# Fluid Engineering Mechanics

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1 Fluid Engineering Mechanics Chapter Fluid Properties: Density, specific volume, specific weight, specific gravity, compressibility, viscosity, measurement of viscosity, Newton's equation of viscosity, Surface tension, capillarity and pressure Dr. Muhammad Ashraf Javid Assistant Professor Department of Civil and Environmental Engineering

2 Physical Properties of Fluids Density Viscosity Specific Volume Surface Tension Specific Weight Pressure Specific Gravity Buoyancy Compressibility

3 Density It is also termed as specific mass or mass density. It is the mass of substance per unit volume.i.e., mass of fluid per unit volume. It is designated with symbol of ρ (rho) ρ =mass/volume =M/L 3 Fundamental Units=kg/m 3, slug/m 3, g/cm 3 F Ma M 3 L M FT 4 L F a FT L Note: Density of water at 4 o c=000kg/m 3,.938slug/ft 3, g/cm 3 3

4 Specific Volume It is defined as volume of substance per unit mass. It is designated with υ. 3 volume / mass L / M Fundamental Units=m 3 /kg, m 3 /slug, cm 3 /g F Ma 4 L FT 4

5 Relationship Between Density and Specific Volume volume / mass L mass/ volume M 3 / M / L 3 / / 5

6 Specific Weight It is the weight of substance per unit volume or say it is the weight of fluid per unit volume. It is designated by γ (gamma). weight volume W L Mg L 3 3 ML L T M L T W g 3 T Mg L Note: Specific weight of water at 4 o c=980n/m 3, 6.4lb/ft 3, 98dyne/cm 3 6

7 Relation Between and Mass volume & weight volume W Mg g 7

8 Effect of Temperature and Pressure on Specific Weight 8 As the equation of state for a perfect gas is given by P RT Where P =absolute pressure υ =specific volume T =absolute temperature R =gas constant For perfect gases mr=83 N-m/(kg-k) Where m=molecular weight of gas P RT P RT P / g gp RT RT (constant) P P T &,constant T g g R

9 Effect of Temperature and Pressure on Specific Weight Since weight volume W V Assuming constant pressure T n n 0, Assuming constant temperature n P n 0, 9

10 Specific Gravity (Relative Density) It is the ratio of density of a substance and density of water at 4 o C. It is the ratio of specific weight of substance and specific weight of water at 4 o C. It is the ratio of weight of substance and weight of an equal volume of water at 4 o C. S fluid water fluid water W W fluid water Remember: P & T S fluid water at T at T o o C C fluid water at T at T o o C C W W fluid water at T at T o o C C 0 Note: Specific gravity of liquid is measured w.r.t. water while for cases of gases it is measured w.r.t. standard gas (i.e. air)

11 Class Problems

12

13 Compressibility Compressible fluids Incompressible fluids In fluid mechanics we deal with both compressible and incompressible fluids of either variable or constant density. Although there is no such thing in reality as incompressible fluid, we use this terms where the change in density with pressure is so small as to be negligible. This is usually the case with liquids. Ordinarily, we consider the liquids as incompressible. We may consider the gases to be incompressible when the pressure variation is small compared with absolute pressure.

14 Compressibility Compressible fluids Fluids which can be compressed. Fluid in which there is a change in volume with change in pressure P P P P v v v v 4 As a result of change in volume, density and specific weight of fluid also changes. Hence, for compressible fluids, v v

15 Compressibility Incompressible fluids Fluids which can not be compressed. Fluid in which there is no change in volume with change in pressure P P P P v v v v v 5 As a result, no-change in volume, density and specific weight of fluid. Hence, for incompressible fluids, v v

16 Compressibility (Volumetric Strain) 6 Volumetric Strain is the ratio of change in volume and original volume. P P v v P P v v / / / v dv v v v M v M v M v Volumetric strain=change in specific volume/original specific volume. d

17 Compressibility Bulk Modulus or Volume Modulus of Elasticity (Ev): It is defined as ratio of volumetric stress to volumetric strain E v = volumetric stress/volumetric strain E v =change in pressure/compressibility E v dp dv v E v dp d 7

18 Viscosity The viscosity of a fluid is a measure of its resistance to shear or angular deformation. It is the property of a fluid by mixture of which it offers resistance to deformation under the influence of shear forces. It depends upon the cohesion and molecular momentum exchange between fluid layers. It can also be defined as internal resistance offered by fluid to flow. It is denoted by μ. It is also termed as coefficient of viscosity or absolute viscosity or dynamic viscosity or molecular viscosity. 8

19 Factor affecting viscosity. Cohesion. Molecular momentum. Cohesion: It is the attraction between molecules of fluid. More the molecular attraction (cohesion) more is the viscosity (resistance to flow) of fluid. It is dominant in liquids.. Molecular momentum: Molecules in any fluid change their position with time and is known as molecular activity. More the molecular activity more will be viscosity of the fluid. It is dominant in gases A B 9

20 Effect of temperature on viscosity 0 For Liquids: In case of liquids, cohesion (molecular attraction is dominant). Therefore, if the temperature of liquid is increased, its cohesion and hence viscosity will decrease. For Gases: T In gases momentum exchange is dominant. Therefore, if the temperature of gases is increases, its momentum exchange will increase and hence viscosity will increase. T

21 Kinematic Viscosity It is ratio of absolute viscosity and density of fluid. It is denoted by (nu)

22 Newton s Equation of Viscosity Consider two parallel plates, in which lower plate is fixed and upper is moving with uniform velocity U under the influence of force F. Space between the plates is filled with a fluid having viscosity, μ. U u Moving plate Force, F Y dy du Fixed plate F= Applied force (shearing force) A= Contact area of plate(resisting area) Y=gap/space between plates U= Velocity of plate As the upper plate moves, fluid also moves in the direction of applied force due to adhesion.

23 Newton s Equation of Viscosity Factors affecting Force, F Hence, ( i) F A; ( ii) F U; ( iii) F F AU Y Where, μ is coefficient of viscosity F AU Y Y Assuming linear velocity profile (as shown in figure) F A U Y du dy 3

24 Newton s Equation of Viscosity At boundaries the particles of fluid adhere to wall and so their velocities are zero relative to wall. This so called non-slip condition occurs in viscous fluids 4

25 Newton s Equation of Viscosity dy du The above equation is called as Newton s equation of viscosity. The equation shows that the shearing stress is directly proportional to the velocity gradient. In the above equation du/dy= velocity gradient or rate of change of deformation or shear rate μ = absolute viscosity τ=shear stress 5

26 Dimensional Analysis of Viscosity Viscosity Kinematic Viscosity F A FT L Y U L FL ( L / T ) This expression is used to write fundamental unit of viscosity M LT M 3 L L T MLT L M LT T F MLT 6

27 Unit of Viscosity Viscosity M / LT Widely used unit is Poise =0.N.s/m Kinematic Viscosity SI BG CGS N-s/m Lb-s/ft Dyne-s/cm L /T (Poise, P) Kg/(m-s) Slug/(ft-s) g/(cm-s) SI BG CGS m /s ft /s cm /s (stoke) 7 Widely used unit is Stoke=0-4 m /s

28 Problem x x0 m / s 8

29 Problem A flat plate 00mm x 750mm slide on oil (μ =0.85N.s/m ) over a large surface as shown in fig. What force, F, is required to drag the plate at a velocity u of.m/s if the thickness of the separating oil film is 0.6mm? F F F F A A U Y du dy U A Y 55N du dy A 0.85 U Y A. 0.6 / Here, t = Y 9

30 Problem A space 6mm wide between two large plane surfaces is filled with SAE 30 Western lubricating oil at 35 o C(Fig). What force is required to drag a very thin plate of 0.4m area between the surfaces at a speed u=0.5m/s (a) if this plate is equally spaced between the two surfaces? (b) if t=5mm? Solution: Y=6mm A=0.4m u=0.5m/s T=35 o C μ =0.8N.s/m (from figure A.) (a) F=? If Y=8mm F A U Y du dy 8mm 30 F F A F A

31 Solution to Problem 3 8mm N A y u A y u F A A F F F / /

32 Solution to problem 3 (b): F=? If t =5mm y =, y =5mm N A y u A y u F A A F F F / / mm

33 33

34 Shear Stress ~ Velocity gradient curve Ideal fluid Newtownian Fluid Non-Newtownian fluid Ideal plastic Real solid Ideal solid/elastic solid Real solid 34

35 Shear Stress ~ Velocity gradient curve Ideal Fluid: The fluid which does not offer resistance to flow 0 0 Newtownian Fluid: Fluid which obey Newtown s law of viscosity du dy slope of curve ( ~ du / dy )is constant Non-Newtonian fluid: Fluid which does not obey Newtown s Law of viscosity du dy slope of curve ( ~ du / dy )changing continuously 35

36 Shear Stress ~ Velocity gradient curve Ideal Solid: solid which can never be deformed under the action of force du 0 dy Real solid: solid which can be deformed under action of forces Ideal Plastic: These are substances which offer resistance to shear forces without deformation upon a certain extent but if the load is further increased then they deform Real Plastic: These are substances in which there is deformation with the application of force and it increases with increase in applied load. 36

37 Problem 37

38 38

39 Exercise Problems 39

40 Measurement of Viscosity The following devices are used for the measurement of viscosity. Tube type viscometer. Rotational type viscometer 3. Falling sphere type viscometer 40

41 Falling Sphere Viscometer It consists of a tall transparent tube or cylinder and a sphere of known diameter. The sphere is dropped inside the tube containing liquid and time of fall of sphere between two points (say A and B) is recorded to estimate the fall velocity (s/t)of sphere inside liquid. Where s = distance between point A and B and t is the time of travel. From this velocity of fall, viscosity is estimated from the expression of fall sphere type viscometer. W FB FD Dt D Fig. Falling Tube type viscometer s A B 4

42 Falling Sphere Viscometer D=D s =Diameter of sphere Dt D t =Diameter of tube or cylinder D V t =velocity of sphere in tube (s/t) s=distance between points A and B t=time taken by sphere to cover distance (s) W=weight of sphere=γ*(vol) W F B s A = γ s (πd 3 /6) F D F B =Force of Buoyancy = γ L (πd 3 /6) F D =Drag force = (3πμVD) Stoke s Law B Note: V is not equal to V t Fig. Falling Tube type viscometer 4

43 Falling Sphere Viscometer 43 Buoyancy: It is the resultant upward thrust exerted by the fluid on a sphere. It is the tendency of fluid to lift the body and it is equal to weight of volume of fluid displaced by the body (Archimedes Principal). Drag Force: It is a resisting force generated by the liquid on the moving object which is acting in the opposite direction of movement. V t =velocity of sphere in tube with wall effect V=velocity of sphere in tube without wall effect V>Vt t t t D D D D V V 3 D t D if ; 0; 3 3 D VD D W F F F S L D B y

44 Falling Sphere Viscometer 44 The above equation is governing equation for falling sphere type viscometer. For a particular temperature, D, γ s and γ L are constant. So we can write ; Thus, velocity of fall is inversely proportional to viscosity and is indicative of viscosity in falling sphere type viscometer. Note: This method can only be used for transparent liquids L S L S L S V D D D V D D VD V

45 Problem:..0 45

46 Problem:..0 46

47 Tube Type Viscometer In tube type viscometer, liquid is placed in a container to a certain level. Valve in the bottom is opened to fill the flask of known volume. H h Time taken to fill the flask is recorded which gives measure of viscosity of liquid H 47 H =Average imposed head causing flow=h+l-h/ V L =volume of flask D=Diameter of tube L=length of tube h=fall of liquid level in container to fill liquid in flask V

48 Tube Type Viscometer Lets consider two points and and apply energy equation. z P V g z P V g H L H h Where, g are potential, pressure, and velocity head and P V z, & H L =head loss V H' g H L The head loss in tube type viscometer is due to friction loss in tube and is represented by (i) Datum V H 48 H H 3L V D L F (ii)

49 Tube Type Viscometer 49 Eq (ii) is a Hagen Poiseulli law for laminar flows in tube. Moreover, for laminar flow in tube viscometer V<< therefore V can be neglected. Hence Eq (i) becomes as H' 0 0 H' H ' Q 3L V D 3L D Q A 4 D H ' 8L 3L V D H H 3L V D L F Q D 4 H ' 8QL Equation for tube type viscometer A AV 4 D A=Cross-sectional area of tube and V is average velocity

50 Tube Type Viscometer D 4 H ' 8QL D 4 g 8 VL t K t t H' t L Where Q V L Volume time k t = constant of tube type viscometer d 4 V t h L 50 Note: Equation of tube type viscometer is applicable for laminar flows. For flows in pipes, flow will be laminar if Re 000 and flow will be turbulent if Re>4000 Reynolds no Re VD Re= Transition flow 000

51 Problem;..7 Solution: D=0.040in=.040/ ft L=3.in=3./ft V L =60mL=0.00 ft 3 H =(0+9.5)/=9.75 in t=8.7sec =H D 4 g 8 VL H' t L / ft / s 5

52 Problem 5

53 Rotational Type Viscometer It consists of two concentric cylinders. Small cylinder is placed inside the bigger cylinder. The gap (space) between cylinders is filled with liquid up to a height, h,. Δr r r h Then either inner or outer cylinder is fixed and other is rotated by applying a constant torque. Revolution per minute (RPM) is measured which is indicative of viscosity. 53

54 Rotational Type Viscometer r =outer radius of inner cylinder r =inner radius of outer cylinder r=mean radius=(r +r )/ Δr=gap (space) between cylinders h=height of liquid F=shearing force F A T=Applied torque (mean torque)=f x r Δr r r h T Ar F A T du Ar dy T u r Ar u r du dy () 54

55 Rotational Type Viscometer Resisting area u N r 60 Where N=RPM(Revolution per minute) ω=angular frequency A r h Angular Velocity u r revolution radian revolution/s radian/s N revolution/s N radian/s N revolution/min N / 60 radian/min Δr r r h

56 Rotational Type Viscometer Now substituting the values of A and u in Eq. () T T T rn 60r Where K r =Rotational viscometer constant K r 3 r h N 5r K N r 3 r h 5 r rh r By re-arranging the formula, the absolute viscosity using rotational type viscometer can be obtained as T kr N For a particular viscometer, both T and Kr are constant and therefore N 56

57 Problem:..6 Solution: Rotation type viscometer h=300mm=0.3m OD of inner cylinder=00mm=0.m Δr r = 50mm r = 5mm ID of outer cylinder=0mm=0.0m h=300mm Δr=(0-00)/=mm=0.00m Torque=T= 8 N-m N=/4 rev/s=60/4 RPM Neglect mechanical friction 57

58 Problem:..6 T K K r r N 3 r h 5r T K r N 0.4Ns / m 58

59 Surface Tension The tension force created at the imaginary thin surface due to unbalanced-molecular attraction is termed as surface tension. B v A Molecule A in figure above is situated at a certain depth below the surface. It is acted upon by equal force from all sides whereas molecule B (situated at the surface) is acted upon by unbalanced forces from below. Thus a tight skin/film/surface is formed at the surface due to inward molecular pull. 59

60 Types of molecular attraction Cohesion: It is the attraction force between the molecules of same material Adhesion: It is the attraction force between the molecules of different materials Surface tension depends upon the relative magnitude of cohesion and adhesion but primarily it depend upon the cohesion. With the increase in temperature cohesion reduces and hence surface tension also reduces. Concept of surface tension is used in capillarity action 60

61 Capillarity It is the rise or fall of a liquid in a small diameter (< 0.5 ) tube due to surface tension and adhesion between liquid and solid. For capillary action diameter of tube is less than 0.5inch while for large diameter tubes this phenomenon become negligible. θ D h σ D θ h v v σ Water θ<90 Mercury θ>90 The curved surface that develops in tube is called meniscus 6

62 Capillarity D D= diameter of tube γ=specific weight of liquid h=capillary rise/fall θ=angle of contact or contact angle σ=force of surface tension per unit length v θ h σ Derivation of expression for capillary rise/fall Let s take Fy 0 Weight of column of liquid acting downward= vol 4 D h Vertical component of force of surface tension= D cos 6

63 Capillarity Equating both equations 4 D cos D h D cos D h 4 hd 4 cos or h 4cos D The above equation is used to compute capillary rise/fall. Note: Fall has ve sign 63

64 Problem..9 Solution: At 50 o F With θ=0 o lb / ft, lb / 4 cos h D ft.74in / ft True static height= in=5.6in 64

65 Surface Tension of water Surface Tension of Water in Temperature contact with Air - t - ( o - σ - F) (0-3 lb/ft)

66 Vapor Pressure of liquids Vapor Pressure: It is the pressure at which liquid transforms into vapors or it is the pressure exerted by vapors of liquid. All the liquids have tendency to release their molecules in the space above their surface. If the liquid in container have limited space above it, then the surface is filled with the vapors. These vapors when released from liquid exert pressure known as vapor pressure. It is the function of temperature. More the temperature more will be vapor pressure. 66

67 Vapor Pressure of liquids Saturated vapor pressure: It is the vapor pressure that corresponds to the dynamic equilibrium conditions (saturation) i.e., when rate of evaporation becomes equal to rate of condensation. Boiling vapor pressure: The pressure at which vapor pressure becomes equal to atmospheric pressure. 67

68 Effect of Temperature and Pressure on E v of Water Ev(max) Ev Ev Pressure is constant Temperature is constant Temp, T 50 o C Pressure, dp 68

69 Relation Between E v and Compressibility As Ev dp dv v Ev dv v Ev compressibility 69

70 Problem / Ev dp d dp 70 / g

71 Solution 7

72 Sample MCQs.Specific gravity of a liquid is equal to (a). Ratio of mass density of water to mass density of liquid (b). inverse of mass density (c). Ratio of specific weight of liquid to specific weight of water (d). None of all. What happens to the viscosity of a liquid when its temperature is raised? (a). The viscosity of the liquid increases (b). The viscosity of the liquid stays the same (c). The viscosity of the liquid decreases (d).the temperature of a liquid does not rise 3. What happens to the specific weight of a liquid when its temperature is raised? (a). It increases (b). It stays the same (c). It decreases (d). The temperature of a liquid does not rise 7

73 Sample MCQs 4. In a falling sphere viscometer, according to force balance we have (a). Weight=Drag+Buoyancy (b). Drag=Weight - Buoyancy (c). Buoyancy=Weight +Drag (d). None of all 5. The kinematic viscosity is (a). Multiplication of dynamic viscosity and density (b). division of dynamic viscosity by density (c). Multiplication of dynamic viscosity and pressure (d). None of the above 6. As a result of capillary action 73 (a) liquid rise in capillary tube (b) liquid falls down in capillary tube Both (a) and (b) (d) None of all

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