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1 FLUID MECHANICS Gaza CHAPTER 1 Fluids and their Properties Dr. Khalil Mahmoud ALASTAL

2 Objectives of this Chapter: Define the nature of a fluid. Show where fluid mechanics concepts are common with those of solid mechanics and indicate some fundamental areas of difference. Introduce viscosity and show what are Newtonian and non-newtonian fluids Define the appropriate physical properties and show how these allow differentiation between solids and fluids as well as between liquids and gases. K. ALASTAL 2

3 1.1 Fluids : We normally recognize three states of matter: Solid. Liquid. Gas. have common characteristic and both called fluids Fluids in contrast to solids, lack the ability to resist deformation. They flow (move) under the action of such forces, and deforms continuously as the force is applied. A solid can resist deformation, which may cause some displacement but the solid does not continue to move indefinitely. K. ALASTAL 3

4 1.1 Fluids: (cont.) The deformation is caused by shearing forces which act tangentially to a surface. F acting tangentially on a rectangular (solid lined) element ABDC. This is a shearing force and produces the (dashed lined) rhombus element AB'C'D. K. ALASTAL 4

5 1.1 Fluids: (cont.) A Fluid is a substance which deforms continuously, or flows, when subjected to shearing forces. If a fluid is at rest there are no shearing forces acting. All forces must be perpendicular to the planes which they are acting. Fluid Flow Fluid Rest Shear stress Yes Shear stress No K. ALASTAL 5

6 1.1 Fluids: (cont.) There are two aspects of fluid mechanics which make it different to solid mechanics: The nature of fluid is much different to that of solid. In fluids we usually deal with continuous streams of fluid without a beginning or end. In solids we only consider individual elements. K. ALASTAL 6

7 1.2 Shear stress in moving fluid: If fluid is in motion, shear stresses are developed. This occurs if the particles of the fluid move relative to each other with different velocities, causing the shape of the fluid to become distorted. On the other hand, if the velocity of the fluid is the same at every point, no shear stress will be produced, the fluid particles are at rest relative to each other. Fluid particles Moving plate Fixed surface Shear force New particle position K. ALASTAL 7

8 1.2 Shear stress in moving fluid: In practice we are concerned with flow past solid boundaries; aeroplanes, cars, pipe walls, river channels Fluid next to the wall will have zero velocity. (The fluid sticks to the wall.) Moving away from the wall velocity increases to a maximum. Plotting the velocity across the section gives velocity profile Change in velocity with distance is velocity gradient Velocity gradient = dv dy velocity profile K. ALASTAL 8

9 1.2 Shear stress in moving fluid: Because particles of fluid next to each other are moving with different velocities there are shear forces in the moving fluid i.e. shear forces are normally present in a moving fluid. On the other hand, if a fluid is a long way from the boundary and all the particles are traveling with the same velocity. Therefore, the fluid particles are at rest relative to each other. K. ALASTAL 9

10 1.2 Shear stress in moving fluid: The shearing force acts on: Area = A = BC s τ = Shear Stress = F A The deformation which shear stress causes is measured by the angle, and is known as shear strain. In a solid shear strain,, is constant for a fixed shear stress t. In a fluid increases for as long as t is applied - the fluid flows. K. ALASTAL 10

11 1.2 Shear stress in moving fluid: It has been found experimentally that: the rate of shear strain (shear strain per unit time, /time) is directly proportional to the shear stress. Shear strain, Rate of x y shear strain x yt v y (Velocity gradient) v t constant y Dynamic Viscosity τ = μ dv dy K. ALASTAL 11

12 1.2 Shear stress in moving fluid: τ = μ dv dy This is known as Newton s law of viscosity. m : depends on fluid under consideration K. ALASTAL 12

13 1.3 Differences between solids & fluid Solid The strain is a function of the applied stress The strain is independent of the time over which the force is applied and (if the elastic limit is not reached) the deformation disappears when the force is removed Fluid The rate of strain is proportional to the applied stress. Continues to flow as long as the force is applied and will not recover its original form when the force is removed. t Solid F A t Fluid F V m A h K. ALASTAL 13

14 1.4 Newtonian & Non-Newtonian Fluid Newtonian Fluids Fluid obey Newton s law of viscosity refer Newtonian fluids τ = μ dv dy A linear relationship between shear stress and the velocity gradient. The viscosity m is constant. Most common fluids are Newtonian. Example: Air, Water, Oil, Gasoline, Alcohol, Kerosene, Benzene, Glycerine K. ALASTAL 14

15 1.4 Newtonian & Non-Newtonian Fluid Non-Newtonian Fluids Fluid Do not obey Newton s law of viscosity Slope of the curves for non-newtonian fluids varies. There are several categories of Non-Newtonian Fluid. These categories are based on the relationship between shear stress and the velocity gradient. The general relationships is: τ = A + B dv dy where A, B and n are constants. For Newtonian fluids A = 0, B = m and n = 1. Non- Newtonian fluids n K. ALASTAL 15

16 1.4 Newtonian & Non-Newtonian Fluid Plastic: Shear stress must reach a certain minimum before flow commences. Bingham plastic: As with the plastic above, a minimum shear stress must be achieved. With this classification n = 1. An example is sewage sludge, toothpaste, and jellies. K. ALASTAL 16

17 1.4 Newtonian & Non-Newtonian Fluid Pseudo-plastic: No minimum shear stress necessary and the viscosity decreases with rate of shear, e.g. colloidial substances like clay, milk and cement. Dilatant substances; Viscosity increases with rate of shear e.g. quicksand. Thixotropic substances: Viscosity decreases with length of time shear force is applied e.g. thixotropic jelly paints. Rheopectic substances: Viscosity increases with length of time shear force is applied Viscoelastic materials: Similar to Newtonian but if there is a sudden large change in shear they behave like plastic. K. ALASTAL 17

18 1.5 Liquids and Gases: Liquid Difficult to compress and often regarded as incompressible Occupies a fixed volume and will take the shape of the container A free surface is formed if the volume of container is greater than the liquid. Gases Easily to compress changes of volume is large, cannot normally be neglected and are related to temperature No fixed volume, it changes volume to expand to fill the containing vessels Completely fill the vessel so that no free surface is formed. K. ALASTAL 18

19 1.8 Properties of Fluids: There are many ways of expressing density: A. Mass density: r = Mass per unit volume ρ = M V Where: M = mass of fluid, V= volume of fluid Units: kg/m 3 Dimensions: ML -3 Typical values: Water = 1000 kg/m 3, Mercury = kg/m 3 Air = 1.23 kg/m 3, Paraffin Oil = 800kg/m 3 K. ALASTAL 19

20 1.8 Properties of Fluids: B. Specific Weight : Specific Weight w is defined as: w = g = weight per unit volume γ = ρg = W V Where: W = weight of fluid, V= volume of fluid Units: N/m 3 Dimensions: ML -2 T -2 Typical values: Water = 9814 N/m 3, Mercury = N/m 3 Air = N/m 3, Paraffin Oil = 7851 N/m 3 K. ALASTAL 20

21 1.8 Properties of Fluids: B. Relative Density : Relative Density (or specific gravity) s, is defined as the ratio of mass density of a substance to some standard mass density. For solids and liquids this standard mass density is the maximum mass density for water (which occurs at 4 o C) ς = SG = ρ Sub ρ water at 4 C Units: none, as it is a ratio Dimensions: 1 Typical values: Water = 1, Mercury = 13.5, Paraffin Oil = 0.8 K. ALASTAL 21

22 1.8 Properties of Fluids: D. Specific Volume : Specific Volume, υ, is defined as the reciprocal of mass density. Units: m 3 /kg υ = 1 ρ K. ALASTAL 22

23 1.9 Properties of Fluids: Viscosity Viscosity, m, is the property of a fluid, due to cohesion and interaction between molecules, which offers resistance to shear deformation. The ease with which a fluid pours is an indication of its viscosity. (The viscosity is measure of the fluidity of the fluid). Fluid with a high viscosity such as oil deforms more slowly than fluid with a low viscosity such as water. There are two ways of expressing viscosity: A. Coefficient of Dynamic Viscosity B. Coefficient of Kinematic Viscosity K. ALASTAL 23

24 A. Coefficient of Dynamic Viscosity The Coefficient of Dynamic Viscosity, m, is defined as the shear force, per unit area, (or shear stress t), required to drag one layer of fluid with unit velocity past another layer a unit distance away. μ = τ dv dy = Force Area Velocity Distance = Force Time Area = Mass Length Area Units: N.s/m 2 or kg/m.s (kg m -1 s -1 ) (Note that m is often expressed in Poise, P, where 10 P = 1 N.s/m 2 ) Dimensions: ML -1 T -1 Typical values: Water =1.14(10-3 ) Ns/m 2, Air =1.78(10-5 ) Ns/m 2 Mercury =1.552 Ns/m 2, Paraffin Oil =1.9 Ns/m 2 K. ALASTAL 24

25 B. Coefficient of Kinematic Viscosity Kinematic Viscosity, n, is defined as the ratio of dynamic viscosity to mass density. ν = μ ρ Units: m 2 /s (Note that n is often expressed in Stokes, St, where 10 4 St = 1 m 2 /s) Dimensions: ML -1 T -1 Typical values: Water =1.14(10-6 ) m 2 /s, Air =1.46(10-5 ) m 2 /s, Mercury =1.145(10-4 ) m 2 /s, Paraffin Oil =2.375(10-3 ) m 2 /s K. ALASTAL 25

26 1.12 Surface Tension Liquid droplets behave like small spherical balloons filled with liquid, and the surface of the liquid acts like a stretched elastic membrane under tension. The pulling force that causes this is due to the attractive forces between molecules called surface tension s In the body of the liquid, a molecule is surrounded by other molecules and intermolecular forces are symmetrical and in equilibrium. At the surface of the liquid, a molecule has this force acting only through 180. Thus, The surface molecules being pulled inward towards the bulk of the liquid. This effect causes the liquid surface to behave like a very thin membrane under tension. Surface tension is defined as force per unit length, and its unit is N/m. K. ALASTAL 26

27 1.12 Surface Tension: Spherical drop Real Fluid Drops Mathematical Model R is the radius of the droplet, s is the surface tension, Dp is the pressure difference between the inside and outside pressure. The force developed around the edge due to surface tension along the line: F σ = 2πRσ This force is balanced by the force produced from the pressure difference Dp: F p = πr 2 p K. ALASTAL 27

28 1.12 Surface Tension: Spherical drop Now, equating the Surface Tension Force to the Pressure Force, we can estimate p = pi p e : p = 2ς R This indicates that the internal pressure in the droplet is greater that the external pressure since the right hand side is entirely positive. The surface tension forces is neglected in many engineering problems since it is very small. K. ALASTAL 28

29 1.12 Surface Tension Bug Problem Cross-section of bug leg F q F=surface tension on 1 side of leg K. ALASTAL 29

30 Approximate Physical Properties of Common Liquids at Atmospheric Pressure K. ALASTAL 30

31 1.13 Capillarity: Capillary effect is the rise or fall of a liquid in a small-diameter tube. Capillary action in small tubes which involve a liquid-gas-solid interface is caused by surface tension. Wetted Non-Wetted Adhesion Cohesion Adhesion Cohesion Adhesion > Cohesion Water Cohesion > Adhesion Mercury K. ALASTAL 31

32 1.13 Capillarity: H: is the height, R: is the radius of the tube (d=2r) q : is the angle of contact between liquid and solid. The weight of the fluid is balanced with the vertical force caused by surface tension. q H =pds d πdς cos θ = ρg π 4 d2 H =d H = 4ς cos θ ρgd For clean glass in contact with water, q 0, and thus as R decreases, H increases, giving a higher rise. (H= 5mm in a tube of 5mm diameter in water) For a clean glass in contact with Mercury, q 130, and thus H is negative or there is a push down of the fluid. (H= -1.4mm in a tube of 5mm diameter in mercury) K. ALASTAL 32

33 1.14 Compressibity & Bulk Modulus: All materials are compressible under the application of an external force The compressibility of a fluid is expressed by its bulk modulus K, which describes the variation of volume with change of pressure: K = Change in Pressure Volumetric Strain Thus, if the pressure intensity of a volume of fluid, V, is increased by Δp and the volume is changed by ΔV, then: K = p V/V or K = ρ p ρ The ve sign indicate that the volume decrease as the pressure increase. Typical values: Water = 2.05x10 9 N/m 2 ; Oil = 1.62x10 9 N/m 2 K. ALASTAL 33

34 1.14 Compressibity & Bulk Modulus: Large values of the bulk modulus indicate incompressibility. Incompressibility indicates large pressures are needed to compress the volume slightly. Most liquids are incompressible for most practical engineering problems. Example: 1 MPa pressure change = 0.05% volume change, Water is relatively incompressible. K. ALASTAL 34

35 Example: A reservoir of oil has a mass of 825 kg. The reservoir has a volume of m 3. Compute the density, specific weight, and specific gravity of the oil. Solution: r oil mass volume m V kg/ m 3 g oil weight volume mg V rg N / m 3 SG oil r r oil C K. ALASTAL 35

36 Example: Water has a surface tension of 0.4 N/m. In a 3-mm diameter vertical tube, if the liquid rises 6 mm above the liquid outside the tube, calculate the wetting angle. Solution Capillary rise due to surface tension is given by; 4s cosq h rgd cosq gdh 4s q = 83.7 K. ALASTAL 36

37 Example: Given: Pressure of 2 MPa is applied to a mass of water that initially filled 1000-cm 3 volume. Find: Volume after the pressure is applied. Solution: K = 2.2x10 9 Pa V V Dp K DV / V Dp DV V K 6 2x10 Pa 1000cm 9 2.2x10 Pa final final 0.909cm V DV cm K. ALASTAL 37

38 Example: Find: Capillary rise between two vertical glass plates 1 mm apart. s = 7.3x10-2 N/m l is into the page Solution: F vertical 2sl hltg 0 h h 0 2s tg h 14.9 mm 2 2*7.3x * m s h q s t K. ALASTAL 38

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