Introduction:

2. Introduction: • Fluid mechanics is an exciting and fascinating subject with unlimited practical applications ranging from: • microscopic biological systems • to • automobiles, airplanes, and spacecraft propulsion.

3. Fluid Mechanics: Fluid mechanics is defined as the science that deals with the behavior of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries. Fluid mechanics is also referred to as fluid dynamics by considering fluids at rest as a special case of motion with zero velocity.

4. A substance in the liquid or gas phase is referred to as a fluid. What is a Fluid? Distinction between a solid and a fluid is made on the basis of the substance’s ability to resist an applied shear (or tangential) stress that tends to change its shape. A solid can resist an applied shear stress by deforming, whereas a fluid deforms continuously under the influence of shear stress, no matter how small.

5. Application Areas of Fluid Mechanics

6. Application Areas of Fluid Mechanics • Design of modern engineering systems from vacuum cleaners to supersonic • aircraft. • All artificial hearts, breathing machines, and dialysis systems are designed • using fluid dynamics. • The piping systems for cold water, natural gas, and sewage for an individual • house and the entire city are designed primarily on the basis of fluid mechanics. • the piping and ducting network of heating and air-conditioning systems. • A refrigerator involves tubes through which the refrigerant flows, a compressor that pressurizes the refrigerant, and two heat exchangers where the refrigerant absorbs and rejects heat.

7. On a broader scale: Fluid mechanics plays a major part in the design and analysis of: • aircraft, boats, submarines, rockets, jet engines, wind turbines, • biomedical devices, the cooling of electronic components, and the transportation of water, crude oil, and natural gas. • It is also considered in the design of buildings, bridges, and even billboards to make sure that the structures can withstand wind loading. • Numerous natural phenomena such as the rain cycle, weather patterns, the rise of ground water to the top of trees, winds, ocean waves, and currents in large water bodies are also governed by the principles of fluid mechanics.

8. Properties of Fluids Any characteristic of a system is called property. Some familiar properties are pressure P, temperature T, volume V, and mass m. Properties are considered to be either intensive or extensive. • Intensive properties are those that are independent of the mass of a system, such as temperature, pressure, and density.

9. Extensive properties are those whose values depend on the size—or extent—of the system. Total mass, total volume V, and total momentum are some examples of extensive properties.

10. Criteria to differentiate intensive and extensive properties

11. It means to disregard the atomic nature of a substance and view it as a continuous, homogeneous matter with no holes. Continuum The continuum idealization allows us to treat properties as point functions and to assume that the properties vary continually in space with no jump discontinuities.

12. Properties of Fluids • Density- Density is defined as mass per unit volume. That is, Density: • Specific volume- The reciprocal of density is the specific volume v, which is defined as volume per unit mass. That is,

13. Specific weight-The weight of a unit volume of a substance is called specific weight and is expressed as • Specific gravity- defined as the ratio of the density of a substance to the density of some standard substance at a specified temperature (usually water at 4°C, for which density = 1000 kg/m3). That is,

14. Vapour Pressure: The vapour pressure(Pv) of a pure substance is defined as the pressure exerted by its vapour in phase equilibrium with its liquid at a given temperature.

15. Cavitation: Sometimes in liquid-flow systems(such as pumps, turbines etc)the liquid pressure drop below the vapour pressure at some locations. • results in unplanned vaporization. • vapor bubbles collapse as they are swept away from the low pressure regions, generating highly destructive, extremely high-pressure waves. • Results in drop in performance and even the erosion of impeller blades of a pump.

16. Cavitation damage

17. A fluid contracts when more pressure is applied on it and expands when the pressure acting on it is reduced. • Coefficient Of Compressibility(κ) 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. • The coefficient of compressibility of a truly incompressible substance • (v = constant) is infinity. • Coefficient of Volume Expansion(β) represents the variation of the density of a fluid with temperature at constant pressure.

18. some insects can land on water or even walk on water and that small steel needles can float on water • water beads up into small drops on flower petals • Surface tension Some consequences of surface tension

19. The surface of the liquid acts like a stretched elastic membrane under tension. The pulling force that causes this tension acts parallel to the surface and is due to the attractive forces between the molecules of the liquid. The magnitude of this force per unit length is called surface tension(σs ) and is usually expressed in the unit N/m. • This effect is also called surface energy and is expressed in the equivalent unit of N-m/m2or J/m2. In this case, σs represents the stretching work that needs to be done to increase the surface area of the liquid by a unit amount.

20. To determine excess pressure inside a drop and bubble:

21. Capillary Effect • It 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 capillary effect is also partially responsible for the rise of water to the top of tall trees.

22. Capillary rise of water and the capillary fall of mercury in a small-diameter glass tube :

23. Contact angle or wetting angle(Φ) Contact(or wetting) angle ϕ, defined as the angle that the tangent to the liquid surface makes with the solid surface at the point of contact. The curved free surface of a liquid in a capillary tube is called the meniscus. Contact Angle If, Φ <90o then fluid wets the surface. If, Φ >90o then fluid does not wet the surface. Contact angle(ϕ) for wetting and non-wetting fluids

24. To Calculate The Magnitude of the Capillary Rise in a Circular Tube Proof: Equating the vertical component of the surface tension force to the weight gives:

25. This relation is also valid for non-wetting liquids (such as mercury in glass) and gives the capillary drop. In this case cosΦ>90° and thus cosΦ<0, which makes h negative. • Therefore, a negative value of capillary rise corresponds to a capillary drop.

26. A 0.6-mm-diameter glass tube is inserted into water at 20°C in a cup. Determine the capillary rise of water in the tube. Note: The surface tension of water at 20°C is 0.073 N/m. The contact angle of water with glass is 0°.

27. There is a property that represents the internal resistance of a fluid to motion or the “fluidity,” and that property is the viscosity. The force a flowing fluid exerts on a body in the flow direction is called the drag force. • Viscosity Courtesy: Fluid Mechanics by Cengel and Cimbala

28. Consider a fluid layer between two very large parallel plates (or equivalently, two parallel plates immersed in a large body of a fluid) separated by a distance. To obtain a relation for viscosity:

29. Now a constant parallel force F is applied to the upper plate while the lower plate is held fixed. • After the initial transients, it is observed that the upper plate moves continuously under the influence of this force at a constant velocity V.

30. In steady laminar flow, the fluid velocity between the plates varies linearly between 0 and V, and thus the velocity profile and the velocity gradient are: where y is the vertical distance from the lower plate.

31. During a differential time interval dt, the sides of fluid particles along a vertical line MN rotate through a differential angle dβ while the upper plate moves a differential distance da=V dt. The angular displacement or deformation (or shear strain) can be expressed as:

32. Rearranging, the rate of deformation under the influence of shear stress τ becomes • It can be verified experimentally that for most • fluids the rate of deformation (and thus the • velocity gradient) is directly proportional to the • shear stressτ ,

33. These fluids for which the rate of deformation is proportional to the shear stress are called Newtonian fluids after Sir Isaac Newton, who expressed it first in 1687. • Most common fluids such as water, air, gasoline, and oils are Newtonian fluids. • Blood and liquid plastics are examples of non-Newtonian fluids.

34. In one-dimensional shear flow of Newtonian fluids, shear stress can be expressed by the linear relationship: • where the constant of proportionality μ is called • the coefficient of viscosity or the dynamic • (or absolute) viscosity of the fluid, whose unit is • kg/m · s, or equivalently, N·s/m2 (or Pa-s where Pa • is the pressure unit pascal).

35. A common viscosity unit is poise, which is equivalent to 0.1 Pa-s (or centipoise, which is one-hundredth of a poise). • The viscosity of water at 20°C is 1 centipoise, and thus the unit centipoise serves as a useful reference.

37. Non-Newtonian fluids: • For non-Newtonian fluids, the relationship between shear stress and rate of deformation is not linear, as shown in Fig.

38. The slope of the curve on the τ versus du/dy chart is referred to as the apparent viscosity of the fluid. • Fluids for which the apparent viscosity increases with the rate of deformation (such as solutions with suspended starch or sand) are referred to as dilatant or shear thickening fluids,

39. The fluid which becomes less viscous as it is sheared harder, such as some paints, polymer solutions, and fluids with suspended particles) are referred to as pseudoplastic or shear thinning fluids. • Some materials such as toothpaste can resist a finite shear stress and thus behave as a solid, but deform continuously when the shear stress exceeds the yield stress and thus behave as a fluid. Such materials are referred to as Bingham plastics.

40. The viscosity of a fluid is to be measured by a viscometer constructed of two 40-cm-long concentric cylinders. The outer diameter of the inner cylinder is 12 cm, and the gap between the two cylinders is 0.15 cm. The inner cylinder is rotated at 300 rpm, and the torque is measured to be 1.8 N-m. Determine the viscosity of the fluid.

41. Pressure is the normal compressive force per unit • area. • Pressure at any point in a fluid is the same in all • directions. • It has magnitude but not a specific direction, and • thus it is a scalar quantity. • Pressure

42. Fluid statics deals with problems associated with fluids at rest. • In fluid statics, there is no relative motion between adjacent fluid layers, and • thus there are no shear (tangential) stresses in the fluid trying to deform it. • The only stress we deal with in fluid statics is the normal stress, which is the • pressure, and the variation of pressure is due only to the weight of the fluid. • Fluid statics is used to determine the forces acting on floating or submerged • bodies and the forces developed by devices like hydraulic presses • and car jacks. • The design of many engineering systems such as water dams and liquid • storage tanks requires the determination of the forces acting on the surfaces • using fluid statics. INTRODUCTION TO FLUID STATICS

43. To determine the variation of pressure in a fluid at rest :

44. Pascal’s law The pressure applied to a confined fluid increases the pressure throughout by the same amount. This is called Pascal’s law, after Blaise Pascal (1623–1662). Pascal also knew that the force applied by a fluid is proportional to the surface area. Hydraulic brakes and lifts etc are based on Pascal’s law.

45. Hydraulic lifts: The area ratio A2/A1 is called the ideal mechanical advantage of the hydraulic lift.

46. A device based on this principle is called a manometer, and it is commonly used to measure small and moderate pressure differences. A manometer mainly consists of a glass or plastic U-tube containing one or more fluids such as mercury, water, alcohol, or oil. To keep the size of the manometer to a manageable level, heavy fluids such as mercury are used if large pressure differences are anticipated. THE MANOMETER

47. Here, C.P. = Center of Pressure C.A. = Center of Area Hydrostatic force on an inclined plane:

48. Force on the elemental area dA: dFP = P dA = (ρgh )dA Force on the entire surface area: Where, Pressure Force on an inclined plane

49. Let, a surface be submerged in a liquid inclined at an angle,θ with the free surface of the liquid. Pressure on the elemental area = ρgh Hence, Force on the elemental area, dF = (ρgh)dA Moment of this force about x-axis dM = y dF = y (ρgh)dA Hence, moment of all such forces on the surface about x-axis, To locate the centre of pressure of an inclined submerged surface:

50. Hence, moment of all such forces on the surface about x-axis, ---------(1) = second moment of area Moment of hydrostatic force on the surface about x-axis ---------(2)