Fluid Mechanics lecture CHAPTER ONE.pptx

Fluid Mechanics and Dynamics

1. Fluid Mechanics BMAE 210 BY: BERNICE DOGBEY REGIONAL MARITIME UNIVERSITY FACULTY OF ENGINEERING & APPLIED SCIENCES MARINE ENGINEERING DEPARTMENT

2. COURSE OUTLINE Chapter I: Concept of Fluid s Chapter II: Principles of Fluid Statics Chapter III: F orces on Submerged Surfaces Chapter IV: Fluid Dynamics Chapter V: Flow Through Pipes Chapter VI: Pumps Chapter VI: Dimensional Analysis and Similitude.

3. Referrence Books 1. Durgaiah , D.R., Fluid Mechanics and Machinery 2. Massey B.S., Mechanics of Fluids 3. Dr. Fogiel , M., Fluid Mechanics: A complete solution guide to any textbooks 4. Douglas J. S., Solving Problems in Fluid Mechanics Vol 1 5. Douglas J. S., Solving Problems in Fluid Mechanics Vol 2 6. Khurmi R. J., A textbook of hydraulics and hydraulic machines 7. Hannah J. & Hillier M. J., Applied Mechanics

4. CHAPTER - 1 CONCEPT OF FLUIDS

5. Concept Of Fluids Definition Of Fluid Concept Of Continuum Properties of Fluids Density, Specific Weight, Specific Gravity Viscosity & Viscosity Laws Types Of Fluids – Newtonian and Non-Newtonian Surface Tension Capillarity Compressibility

6. What is A Fluid ? The Fluid is the Substance which undergoes continuous deformation under the action of infinitely small Shear stress. It is a substance c apable of flowing. It has no shape of its own but takes the volume of its container. Fluids consist of Liquids and Gases

7. Solids, Liquids and Gases

8. Solids, Liquids & Gases Temperature And Pressure Adding heat to a substance increases its temperature . But what is actually going on ? The jiggling of the atoms or molecules in the substance become more energetic as temperature increases . This is usually referred to as ‘ Internal Energy ’ in thermodynamics . When the atoms bounce off the wall, they exert a force on the walls. The change in momentum ( m∆ ν )causes the pressure . In a sealed container, pressure increases as temperature increases.

9. Fluid As A Continuum Fluids are aggregations of molecules; widely spaced for a gas and closely spaced for liquids. Distance between the molecules is very large compared to the molecular diameter. The number of molecules involved is immense and the separation between them is normally negligible. Under these conditions, fluid can be treated as continuum and the properties at any point can be treated as bulk behavior of the fluids .

10. Ideal Fluid and Real Fluids Ideal Fluids The fluids which has no viscosity, no surface tension and is ideally incompressible is called as Ideal Fluids. Ideal fluids are imaginary fluids and do not exist in nature Water and Air are treated as Ideal fluid for study purpose. Real Fluids The Fluids existing in nature which has viscosity, surface tension and can be incompressible and compressible in nature are called as real fluids.

11. Fluid Mechanics The branch of science and engineering which deals with the study of fluid at rest or in motion and their interactions with solids or other fluids at the boundaries. . Three branches of Fluid Mechanics Fluid Statics - The study of fluid at rest Fluid Kinematics - The study of fluid in motion without considering the forces acting on them . Fluid Dynamics - The study of fluid in motion considering the forces acting on them .

12. Application of Fluid Mechanics

13. Units and Dimensions Unit - Standards in terms of which the physical quantities are specified . Primary/Fundamental Quantities Mass, Length ,Time, Temperature. Secondary / Derived Quantities Velocity, density, acceleration.etc Dimensions - The representation of secondary or derived quantities in terms of primary or fundamental quantities. Notations of Dimensions: [ M a L b T c ]

14. No-Slip Condition The No-Slip Condition Fluid flow is often confined by solid surfaces, and it is important to understand how the presence of solid surfaces affects fluid flow. All experimental observations indicate that a fluid in motion comes to a complete stop at the surface and assumes a zero velocity relative to the surface. That is, a fluid in direct contact with a solid “sticks” to the surface, and there is no slip. This is known as the no-slip condition .

15. No-Slip Condition The fluid property responsible for the no-slip condition and the development of the boundary layer is viscosity . The region between the solid surface and the fluid is called the boundary layer.

16. Properties of Fluid s – Mass / Mass Density Mass : - The quantitative measure of matter is called as mass. Denoted by ‘ m ’ Unit : kilogram ( Kg ) . Dimensions : [M 1 L T ] Mass Density : The mass per unit volume of the body. Denoted by ‘ ρ ’ (rho ) Unit : Kg/m 3 Dimensions : [M 1 L - 3 T ] For Water: Mass density of fresh water is 1000 kg/m 3 For Mercury: Mass density is 13600 kg/m 3 For Petrol: Mass density is 730 kg/m 3 𝒎𝒂𝒔𝒔 𝑴𝒂𝒔𝒔 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 ( ρ ) = 𝒗𝒐𝒍𝒖𝒎𝒆

17. Properties of Fluids – Density of Gases Density of Ideal Gases Any equation that relates temperature, pressure and density (or specific volume) of a substance is known as the Equation of State . The equation of state for substances in the gas phase is the Ideal Gas Equation of State , expressed as Where, P = Absolute Pressure = Specific Volume T = Temperature (K) R = Gas Constant    

18. Properties of Fluids - Specific Weight Specific Weight/Weight Density : The weight per unit volume of the body. Specific weight = 𝒘𝒆𝒊𝒈𝒉𝒕 𝒗𝒐𝒍𝒖𝒎𝒆 Denoted by ‘ γ ’ (Gamma) Unit : N/m 3 Dimensions : [M 1 L - 2 T - 2 ] Relation between mass density and specific weight: γ = = 𝒘𝒆𝒊𝒈𝒉𝒕 𝒎 *g 𝒗𝒐𝒍𝒖𝒎𝒆 𝒗𝒐𝒍𝒖𝒎𝒆  

19. Properties of Fluids - Specific Volume Specific Volume : - The ratio of the volume of a fluid per unit mass of the fluid is called its specific volume. It is denoted by s   Specific volume = 𝒗𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒍𝒖𝒊𝒅 𝟏 = 𝒎𝒂𝒔𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒍𝒖𝒊𝒅 𝒎𝒂𝒔𝒔 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 s s = 𝑽 = 𝟏   𝒎 𝞀 Unit : m 3 / Kg. Dimensions : [M - 1 L 3 T ]

20. Properties of Fluids - Specific G ravity Specific gravity : The ratio of specific weight /density of the fluid to the specific weight /density of a standard fluid . Denoted by ‘ s ’ Specific gravity = 𝒔𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝒘𝒆𝒊𝒈𝒉𝒕 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒍𝒖𝒊𝒅 𝒔𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝒘𝒆𝒊𝒈𝒉𝒕 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒇𝒍𝒖𝒊𝒅 s = It is unit less and dimensionless Specific gravity of water(s w ) is 1. Υ 𝒇𝒍𝒖𝒊𝒅 Υ 𝒘𝒂𝒕𝒆𝒓 OR  

21. Worked Examples A reservoir of glycerin has a mass of 1,200kg and a volume of 300m3. Calculate its density, specific gravity and specific volume. Calculate the specific weight, density and specific gravity of 1L of petrol which weighs 7N. The density of a liquid is 2.93g/cm3. What is its specific gravity, specific volume and specific weight? Calculate the density, specific weight and weight of 1L liquid of specific gravity 0.8. The specific gravity of ice is 0.9, calculate the weight density of the ice. The mass of a fluid system is 4kg, its density is 2g/cm3 and g=9.81m/s2. Determine the Specific volume, Specific weight and total weight of the fluid. If 25L of an oil weighs 425g, what is the density and specific gravity of the oil.

22. Worked Examples 8. Determine the density, specific gravity, and mass of the air in a room whose dimensions are 4 m x 5 m x 6 m at 100 kPa and 25°C. R = 0.287 kPa.m / kg.K 9. What is the volume of a solution that weighs 45N and has a specific gravity of 0.78? If 0.5m3 of a liquid has a density of 1.8 g/cm3, what is the weight of the liquid? What is the specific weight of air at 48kPa and 21 C. R = 0.287 kPa.m / kg.K . A mass of 150g of argon is maintained at 200 Pa and 100°F in a tank. What is the volume of the tank? A 100L container is filled with 1 kg of air at a temperature of 27°C. What is the pressure in the container? R = 0.287 kPa.m / kg.K .

23. Properties Of Fluids - Viscosity When two solid bodies in contact move relative to each other, a friction force develops at the contact surface in the direction opposite to motion . The situation is similar when a fluid moves relative to a solid or when two fluids move relative to each other. We move with ease in air, but not so in water. There is a property that represents the internal resistance of a fluid to motion or the “fluidity,” and that property is the viscosity .

24. Properties Of Fluids - Viscosity The force a flowing fluid exerts on a body in the flow direction is called the drag force , and the magnitude of this force depends, in part, on viscosity . Viscosity therefore is defined as the property of the fluid by virtue of which the fluid offers resistance to deformation under the action of shear stress.

25. Properties Of Fluids - Viscosity In fluid mechanics, shear stress, defined as a tangential force per unit area, is used rather than force itself. In a simple shear flow such as this, the shear stress is directly proportional to the slope of the velocity profile . Newton’s Law of Viscosity states that “ the shear Stress is directly proportional to the rate of Shear stain or Velocity Gradient .”

26. Properties Of Fluids - Viscosity

27. Units and Dimensions of Viscosity Unit Dimension :   The popular unit for absolute viscosity is Poise named in honour of Poiseuille . 1 Poise = 0.1 kg/ ms Centipoise ( cP ) is also used more frequently as 1 cP = 0.001 kg/ ms

28. Kinematic Viscosity Kinematic Viscosity : It is defined as the ratio of the absolute viscosity to mass density of the fluid. It is denoted by Greek Symbol ν ( nu ). [ 𝝂] = [ M L 2 T - 1 ] Units : m 2 /s Dimension : The popular unit used is stokes m 2 /s  

29. Effect of Temperature on Viscosity of the fluids In case of Liquids , viscosity is due to molecular attraction (cohesive forces) Thus, Viscosity of Liquid decreases with increase in temperature. In case of gases, the viscosity is due to molecular momentum exchange. Thus, viscosity of gases increase with increase in temperature

30. Classification Of Fluids Based On The Viscosity Based on Newton’s Law of Viscosity, The fluids are classified as : 1. Newtonian fluid s Fluids for which the rate of deformation is linearly proportional to the shear stress. τ = 𝝁 𝒅𝒖 𝒅𝒚 Eg . Glycerine, kerosene, air , water, alcohol This means that the proportionality parameter, µ , is constant. The viscosity at any given temperature and pressure is constant and is independent of the rate of deformation.

31. Classification Of Fluids Based On The Viscosity Non Newtonian Fluid s Fluids for which the relationship between shear stress and rate of deformation is not linear. In non Newtonian fluids the viscosity will vary with variation in the rate of deformation. τ = 𝝁 𝒅𝒖 𝒅𝒚 𝒏 Non Newtonian fluids can be further classified as simple non Newtonian, ideal plastic and shear thinning, shear thickening and real plastic fluids. Linear relationship between shear stress and rate of deformation (du/ dy ) does not exist. In plastics, up to a certain value of applied shear stress there is no flow. After this limit it has a constant viscosity at any given temperature.

32. Classification Of Fluids Based On The Viscosity a. Dilatant fluids ( n > 1) – viscosity increases with increasing rate of deformation (Shear Thickening) Eg . Starch,butter, wet sand. b. Pseudoplastic fluids ( n < 1 ) – viscosity decreases with increasing rate of deformation (Shear Thinning) Eg. paint , ketchup, syrups, blood , milk, toothpaste .

33. Examples

34. Examples

35. Properties Of Fluids - Surface Tension All liquids exhibit a free surface known as meniscus when in contact with vapour or gas. Liquid molecules exhibit cohesive forces binding them with each other. The molecules below the surface are generally free to move within the liquid and they move at random. When they reach the surface they reach a dead end in the sense that no molecules are present in great numbers above the surface to attract or pull them out of the surface.

36. Properties Of Fluids - Surface Tension So they stop and return back into the liquid. A thin layer of few atomic thickness at the surface formed by the cohesive bond between atoms slows down and sends back the molecules reaching the surface. This cohesive bond exhibits a tensile strength for the surface layer and this is known as surface tension . Surface tension decreases with temperature. Contaminants as well as detergents also decrease surface tension.

37. Properties Of Fluids - Surface Tension Surface Tension therefore i s defined as the tensile force acting on the surface of the liquid in contact with the gas or on the surface between two immiscible liquids such that the contact surface behaves like a membrane under tension . Surface tension may also be defined as the work in N/m required to create unit surface of the liquid. The work is actually required for pulling up the molecules with lower energy from below, to form the surface . Another definition for surface tension is the force required to keep unit length of the surface film in equilibrium (N/m ). The formation of bubbles, droplets and free jets are due to the surface tension of the liquid .

38. Properties Of Fluids - Surface Tension It is denoted by Greek Symbol σ We are going to be looking at Surface Tension on Liquid droplet Surface Tension on Hollow Bubble Surface Tension on Liquid jet

39. Surface Tension on Liquid Droplet

40. Surface Tension on the Hollow Bubble Surface Tension on an Hollow Bubble : A hollow bubble like a soap bubble in air has two surfaces in contact with air; one inside and the other outside. Thus two surfaces are subjected to surface tension. In this case, we have;

41. Surface Tension on the Liquid Jet

42. Properties of Fluids - Capillarity Another interesting consequence of surface tension is the capillary effect , which is the rise or fall of a liquid in a small-diameter tube inserted into the liquid . Such narrow tubes or confined flow channels are called capillaries . The rise of kerosene through a cotton wick inserted into the reservoir of a kerosene lamp is due to this effect. The capillary effect is also partially responsible for the rise of water to the top of tall trees. The curved free surface of a liquid in a capillary tube is called the meniscus .

43. Properties of Fluids - Capillarity Capillarity can therefore be defined as the phenomenon of rise and fall in the level of liquid surface relative to adjacent liquid surface when the small tube is held vertically in the liquid. The rise in the level of liquid in the tube is called as capillary rise . The fall in the level of the liquid in the tube is called as capillary depression .

44. Properties of Fluids - Capillarity It is commonly observed that water in a glass container curves up slightly at the edges where it touches the glass surface but the opposite occurs for mercury; it curves down at the edges. This effect is usually expressed by saying that water wets the glass (by sticking to it) while mercury does not . The strength of the capillary effect is quantified by the contact (or wetting) angle , defined as the angle that the tangent to the liquid surface makes with the solid surface at the point of contact .

45. Properties of Fluids - Capillarity A liquid is said to wet the surface when Ø < 90 ° and not to wet the surface when Ø > 90 °. In atmospheric air , the contact angle of water with glass is nearly zero, Ø ≈ 0 °. Therefore, the surface tension force acts upward on water in a glass tube along the circumference, tending to pull the water up. The contact angle for mercury–glass is 130° and 26° for kerosene–glass in air . Note that the contact angle, in general, is different in different environments (such as another gas or liquid in place of  air).

46. Properties of Fluids - Capillarity The phenomenon of the capillary effect can further be explained by considering cohesive forces (the forces between like molecules, such as water and water) and adhesive forces (the forces between unlike molecules, such as water and glass ). The liquid molecules at the solid–liquid interface are subjected to both cohesive forces by other liquid molecules and adhesive forces by the molecules of the solid. The relative magnitudes of these forces determine whether a liquid wets a solid surface or not. T he water molecules are more strongly attracted to the glass molecules than they are to other water molecules, and thus water tends to rise along the glass surface . The opposite occurs for mercury, which causes the liquid surface near the glass wall to be suppressed.

47. Capillarity

48. Capillarity For Equilibrium, Equating above equations The value of angle of contact θ , for water is .

49. Examples Water rises to a height of 4.5cm in a capillary tube of radius r. Find r, assuming the surface tension of water is 0.073 N/m. Take the angle of contact in the glass as A liquid of density , rises to a height of 7mm in a capillary tube of internal diameter 2mm. If the angle of contact of the liquid to the glass is , find the surface tension of the liquid. A capillary tube of radius 0.05cm is dipped vertically into a liquid of surface tension 0.04N/m and density 0.8 . Calculate the height of capillary rise, if the angle of contact is A capillary tube 0.12mm in diameter has its lower end immersed in liquid with density . Calculate the height of capillary rise if σ . Find the angle of contact to a capillary tube of radius 0.0005m, having a density of 680 Given that the liquid has a surface tension of 0.062 and a capillary rise is 5.2cm.  

50. Properties of Fluids - Bulk Modulus & Compressibility The volume (or density) of a fluid changes with a change in its temperature or pressure. Fluids usually expand as they are heated or depressurized and contract as they are cooled or pressurized. But the amount of volume change is different for different fluids. That is, fluids act like elastic solids with respect to pressure.

51. Properties of Fluids - Bulk Modulus & Compressibility Therefore, it is appropriate to define a coefficient of compressibility, k (also called the bulk modulus of elasticity ) for fluids as k =−V , Pa; T= constant   T he coefficient of compressibility represents the change in pressure corresponding to a fractional change in volume or density of the fluid while the temperature remains constant.

52. Properties of Fluids - Bulk Modulus & Compressibility Bulk Modulus is the measure of ability of a substance to withstand changes in volume when it undergoes compression on all sides. A large value of ‘k’ indicates that a large change in pressure is needed to cause a small fractional change in volume, and thus a fluid with a large k, is essentially incompressible. This is typical for liquids, and explains why liquids are usually considered to be incompressible.

53. Properties of Fluids - Bulk Modulus & Compressibility For an ideal gas, ; Therefore ,   Therefore, the coefficient of compressibility of an ideal gas is equal to its absolute pressure, and so k , of the gas increases with increasing pressure.

54. Properties of Fluids - Bulk Modulus & Compressibility The value for K water = 2.05 x 10 9 N/m 2 For Air, K air = 1.02 x 10 5 N/m 2 Compressibility of the fluid is expressed as the inverse of the Bulk Modulus of the fluid . As K water >> K air, Thus, water is less compressible than air . Thus, water is incompressible as compared to the air.

55. End of Chapter 1