhi 16 (+91) ,

For GATE PSU Chemical Engineering Fluid Mechanics GATE Syllabus Fluid statics, Newtonian and non-newtonian fluids, Bernoulli equation, Macroscopic friction factors, energy balance, dimensional analysis,

### Chapter 1 INTRODUCTION

Chapter 1 INTRODUCTION 1-1 The Fluid. 1-2 Dimensions. 1-3 Units. 1-4 Fluid Properties. 1 1-1 The Fluid: It is the substance that deforms continuously when subjected to a shear stress. Matter Solid Fluid

### Hydrostatic. Pressure distribution in a static fluid and its effects on solid surfaces and on floating and submerged bodies.

Hydrostatic Pressure distribution in a static fluid and its effects on solid surfaces and on floating and submerged bodies. M. Bahrami ENSC 283 Spring 2009 1 Fluid at rest hydrostatic condition: when a

### Fluid Mechanics-61341

An-Najah National University College of Engineering Fluid Mechanics-61341 Chapter [2] Fluid Statics 1 Fluid Mechanics-2nd Semester 2010- [2] Fluid Statics Fluid Statics Problems Fluid statics refers to

### Fluid Mechanics Discussion. Prepared By: Dr.Khalil M. Al-Astal Eng.Ahmed S. Al-Agha Eng.Ruba M. Awad

Discussion Prepared By: Dr.Khalil M. Al-Astal Eng.Ahmed S. Al-Agha Eng.Ruba M. Awad 2014-2015 Chapter (1) Fluids and their Properties Fluids and their Properties Fluids (Liquids or gases) which a substance

### ː ˠ Ǫː Ǫ. ʿǪ ǪȅԹ ˠ. Save from:

ː ˠ Ǫː Ǫ قسم الهندسة الكيمیاویة Ǫ ߧ Ǫلثانیة ʿǪ ǪȅԹ ˠ د. ˡ Զ ބ, د.صلاح سلمان Save from: http://www.uotechnology.edu.iq/dep-chem-eng/index.htm FLUID FLOW- 2 nd Year CHEMICAL ENG. DEPT. LECTURE-TWO-

### Chapter 3 Fluid Statics

Chapter 3 Fluid Statics 3.1 Pressure Pressure : The ratio of normal force to area at a point. Pressure often varies from point to point. Pressure is a scalar quantity; it has magnitude only It produces

### CHARACTERISTIC OF FLUIDS. A fluid is defined as a substance that deforms continuously when acted on by a shearing stress at any magnitude.

CHARACTERISTIC OF FLUIDS A fluid is defined as a substance that deforms continuously when acted on by a shearing stress at any magnitude. In a fluid at rest, normal stress is called pressure. 1 Dimensions,

### Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur. Lecture - 9 Fluid Statics Part VI

Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 9 Fluid Statics Part VI Good morning, I welcome you all to this session of Fluid

### MULTIPLE-CHOICE PROBLEMS:(Two marks per answer) (Circle the Letter Beside the Most Correct Answer in the Questions Below.)

MULTIPLE-CHOICE PROLEMS:(Two marks per answer) (Circle the Letter eside the Most Correct Answer in the Questions elow.) 1. The absolute viscosity µ of a fluid is primarily a function of: a. Density. b.

### Applications of Hydrostatics

Applications of Hydrostatics Pressure measurement with hydrostatics Mercury Barometer - This is a device used to measure the local atmospheric pressure, p a. As seen in the sketch, it is formed by inverting

### Eric G. Paterson. Spring 2005

Eric G. Paterson Department of Mechanical and Nuclear Engineering Pennsylvania State University Spring 2005 Reading and Homework Read Chapter 3. Homework Set #2 has been posted. Due date: Friday 21 January.

### Fluid Mechanics. du dy

FLUID MECHANICS Technical English - I 1 th week Fluid Mechanics FLUID STATICS FLUID DYNAMICS Fluid Statics or Hydrostatics is the study of fluids at rest. The main equation required for this is Newton's

### Fluid Mechanics Testbank By David Admiraal

Fluid Mechanics Testbank By David Admiraal This testbank was created for an introductory fluid mechanics class. The primary intentions of the testbank are to help students improve their performance on

### CE MECHANICS OF FLUIDS

CE60 - MECHANICS OF FLUIDS (FOR III SEMESTER) UNIT II FLUID STATICS & KINEMATICS PREPARED BY R.SURYA, M.E Assistant Professor DEPARTMENT OF CIVIL ENGINEERING DEPARTMENT OF CIVIL ENGINEERING SRI VIDYA COLLEGE

### storage tank, or the hull of a ship at rest, is subjected to fluid pressure distributed over its surface.

Hydrostatic Forces on Submerged Plane Surfaces Hydrostatic forces mean forces exerted by fluid at rest. - A plate exposed to a liquid, such as a gate valve in a dam, the wall of a liquid storage tank,

### Approximate physical properties of selected fluids All properties are given at pressure kn/m 2 and temperature 15 C.

Appendix FLUID MECHANICS Approximate physical properties of selected fluids All properties are given at pressure 101. kn/m and temperature 15 C. Liquids Density (kg/m ) Dynamic viscosity (N s/m ) Surface

### ACE Engineering College

ACE Engineering College Ankushapur (V), Ghatkesar (M), R.R.Dist 501 301. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MECHANICS OF FLUIDS & HYDRAULIC

### Hydrostatics. ENGR 5961 Fluid Mechanics I: Dr. Y.S. Muzychka

1 Hydrostatics 2 Introduction In Fluid Mechanics hydrostatics considers fluids at rest: typically fluid pressure on stationary bodies and surfaces, pressure measurements, buoyancy and flotation, and fluid

### Chapter 14. Lecture 1 Fluid Mechanics. Dr. Armen Kocharian

Chapter 14 Lecture 1 Fluid Mechanics Dr. Armen Kocharian States of Matter Solid Has a definite volume and shape Liquid Has a definite volume but not a definite shape Gas unconfined Has neither a definite

### Chapter 14. Fluid Mechanics

Chapter 14 Fluid Mechanics States of Matter Solid Has a definite volume and shape Liquid Has a definite volume but not a definite shape Gas unconfined Has neither a definite volume nor shape All of these

### MULTIPLE-CHOICE PROBLEMS :(Two marks per answer) (Circle the Letter Beside the Most Correct Answer in the Questions Below.)

Test Midterm 1 F2013 MULTIPLE-CHOICE PROBLEMS :(Two marks per answer) (Circle the Letter Beside the Most Correct nswer in the Questions Below.) 1. The absolute viscosity µ of a fluid is primarily a function

### 10.52 Mechanics of Fluids Spring 2006 Problem Set 3

10.52 Mechanics of Fluids Spring 2006 Problem Set 3 Problem 1 Mass transfer studies involving the transport of a solute from a gas to a liquid often involve the use of a laminar jet of liquid. The situation

### 1 Fluid Statics. 1.1 Fluid Properties. Fluid

1 Fluid Statics 1.1 Fluid Properties Fluid A fluid is a substance, which deforms when subjected to a force. A fluid can offer no permanent resistance to any force causing change of shape. Fluid flow under

### CE MECHANICS OF FLUIDS UNIT I

CE 6303- MECHANICS OF FLUIDS UNIT I 1. Define specific volume of a fluid and write its unit [N/D-14][M/J-11] Volume per unit mass of a fluid is called specific volume. Unit: m3 / kg. 2. Name the devices

### States of matter. Density high > high >> low (pressure dependent)

Fluids States of matter Solids Fluids crystalline amorphous liquids gasses Inter-atomic forces strong > strong >> very weak Density high > high >> low (pressure dependent) Density is an important material

### 11.1 Mass Density. Fluids are materials that can flow, and they include both gases and liquids. The mass density of a liquid or gas is an

Chapter 11 Fluids 11.1 Mass Density Fluids are materials that can flow, and they include both gases and liquids. The mass density of a liquid or gas is an important factor that determines its behavior

### Fluid Mechanics Abdusselam Altunkaynak

Fluid Mechanics Abdusselam Altunkaynak 2.3.3 Pascal s Law Let s consider a closed container filled with gas. We have seen earlier that the pressure at a point can be determined in relation with the pressure

### Fluid Mechanics. If deformation is small, the stress in a body is proportional to the corresponding

Fluid Mechanics HOOKE'S LAW If deformation is small, the stress in a body is proportional to the corresponding strain. In the elasticity limit stress and strain Stress/strain = Const. = Modulus of elasticity.

### Nicholas J. Giordano. Chapter 10 Fluids

Nicholas J. Giordano www.cengage.com/physics/giordano Chapter 10 Fluids Fluids A fluid may be either a liquid or a gas Some characteristics of a fluid Flows from one place to another Shape varies according

### Chapter 4 DYNAMICS OF FLUID FLOW

Faculty Of Engineering at Shobra nd Year Civil - 016 Chapter 4 DYNAMICS OF FLUID FLOW 4-1 Types of Energy 4- Euler s Equation 4-3 Bernoulli s Equation 4-4 Total Energy Line (TEL) and Hydraulic Grade Line

### Formulae that you may or may not find useful. E v = V. dy dx = v u. y cp y = I xc/a y. Volume of an entire sphere = 4πr3 = πd3

CE30 Test 1 Solution Key Date: 26 Sept. 2017 COVER PAGE Write your name on each sheet of paper that you hand in. Read all questions very carefully. If the problem statement is not clear, you should ask

### NEWTON S LAWS OF MOTION (EQUATION OF MOTION) (Sections )

NEWTON S LAWS OF MOTION (EQUATION OF MOTION) (Sections 13.1-13.3) Today s Objectives: Students will be able to: a) Write the equation of motion for an accelerating body. b) Draw the free-body and kinetic

### R09. d water surface. Prove that the depth of pressure is equal to p +.

Code No:A109210105 R09 SET-1 B.Tech II Year - I Semester Examinations, December 2011 FLUID MECHANICS (CIVIL ENGINEERING) Time: 3 hours Max. Marks: 75 Answer any five questions All questions carry equal

### TOPICS. Density. Pressure. Variation of Pressure with Depth. Pressure Measurements. Buoyant Forces-Archimedes Principle

Lecture 6 Fluids TOPICS Density Pressure Variation of Pressure with Depth Pressure Measurements Buoyant Forces-Archimedes Principle Surface Tension ( External source ) Viscosity ( External source ) Equation

### FE Fluids Review March 23, 2012 Steve Burian (Civil & Environmental Engineering)

Topic: Fluid Properties 1. If 6 m 3 of oil weighs 47 kn, calculate its specific weight, density, and specific gravity. 2. 10.0 L of an incompressible liquid exert a force of 20 N at the earth s surface.

### P = ρ{ g a } + µ 2 V II. FLUID STATICS

II. FLUID STATICS From a force analysis on a triangular fluid element at rest, the following three concepts are easily developed: For a continuous, hydrostatic, shear free fluid: 1. Pressure is constant

### Chapter (6) Energy Equation and Its Applications

Chapter (6) Energy Equation and Its Applications Bernoulli Equation Bernoulli equation is one of the most useful equations in fluid mechanics and hydraulics. And it s a statement of the principle of conservation

### 2.The lines that are tangent to the velocity vectors throughout the flow field are called steady flow lines. True or False A. True B.

CHAPTER 03 1. Write Newton's second law of motion. YOUR ANSWER: F = ma 2.The lines that are tangent to the velocity vectors throughout the flow field are called steady flow lines. True or False 3.Streamwise

### Fluid Mechanics. Spring Course Outline

Fluid Mechanics (Fluidmekanik) Course Code: 1TV024 5 hp Fluid Mechanics Spring 2011 Instruct: Chris Hieronymus Office: Geocentrum Dk255 Phone: 471 2383 email: christoph.hieronymus@geo.uu.se Literature:

### ch-01.qxd 8/4/04 2:33 PM Page 1 Part 1 Basic Principles of Open Channel Flows

ch-01.qxd 8/4/04 2:33 PM Page 1 Part 1 Basic Principles of Open Channel Flows ch-01.qxd 8/4/04 2:33 PM Page 3 Introduction 1 Summary The introduction chapter reviews briefly the basic fluid properties

### Section 1 Changes in Motion. Chapter 4. Preview. Objectives Force Force Diagrams

Section 1 Changes in Motion Preview Objectives Force Force Diagrams Section 1 Changes in Motion Objectives Describe how force affects the motion of an object. Interpret and construct free body diagrams.

### Physics 123 Unit #1 Review

Physics 123 Unit #1 Review I. Definitions & Facts Density Specific gravity (= material / water) Pressure Atmosphere, bar, Pascal Barometer Streamline, laminar flow Turbulence Gauge pressure II. Mathematics

### MECHANICAL PROPERTIES OF FLUIDS:

Important Definitions: MECHANICAL PROPERTIES OF FLUIDS: Fluid: A substance that can flow is called Fluid Both liquids and gases are fluids Pressure: The normal force acting per unit area of a surface is

### CEE 342 Aut Homework #2 Solutions

CEE 4 Aut 005. Homework # Solutions.9. A free-body diagram of the lower hemisphere is shown below. This set of boundaries for the free body is chosen because it isolates the force on the bolts, which is

### M E 320 Professor John M. Cimbala Lecture 05

M E 320 Professor John M. Cimbala Lecture 05 Today, we will: Continue Chapter 3 Pressure and Fluid Statics Discuss applications of fluid statics (barometers and U-tube manometers) Do some example problems

### S.E. (Mech.) (First Sem.) EXAMINATION, (Common to Mech/Sandwich) FLUID MECHANICS (2008 PATTERN) Time : Three Hours Maximum Marks : 100

Total No. of Questions 12] [Total No. of Printed Pages 8 Seat No. [4262]-113 S.E. (Mech.) (First Sem.) EXAMINATION, 2012 (Common to Mech/Sandwich) FLUID MECHANICS (2008 PATTERN) Time : Three Hours Maximum

### UNIT I FLUID PROPERTIES AND STATICS

SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : Fluid Mechanics (16CE106) Year & Sem: II-B.Tech & I-Sem Course & Branch:

### HOMEWORK ASSIGNMENT ON BERNOULLI S EQUATION

AMEE 0 Introduction to Fluid Mechanics Instructor: Marios M. Fyrillas Email: m.fyrillas@frederick.ac.cy HOMEWORK ASSIGNMENT ON BERNOULLI S EQUATION. Conventional spray-guns operate by achieving a low pressure

### NPTEL Quiz Hydraulics

Introduction NPTEL Quiz Hydraulics 1. An ideal fluid is a. One which obeys Newton s law of viscosity b. Frictionless and incompressible c. Very viscous d. Frictionless and compressible 2. The unit of kinematic

### Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur. Lecture - 8 Fluid Statics Part V

Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 8 Fluid Statics Part V Good morning, I welcome you all to the session of fluid mechanics.

### Q1 Give answers to all of the following questions (5 marks each):

FLUID MECHANICS First Year Exam Solutions 03 Q Give answers to all of the following questions (5 marks each): (a) A cylinder of m in diameter is made with material of relative density 0.5. It is moored

### Physics 107 HOMEWORK ASSIGNMENT #9

Physics 07 HOMEORK ASSIGNMENT #9 Cutnell & Johnson, 7 th edition Chapter : Problems 6, 8, 33, 40, 44 *6 A 58-kg skier is going down a slope oriented 35 above the horizontal. The area of each ski in contact

### CIE Physics IGCSE. Topic 1: General Physics

CIE Physics IGCSE Topic 1: General Physics Summary Notes Length and time A ruler (rule) is used to measure the length of an object between 1mm and 1m. The volume of an object of irregular shape can be

### Civil Engineering Hydraulics Mechanics of Fluids. Pressure and Fluid Statics. The fastest healing part of the body is the tongue.

Civil Engineering Hydraulics Mechanics of Fluids and Fluid Statics The fastest healing part of the body is the tongue. Common Units 2 In order to be able to discuss and analyze fluid problems we need to

### CHAPTER 13. Liquids FLUIDS FLUIDS. Gases. Density! Bulk modulus! Compressibility. To begin with... some important definitions...

CHAPTER 13 FLUIDS Density! Bulk modulus! Compressibility Pressure in a fluid! Hydraulic lift! Hydrostatic paradox Measurement of pressure! Manometers and barometers Buoyancy and Archimedes Principle! Upthrust!

### Lagrangian description from the perspective of a parcel moving within the flow. Streamline Eulerian, tangent line to instantaneous velocity field.

Chapter 2 Hydrostatics 2.1 Review Eulerian description from the perspective of fixed points within a reference frame. Lagrangian description from the perspective of a parcel moving within the flow. Streamline

### HOW TO GET A GOOD GRADE ON THE MME 2273B FLUID MECHANICS 1 EXAM. Common mistakes made on the final exam and how to avoid them

HOW TO GET A GOOD GRADE ON THE MME 2273B FLUID MECHANICS 1 EXAM Common mistakes made on the final exam and how to avoid them HOW TO GET A GOOD GRADE ON THE MME 2273B EXAM Introduction You now have a lot

### CHAPTER 28 PRESSURE IN FLUIDS

CHAPTER 8 PRESSURE IN FLUIDS EXERCISE 18, Page 81 1. A force of 80 N is applied to a piston of a hydraulic system of cross-sectional area 0.010 m. Determine the pressure produced by the piston in the hydraulic

### 3.1 CONDITIONS FOR RIGID-BODY EQUILIBRIUM

3.1 CONDITIONS FOR RIGID-BODY EQUILIBRIUM Consider rigid body fixed in the x, y and z reference and is either at rest or moves with reference at constant velocity Two types of forces that act on it, the

### CH 10: PRESSURE, GRAVITY AND MOMENTS

CH 10: PRESSURE, GRAVITY AND MOMENTS Exercise 10.1: Page 104 1. Convert each of the following to kg: (i) 200 g (ii) 4 g (iii) 2 x 10 5 g (iv) 24 mg 2. Convert each of the following to m 3 : (i) 1 cm 3

### s and FE X. A. Flow measurement B. properties C. statics D. impulse, and momentum equations E. Pipe and other internal flow 7% of FE Morning Session I

Fundamentals of Engineering (FE) Exam General Section Steven Burian Civil & Environmental Engineering October 26, 2010 s and FE X. A. Flow measurement B. properties C. statics D. impulse, and momentum

### AMME2261: Fluid Mechanics 1 Course Notes

Module 1 Introduction and Fluid Properties Introduction Matter can be one of two states: solid or fluid. A fluid is a substance that deforms continuously under the application of a shear stress, no matter

### SECOND ENGINEER REG. III/2 APPLIED MECHANICS

SECOND ENGINEER REG. III/2 APPLIED MECHANICS LIST OF TOPICS Static s Friction Kinematics Dynamics Machines Strength of Materials Hydrostatics Hydrodynamics A STATICS 1 Solves problems involving forces

### ME-B41 Lab 1: Hydrostatics. Experimental Procedures

ME-B41 Lab 1: Hydrostatics In this lab you will do four brief experiments related to the following topics: manometry, buoyancy, forces on submerged planes, and hydraulics (a hydraulic jack). Each experiment

### Chapter 8. Rotational Equilibrium and Rotational Dynamics. 1. Torque. 2. Torque and Equilibrium. 3. Center of Mass and Center of Gravity

Chapter 8 Rotational Equilibrium and Rotational Dynamics 1. Torque 2. Torque and Equilibrium 3. Center of Mass and Center of Gravity 4. Torque and angular acceleration 5. Rotational Kinetic energy 6. Angular

### JNTU World. Subject Code: R13110/R13

Set No - 1 I B. Tech I Semester Regular Examinations Feb./Mar. - 2014 ENGINEERING MECHANICS (Common to CE, ME, CSE, PCE, IT, Chem E, Aero E, AME, Min E, PE, Metal E) Time: 3 hours Max. Marks: 70 Question

### Fluid Mechanics Indian Institute of Technology, Kanpur Prof. Viswanathan Shankar Department of chemical Engineering. Lecture No.

Fluid Mechanics Indian Institute of Technology, Kanpur Prof. Viswanathan Shankar Department of chemical Engineering. Lecture No. # 05 Welcome to this fifth lecture on this nptel course on fluid mechanics

### Chapter -5(Section-1) Friction in Solids and Liquids

Chapter -5(Section-1) Friction in Solids and Liquids Que 1: Define friction. What are its causes? Ans : Friction:- When two bodies are in contact with each other and if one body is made to move then the

### Engineering Mechanics: Statics in SI Units, 12e

Engineering Mechanics: Statics in SI Units, 12e 5 Equilibrium of a Rigid Body Chapter Objectives Develop the equations of equilibrium for a rigid body Concept of the free-body diagram for a rigid body

### Mass of fluid leaving per unit time

5 ENERGY EQUATION OF FLUID MOTION 5.1 Eulerian Approach & Control Volume In order to develop the equations that describe a flow, it is assumed that fluids are subject to certain fundamental laws of physics.

### Unit 21 Couples and Resultants with Couples

Unit 21 Couples and Resultants with Couples Page 21-1 Couples A couple is defined as (21-5) Moment of Couple The coplanar forces F 1 and F 2 make up a couple and the coordinate axes are chosen so that

### 9.1. Basic Concepts of Vectors. Introduction. Prerequisites. Learning Outcomes. Learning Style

Basic Concepts of Vectors 9.1 Introduction In engineering, frequent reference is made to physical quantities, such as force, speed and time. For example, we talk of the speed of a car, and the force in

### Chapter Four Holt Physics. Forces and the Laws of Motion

Chapter Four Holt Physics Forces and the Laws of Motion Physics Force and the study of dynamics 1.Forces - a. Force - a push or a pull. It can change the motion of an object; start or stop movement; and,

### Civil Engineering Department College of Engineering

Civil Engineering Department College of Engineering Course: Soil Mechanics (CE 359) Lecturer: Dr. Frederick Owusu-Nimo FREQUENCY CE 260 Results (2013) 30 25 23 25 26 27 21 20 18 15 14 15 Civil Geological

### Figure 1 Answer: = m

Q1. Figure 1 shows a solid cylindrical steel rod of length =.0 m and diameter D =.0 cm. What will be increase in its length when m = 80 kg block is attached to its bottom end? (Young's modulus of steel

### Chapter 14 - Fluids. -Archimedes, On Floating Bodies. David J. Starling Penn State Hazleton PHYS 213. Chapter 14 - Fluids. Objectives (Ch 14)

Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced. -Archimedes, On Floating Bodies David J.

### MECHANICAL PROPERTIES OF FLUIDS

CHAPTER-10 MECHANICAL PROPERTIES OF FLUIDS QUESTIONS 1 marks questions 1. What are fluids? 2. How are fluids different from solids? 3. Define thrust of a liquid. 4. Define liquid pressure. 5. Is pressure

### Two Hanging Masses. ) by considering just the forces that act on it. Use Newton's 2nd law while

Student View Summary View Diagnostics View Print View with Answers Edit Assignment Settings per Student Exam 2 - Forces [ Print ] Due: 11:59pm on Tuesday, November 1, 2011 Note: To underst how points are

### Thomas Whitham Sixth Form Mechanics in Mathematics

Thomas Whitham Sixth Form Mechanics in Mathematics 6/0/00 Unit M Rectilinear motion with constant acceleration Vertical motion under gravity Particle Dynamics Statics . Rectilinear motion with constant

### Physics 106 Lecture 13. Fluid Mechanics

Physics 106 Lecture 13 Fluid Mechanics SJ 7 th Ed.: Chap 14.1 to 14.5 What is a fluid? Pressure Pressure varies with depth Pascal s principle Methods for measuring pressure Buoyant forces Archimedes principle

### Liquids CHAPTER 13 FLUIDS FLUIDS. Gases. Density! Bulk modulus! Compressibility. To begin with... some important definitions...

CHAPTER 13 FLUIDS FLUIDS Liquids Gases Density! Bulk modulus! Compressibility Pressure in a fluid! Hydraulic lift! Hydrostatic paradox Measurement of pressure! Manometers and barometers Buoyancy and Archimedes

### 5 ENERGY EQUATION OF FLUID MOTION

5 ENERGY EQUATION OF FLUID MOTION 5.1 Introduction In order to develop the equations that describe a flow, it is assumed that fluids are subject to certain fundamental laws of physics. The pertinent laws

### The case where there is no net effect of the forces acting on a rigid body

The case where there is no net effect of the forces acting on a rigid body Outline: Introduction and Definition of Equilibrium Equilibrium in Two-Dimensions Special cases Equilibrium in Three-Dimensions

### Forces and Newton s Laws Reading Notes. Give an example of a force you have experienced continuously all your life.

Forces and Newton s Laws Reading Notes Name: Section 4-1: Force What is force? Give an example of a force you have experienced continuously all your life. Give an example of a situation where an object

### Lesson 6 Review of fundamentals: Fluid flow

Lesson 6 Review of fundamentals: Fluid flow The specific objective of this lesson is to conduct a brief review of the fundamentals of fluid flow and present: A general equation for conservation of mass

### LECTURE NOTES FLUID MECHANICS (ACE005)

LECTURE NOTES ON FLUID MECHANICS (ACE005) B.Tech IV semester (Autonomous) (2017-18) Dr. G. Venkata Ramana Professor. DEPARTMENT OF CIVIL ENGINEERING INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) DUNDIGAL,

### ! =!"#\$% exerted by a fluid (liquid or gas) !"#\$ =!"# FUNDAMENTAL AND MEASURABLE INTENSIVE PROPERTIES PRESSURE, TEMPERATURE AND SPECIFIC VOLUME

FUNDAMENTAL AND MEASURABLE INTENSIVE PROPERTIES PRESSURE, TEMPERATURE AND SPECIFIC VOLUME PRESSURE, P! =!"#\$%!"#! exerted by a fluid (liquid or gas) Thermodynamic importance of pressure One of two independent

### The word thermodynamics is derived from two Greek words Therm which means heat Dynamis which means power

THERMODYNAMICS INTRODUCTION The word thermodynamics is derived from two Greek words Therm which means heat Dynamis which means power Together the spell heat power which fits the time when the forefathers

### Civil Engineering Hydraulics. Pressure and Fluid Statics

Civil Engineering Hydraulics and Fluid Statics Leonard: It wouldn't kill us to meet new people. Sheldon: For the record, it could kill us to meet new people. Common Units 2 In order to be able to discuss