KONJO-FRANCIS-SPHIII-FLUIDS-&-PROPERTIES -OF -MATTER-PPT-2024.pptx

A well developed slide to help students in Mechanics and physics at large

1. LECTURER: KONJO Francis Pote Sindani MSc. Graduation Thesis DEFENSE 2020 SCHOOL OF PREPARATORY - UNIVERSITY OF JUBA SCIENCE-BASED SCHOOLS YEAR: 1. ACADEMIC YEAR: 2023-2024 COURSE/CODE: GENERAL PHYSICS I PART I (MECHANICS). SPH III SPH-III-FLUIDS-&-PROPERTIES-OF-MATTER-84-SLIDES-2023-2024-UoJ-MR. KONJO-F.POTE

2. CONTENTS 1.0 FLUIDS & PROPERTIES OF MATTER : 1.1 Introduction 1.1.1 Models 1.1.2 Laws 1.2 Theories 1.2.1 Physical Quantities 1.2.2 Systems of Unit SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

3. 1.3 DIMENSIONS & DIMENSIONAL ANALYSIS 1.3.1 Definition of Dimensions 1.3.2 Solving Problems involving Dimensions 1.3.3 What is Dimensional Analysis 1.3.4 Finding Dimensional Analysis in the Equations 1.3.5 Significant Figures 1.3. 6 Working with the Numbers SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

4. 1.4. PRESSURE & ARCHIMEDES’ PRINCIPLE 1.4.1 Pressure 1.4.2 Density 1.4.3 Archimedes’ Principle 1.4 .4 Sample Problems SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

5. 1.5 SURFACE TENSION 1.5.1 Surface Tension 1.5.2 Surface Energy 1.5.3 Pressure on Curved Surfaces 1.5.4 Sample Solved Question 1.5.5 Capillaries SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

6. 1.6 BERNOULLI’S EQUATION AND NEWSTON’S LAW OF VISCOSITY 1.6.1 Equation of Continuity 1.6.2 Bernoulli’s Equation 1.6.3 Torricelli’s Equation 1.6.4 Applications of Bernoulli's Principle 1.6.5 The Venturi Tube 1.6. 6 Sample Solved Problems & Exercises SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

7. 1.7 DIFFUSION 1. 7 .1 Brownian Motion 1. 7 .2 Graham’s Law 1. 7 .3 Osmotic Pressure 1. 7 .4 Molecular Weights 1. 7 .5 Osmometer 1. 7 . 6 Exercises SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

8. FLUID MECHANICS SOME TERMINOLOGIES APPLICABLE IN FLUID MECHANICS:- Viscosity.i.e. If a fluid is viscous then it offers a resistance to the motion through it of any solid body. Turbulent Flow.i.e . Disorderly Flow in that the speed and direction of the fluid particles passing any point vary with time. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

9. Line Flow .i.e. The path followed by a particle of the fluid. Streamline .i.e. Is a curve whose tangent at any point is along the direction of the velocity of the fluid at that point. Streamlines never cross. Laminar Flow. i.e. Steady flow in which the velocities of all the particles on any given streamlines are the same, though the particles of different streamlines may move at different speeds. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

10. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Tube of Flow.i.e. A tubular region of a flowing fluid whose boundaries are defined by a set of streamlines. Incompressible Fluid. i.e. It is a fluid in which changes in pressure produce no change in density of the fluid.

11. PRESSURE IN FLUIDS The Pressure, P, on the area given is that area the Normal Force, F, to that area divided by the area, A, i.e. P = F/A. Pressure, P, has dimensions of Force divided by area so that its SI Units are N/ . One N/m2 is called a Pascal (Pa) Another commonly used unit of force is lbs / .   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

12. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS - Pressure in a fluid is Isotropic and it is in all directions. -In a fluid at rest , the force on a given surface is always perpendicular to the surface. -Also, in a fluid at rest , the pressure is the same depth within the fluid. The pressure in a fluid increases with depth in the fluid WHY ? -The pressure in a depth h below the surface of a given fluid at rest is the pressure at the top surface PO, plus the weight per area of the fluid above the depth h:

13. CONTINUATION….. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS P(h) = P(O) + ρ gh where ρ = uniform density of the fluid g = acceleration of gravity

14. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Most liquids have densities that are only slightly dependent on Pressure. This relationship is valid for most applications involving liquids However, gases do have a density that depends on pressure. ⁎ Now, a more general relationship between depth and pressure must be applied: P(h) = P(O) + (y)gdy ρ(y) = the density as a function of depth y within the fluid  

15. EXERCISE A gas has a temperature gradient such that the relationship between pressure and density is given by the relationship : ρ = c P2 in such a gas with a uniform H is Po and the density at the top temperature, the pressure at the top of a column of height of the column is .   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS What are the: ( i ) Pressure, and (ii) Density at the bottom of this column?

16. PRESSURE MEASUREMENTS SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS There are several different instruments that are applied for measuring fluid pressure . One common device is a manometer .

17. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Fig: Showing a Manometer

18. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS A manometer uses the height of a column of fluid supported by the pressure difference at two surfaces of the liquids to determine the pressure difference. The diagram above is of a typical manometer where P2 = the pressure at the lower surface of the liquid and P2 is related to P1 P2 = P1 + ρgh where ρ = the density of the liquid in the device and h = is the difference in vertical position of the two liquids surfaces

19. Pressures are often expressed by the height of the column of liquid supported such a mm of Mercury (mmHg). So it should be easy to see that to take an expression Of the pressure in terms of the height. Of the supported column of liquid and convert into a measurement of Pressure, you need to multiply the height of the column by the density of the fluid and acceleration of gravity. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

20. THE U-SHAPE MANOMETER The U-shape Manometer consists of a U-shape tube containing a liquid . It is used to measure Pressure . The Pressure to be measured ( that of gas ) is applied to one arm of the manometer, the other arm is open to the atmospheric pressure Manometers can be used to measure Pressures of both above and below atmospheric pressures. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

21. CONTINUATION…… Mercury is used as the Manometer liquid unless the pressure being measured is close to atmospheric pressure , in which case a liquid of lower density (e.g. oil, or water ) is more suitable. The pressure registered by the manometer, hρg , is known as the GAUGE PRESSURE . The actual pressure, PA + hρg , is called the ABSOLUTE PRESSURE . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

22. PROBLEM INVOLVING ARCHIMEDE’S PRINCIPLE A raft is made from wood with a density of 378kg/m 3 . The volume of the logs that make up the raft is 3.87m 3 . What is the maximum load that can be placed on the raft before its top surface reaches the surface of the water it is floating in? SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

23. SOLUTION From the Principle of Archimedes’ for the raft to float, the buoyant force must be equal to the weight of the load . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS F B = ( M raft + M load )g = ( + M load )g (Buoyant Force, F B ) (Max raft, ) (Max load, )  

24. CONTINUATION…. Where we have used that the submerged volume is the entire volume of the raft. So, we can solve for the mass of the load. ∴ M load ( max load) = ( = ( 1000kg/m 3 – 378kg/m 3 ) (3.87m 3 ) = 2,407.140000 = 2.41 x kg   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

25. SURFACE TENSION AND CAPILLARITY: A Surface is a boundary between any two materials. If you look up the value of the surface tension ( ϒ ) of a liquid, it will be the surface tension associated with air-liquid interface unless otherwise specified . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

26. CONTINUATION……. There is a surface tension associated with the surface between a liquid and any other material also. The surface tension associated with the boundary between water and glass causes CAPILLARY ACTION . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

27. CONTINUATION….. Capillary action causes water to rise within a small diameter glass tube. To increase the area of the surface of a liquid, it requires moving molecules of the liquid from the interior of the liquid to the surface . Molecules of the liquid at the surface do not have as many molecules surrounding them SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

28. CONTINUATION………. But molecules in the interior of the liquids have a strong attraction for the other molecules in the liquid. WHY????? So, it takes energy to create surface area. Since it takes energy to create surface area , this means that work is done on the liquid as the surface is stretched and so a force, F, is needed to stretch the surface. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

29. CONTINUATION…… ⁎ The force per length of surface being stretched is called the SURFACE TENSION , ( ϒ ) . It can be expressed as ϒ = where , a surface of length, . Its SI Unit is Nm -1 .   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

30. EXERCISE: A thin wire ring of 3cm radius float on the surface of a liquid. The pull required to raise the ring before the film breaks is 30.14 x 10 -3 N more than its weight the surface tension of the liquid (in Nm -1 ) is a) 80 x 10 -3 b) 87 x 10 -3 c) 90 x 10 -3 d) 98 x 10 -3 SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

31. SURFACE TENSION EFFECTS A molecule in the surface of a liquid is subject to intermolecular forces from below but not from above (providing the effects of the molecules of the vapor are ignored). Thus , if the coordination number of the molecules of the interior is n , then that of a surface molecule will be .   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

32. CONTINUATION…….. Therefore, if a molecule of the interior has a potential energy of for instance -0.4eV then a surface molecule, being involved in only half as many bonds will have a potential energy of -0.2eV Therefore, the potential energy of a molecule in the surface exceeds that of one in the interior. All systems arrange themselves in such a way that they have the minimum possible potential energy. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

33. In order that the potential energy associated with the intermolecular tension forces (the surface tension forces) can be a minimum , the number of molecules which reside in the surface has to be a minimum. Therefore: The liquids have the smallest possible surface area, and The average separation of the molecules in the surface of a liquid is greater than that of molecules in the interior. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

34. EXERCISE With the relevant diagrams, explicitly, expound the fact that the Surface behaves like an Elastic Skin in a State of Tension. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

35. SURFACE TENSION AND FREE SURFACE ENERGY The Surface Tension γ of a liquid is defined as the force per unit length acting in the surface and perpendicular to one side of an imaginary line drawn in the surface. SI UNIT The surface tension has a SI Unit of Nm -1 . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

36. FREE SURFACE ENERGY Ϭ Free Surface Energy Ϭ Is defined as the work done in isothermally creating unit area of new surface and its SI Unit is Jm -2 = Nm -1 . Whenever the surface area of a given volume of liquid is increased , work has to be done against the surface tension forces . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

37. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Given the figure below , consider stretching a thin film of liquid on a horizontal frame. Since the film has both an upper and lower surfaces , the force, F , on AB due to surface tension is given by .  

38. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Fig: A thin film of liquid being stretched .

39. CONTINUATION…… If AB is moved a distance X to A′B′ then work has to be done against this force. The surface tension, γ , is free of the area of the film (because as the size of the surface increases more molecules enter it and by so doing it maintains the average molecular separation) but decreases with increasing temperature (because this decreases the binding energy). SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

40. CONTINUATION….. Thus provided AB is moved isothermally to A′ B′ the force on AB will be constant, and therefore, since Work done = Force x Distance Work done = x   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

41. CONTINUATION…… The increase in surface area is 2Lx (upper and lower surfaces), and therefore the work done per unit area increase (the free surface energy Ϭ) is given by Ϭ = i.e. Ϭ =   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

42. CONTINUATION……. Thus , the free surface energy Ϭ is equal to the surface tension This provides a second definition of Surface tension is the work done in isothermally increasing the surface of the liquid by unit area. The SI Unit is Jm -2 = Nm -1 .   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

43. Considering water in a capillary tube rises above the level of the water outside. The effect is known as CAPILLARY RISE and is most marked with narrow tubes. The ability of blotting paper to soak up ink is due to the same effect , the spaces between the fibers act as fine capillary tubes . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS CAPILLARY RISE, MEASUREMENT OF SURFACE TENSION,  

44. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS A liquid whose angle of contact is greater than 90° suffers Capillary depression . Both capillary rise and capillary depression are caused by surface tension and provide a means by which the surface tension γ of a liquid may be measured If a capillary tube is held vertically in liquid which has concave meniscus as in the figure below

45. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Surface tension forces cause the liquid to exert a downward directed force on the walls of the tubes. In accordance with Newton’s Third Law the tube exerts an equal and opposite force on the liquid and it rises in the tube .

46. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS At equilibrium the weight of the liquid which has been lifted up is equal to the vertical component of the force exerted by the tube. The mass of the raised liquid is the product of its density ρ and its volume , therefore its weight is ρ .  

47. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Fig: Liquid in a Capillary Tube (Not to scale)

48. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS The length of the liquid surface in contact with the tube is equal to the circumference of the tube , and therefore the vertical component of the force exerted by the tube is γ cos . Therefore, at equilibrium it is given by   γcos = ρ  

49. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS γ = γ can be determined .  

50. NOTES The weight of the small quantity of liquid in the meniscus has been ignored in deriving equation i.e. γ = h and r are normally measured with a travelling microscope. The tube should be taken at the level of the meniscus in order to measure r.   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

51. CONTINUATION ON NOTES …. are found from tables or measured in separate experiments. Equation i.e. γ = likewise holds for Capillary depression.   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

52. EQUATIONS OF CONITUNITY If a fluid is undergoing steady flow, then the mass of fluid that enters one end of a tube of floe must be equal to the mass that leaves at the other end during the same time. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS EXPRESSION FOR THE EQUATION OF CONTINUITY It is given by =  

53. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Fig: Showing Equation of Continuity

54. CONTINUATION……. and be the cross-sectional areas of the tube of flow at 1 and 2 . and be the densities of the fluid at 1 and 2 respectively. and be the velocities of the respectively of the fluid particles at 1 and 2.   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

55. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

56. BERNOULLI’S PRINCIPLE (EQUATION): It states that for an incompressible non-viscous fluid undergoing steady flow, the pressure plus the kinetic energy per unit volume plus the potential energy per unit volume is constant at all points on a streamline. i.e. P + ρ + ρgh = A Constant where P = the pressure within the fluid ρ = the density of the fluid V = the velocity of the fluid g = the acceleration due to gravity h = the height of the fluid (above some arbitrary reference line)  

57. CONINUATION……….. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

58. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS In the particular case of fluid that is incompressible , under condition of Laminar Flow that is not changing with time, there is a simple relationship, called BERNOULLI’S EQUATION that relates the speed of the fluid, (V) the height (h) of the fluid within a given flow tube.

59. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Bernoulli’s Equation can be derived from the work energy principle and is written as: + + ρg = + + ρg where P = is the Pressure, ρ =density of the fluid, V = the speed of the fluid, g = the acceleration of gravity, and y = the vertical position of the fluid ( measured positive upward). All of the subscripts 1 applies to One location and the subscript 2 applies to another location .  

60. SAMPLE PROBLEMS 1- A fire hose sprays water at a rate of 82 gallons per minute at a speed of 17.4m/s through a nozzle of a diameter 2 . 2cm.The hose has a diameter of 6 . 3cm. What is the pressure in the hose lying on the ground before the water reaches the nozzle held by a fireman at a height of 1 . 4m above the ground? 2- Prove that: P + + ρgh = A Constant   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

61. SAMPLE SOLVED PROBLEM: SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Given that: P + 1/2ρ + ρgh = A Constant Also; = . Find the following from the diagram below : a) b)  

62. SOLUTIONS: ATTENTION: 1- consider it to be 1.2m 2- consider it to be 0.6m 3- let it be also 1.2m 4 - let it be 3.1m 5 - let it be 5m/s 6 - let it be 3atm 7 – ρ is 1000kg/ 8- π is 3.14   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

63. CONTINUATION…….. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

64. CONTINUATION……… SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

65. APPLICATIONS OF BERNOULLI’S PRINCIPLE: FROM TORRICELLI TO SAILBOATS, AIRFOILS, AND TIA For the special of the pressure being identical at point 1 and 2 of a fluid and the velocity of the fluid being zero at point 2, Bernoulli’s equation can be simplied to + ρg = ρg If the above relationship is solved for the velocity of the fluid at point 1, we obtain = . This is known as TORRICELLI’S THREOREM.   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

66. TORRICELLI’S THREOREM Torricelli’s Equation applies approximately to the situation where a fluid is in a large container and leaves the container through a small opening compared to the diameter of the container. In this case, the velocity inside the container far from the small exit opening is small compared to the velocity of the fluid moving through the opening . SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

67. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

68. CONTINUATION….. Consider the fluid to be non-viscous, incompressible (of density ρ) and in horizontal steady flow. Let the and at y on the same streamline as x. Applying Bernoulli’s equation at X and Y gives P x + = P y +   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

69. CONTINUATION…… If the cross-sectional areas at X and Y are , then from the equation of continuity , i.e.   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

70. CONTI…….. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS P x + = P y + ∴ P x - P y = ρ( ) Therefore, by measuring the pressures P x and P y , and knowing that ρ , , it is possible to find the Velocity , of the fluid in the unconstructed ( main) section of the pipe.  

71. SAMPLE SOLVED PROBLEM A venturi tube is used to measure rate of flow of gasoline through a line as shown in the diagram below: SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Figure (a): Venturi Tube

72. CONTI…ON SAMPLE SOLVED QUESTION The larger diameter tube has a 1.22cm and the small diameter has 0.26cm. The gauge pressure in the larger diameter tube is 2.45 x 10 4 N/m 2 and the pressure in the smaller diameter tube is 1.34 x 10 4 N/m 2 . What is flowrate of gasoline through this tube? SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

73. SOLUTION: SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS P 1 + = P 2 +   Due to the equation of Continuity, we know that V 2 = ( )V 1  

74. CONTINUATION ….. We then substitute this information into TORRICELLI’S EQUATION : P 1 + = P 2 + and solve for V 1 : V 1 =   SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

75. CONTINUATION…… SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS

76. CONTINUATION….. SPH-III-(MECHANICS)-2023-2024-UoJ-MR. KONJO-FRANCIS Using this to calculate the flowrate: Q = AV = = /s  

77. Thanks you for your Patience in listening and Participation 29 SPH-III-FLUIDS &-PROPERTIES-OF-MATTER-84-SLIDES-2023-2024-UoJ-MR. KONJO-F.POTE-