Fluid MechanicsFluid MechanicsChapter 1Chapter 1

Fluids and their PropertiesFluids and their Properties

Faculty of Engineering & TechnologyFaculty of Engineering & TechnologyINTI University CollegeINTI University College

EGR 224 Fluid MechanicsEGR 224 Fluid Mechanics

Basic Text: Basic Text: ““Fluid Mechanics” by J. F. Fluid Mechanics” by J. F. DouglasDouglas, J. M. Gasiorek and J. A. Swaffield, J. M. Gasiorek and J. A. Swaffield

55thth edition, 4 edition, 4thth edition, or 3 edition, or 3rdrd edition edition

ChaptersChapters 1, 2, 3, 4, 5, 6, 8,10,14,15 + Handouts 1, 2, 3, 4, 5, 6, 8,10,14,15 + Handouts

ReferencesReferences:: J.F. J.F. DouglasDouglas and R.D. Mathews “Solving Problems in Fluid Mechanics” and R.D. Mathews “Solving Problems in Fluid Mechanics”

Volume 1 and Volume 2Volume 1 and Volume 2

Other text books on “Fluid Mechanics”Other text books on “Fluid Mechanics”

IntroductionIntroduction

Fluid Mechanics Fluid Mechanics is that branch of applied mechanics that is that branch of applied mechanics that is concerned with the statics and dynamics of liquids and is concerned with the statics and dynamics of liquids and gasesgases

Fluid Statics: Fluid Statics: whichwhich treats fluids in the equilibrium state treats fluids in the equilibrium state of no shear stressof no shear stress

Fluid Dynamics: Fluid Dynamics: which treats when portions of fluid are which treats when portions of fluid are in motion relative to other partsin motion relative to other parts

FluidsFluids

In everyday life, we recognize three states of matter: In everyday life, we recognize three states of matter: solid, liquid and gassolid, liquid and gas

Liquids and gases have a common characteristic in Liquids and gases have a common characteristic in which they differ from solidswhich they differ from solids

They are fluids, lacking the ability of solids to offer a They are fluids, lacking the ability of solids to offer a permanent resistance to a deforming force permanent resistance to a deforming force

Definition of Fluids Definition of Fluids

A fluid is a substance which deforms continuously A fluid is a substance which deforms continuously under the action of shearing forces, however small under the action of shearing forces, however small they may be. they may be.

If a fluid is at rest, there can be no shearing forces If a fluid is at rest, there can be no shearing forces acting and, therefore, all forces in the fluid must be acting and, therefore, all forces in the fluid must be perpendicular to the planes upon which they act. perpendicular to the planes upon which they act.

Deformation of fluidsDeformation of fluids

Fluids Fluids flowflow under the action of a force, deforming continuously for under the action of a force, deforming continuously for as long as the force is appliedas long as the force is applied

Deformation is caused by Deformation is caused by shearingshearing forces such as F (Fig. 1.1) and forces such as F (Fig. 1.1) and cause the material originally occupying the space ABCD to deform cause the material originally occupying the space ABCD to deform to AB’C’Dto AB’C’D

Fig. 1.1 Deformation caused by shearing forcesFig. 1.1 Deformation caused by shearing forces

Shear stress in a moving fluidShear stress in a moving fluid

There can be no shear stress in a fluid at restThere can be no shear stress in a fluid at rest Shear stresses are developed when the fluid is in motionShear stresses are developed when the fluid is in motion If the particles of the fluid move relative to each other, they will have If the particles of the fluid move relative to each other, they will have

different velocities, causing the original shape of the fluid to become different velocities, causing the original shape of the fluid to become distorted distorted

Considering successive layers parallel to the boundary (Fig 1.2), the Considering successive layers parallel to the boundary (Fig 1.2), the velocity of the fluid will vary from layer to layer as y increasesvelocity of the fluid will vary from layer to layer as y increases

Fig. 1.2. Variation of velocity with distance from a solid boundaryFig. 1.2. Variation of velocity with distance from a solid boundary

Newton’s Law of ViscosityNewton’s Law of Viscosity

For a small angle, x = For a small angle, x = .y.yshear strain, shear strain, = x/y = x/yrate of shear strain = (x/y).(1/t) rate of shear strain = (x/y).(1/t) = (x/t).(1/y) = (x/t).(1/y) = u/y= u/ywhere u = velocity of the particle at Ewhere u = velocity of the particle at E y = distance from AD.y = distance from AD.

From experimental result, shear stress is proportional to rate of shear strain, From experimental result, shear stress is proportional to rate of shear strain, then then

shear stress, shear stress, = constant . (u/y)= constant . (u/y)where (u/y) = change of velocity with the distance y.where (u/y) = change of velocity with the distance y.

is known as is known as Newton’s law of viscosityNewton’s law of viscosity

where where μμ is the dynamic viscosity of the fluid. is the dynamic viscosity of the fluid.

dydu

Shear stress in a moving fluidShear stress in a moving fluid

Consider a lubricating oil of viscosity Consider a lubricating oil of viscosity undergoes steady shear between a undergoes steady shear between a fixed lower plate and an upper plate moving at a certain speed Vfixed lower plate and an upper plate moving at a certain speed V

y yy y moving platemoving plate u = Vu = V

u = 0u = 0 u u fixed plate velocity profile fixed plate velocity profile

Newton’s law of viscosityNewton’s law of viscosity

where where = shear stress, = shear stress, = dynamic viscosity and = dynamic viscosity and

du/dy = velocity gradient (or) rate of shear straindu/dy = velocity gradient (or) rate of shear strain

dydu

Differences between Solids and FluidsDifferences between Solids and Fluids

For a solid, the strain is a function of the applied stress, For a solid, the strain is a function of the applied stress, providing that the elastic limit is not exceeded providing that the elastic limit is not exceeded

For a fluid, the rate of strain is proportional to the applied For a fluid, the rate of strain is proportional to the applied stressstress

The strain in a solid is independent of the time over which The strain in a solid is independent of the time over which the force is applied, and, if the elastic limit is not exceeded, the force is applied, and, if the elastic limit is not exceeded, the deformation disappears when the force is removed the deformation disappears when the force is removed

A fluid continues to flow as long as the force is applied and A fluid continues to flow as long as the force is applied and will not recover its original form when the force is removed will not recover its original form when the force is removed

Newtonian FluidsNewtonian Fluids Fluids obeying the Fluids obeying the Newton's law of viscosityNewton's law of viscosity and for which and for which

has a constant value are called has a constant value are called Newtonian fluidsNewtonian fluids

Newton's law of viscosity is given byNewton's law of viscosity is given by == (du/dy) (du/dy)

where where = shear stress = shear stress = viscosity of fluid= viscosity of fluid du/dy = shear rate (or) rate of shear strain (or) velocity du/dy = shear rate (or) rate of shear strain (or) velocity

gradientgradient

Non-Newtonian FluidsNon-Newtonian Fluids Fluids which do not obey the Newton’s law of viscosity are Fluids which do not obey the Newton’s law of viscosity are

known as known as non-Newtonian fluidsnon-Newtonian fluids

Bingham Plastic, Bingham Plastic, for which for which thethe shear stress must reach a certain shear stress must reach a certain minimum value before flow commences (e.g. tooth paste, jellies, minimum value before flow commences (e.g. tooth paste, jellies, sewage sludge, etc.) sewage sludge, etc.)

Pseudo-plastic, Pseudo-plastic, for which dynamic viscosity decreases as the rate of for which dynamic viscosity decreases as the rate of shear increases (e.g. polymer solutions, blood, clay, milk, cement, etc.)shear increases (e.g. polymer solutions, blood, clay, milk, cement, etc.)

Dilatant fluids, Dilatant fluids, in which viscosity increases with increasing velocity in which viscosity increases with increasing velocity gradient (e.g. quicksand)gradient (e.g. quicksand)

Newtonian and Non-Newtonian FluidsNewtonian and Non-Newtonian Fluids

Fig. 1.3: Variation of shear stress with velocity gradientFig. 1.3: Variation of shear stress with velocity gradient

Liquids and GasesLiquids and Gases Although liquids and gases both share the common characteristics of Although liquids and gases both share the common characteristics of

fluids, they have many distinctive characteristics of their own.fluids, they have many distinctive characteristics of their own. A liquid is difficult to compress and, for many purposes, may be A liquid is difficult to compress and, for many purposes, may be

regarded as incompressible.regarded as incompressible.A given mass of liquid occupies a fixed volume, and a A given mass of liquid occupies a fixed volume, and a free surfacefree surface is is formed.formed.

A gas is comparatively easy to compressA gas is comparatively easy to compressA given mass of gas has no fixed volume and will expand continuously. A given mass of gas has no fixed volume and will expand continuously. The gas will completely fill any vessel in which it is placed and therefore, The gas will completely fill any vessel in which it is placed and therefore, does not form a free surface.does not form a free surface.

liquidgas

Free surface

System International (SI units)System International (SI units)

Fundamental units:Fundamental units:Mass: kilogramme (kg)Mass: kilogramme (kg)Length: metre (m)Length: metre (m)Time: second (s)Time: second (s)

Derived units:Derived units:All other units are derived from these fundamental units.All other units are derived from these fundamental units.For example,For example,Force = mass.accelerationForce = mass.acceleration = kg.m/s= kg.m/s2 2 = kgm/s = kgm/s2 2 = Newton (N) = Newton (N)

SI units: ExamplesSI units: Examples• Length: metre (m) Length: metre (m) • Area: square metre (mArea: square metre (m22))• Volume: cubic metre (mVolume: cubic metre (m33))

• Volume rate of flow: cubic metres per second (mVolume rate of flow: cubic metres per second (m33/s)/s)• Volume rate of discharge: cubic metres per second (mVolume rate of discharge: cubic metres per second (m33/s)/s)• Flow rate: (mFlow rate: (m33/s)/s)• Discharge: (mDischarge: (m33/s)/s)

• Velocity: metre per second (m/s)Velocity: metre per second (m/s)• Acceleration: metre per square second (m/sAcceleration: metre per square second (m/s2 2 ))

SI units: ExamplesSI units: Examples• Mass: (kg)Mass: (kg)

• Mass density: kilogrammes per cubic metre (kg/mMass density: kilogrammes per cubic metre (kg/m33))

• Weight: Newton (N)Weight: Newton (N)• Force: Newton (N)Force: Newton (N)

• Pressure = Force/Area (N/mPressure = Force/Area (N/m22))

• Work, Energy = Force x Distance (Nm) = (J)Work, Energy = Force x Distance (Nm) = (J)

• Power = Work/time (J/s) = (W)Power = Work/time (J/s) = (W)

Properties of FluidsProperties of Fluids DensityDensity

Mass density Mass density is defined as the mass of the substance is defined as the mass of the substance per unit volume (mass/volume)per unit volume (mass/volume)

Units: kilograms per cubic meter (kg/mUnits: kilograms per cubic meter (kg/m33))water, 1000 kg/mwater, 1000 kg/m33 air, 1.23 kg/mair, 1.23 kg/m33

Specific weightSpecific weightSpecific weight w is defined as the weight per unit volume Specific weight w is defined as the weight per unit volume

w = w = gg

Units: Newtons per cubic meter (N/mUnits: Newtons per cubic meter (N/m33))

Properties of FluidsProperties of Fluids

Relative densityRelative density Relative densityRelative density (or) (or) specific gravityspecific gravity is the ratio of is the ratio of

density of a substance to density of waterdensity of a substance to density of water

= = substancesubstance / / waterwater

No units: (dimensionless)No units: (dimensionless)

Specific volumeSpecific volume Specific volume is defined as the reciprocal of mass density Specific volume is defined as the reciprocal of mass density

(m (m33/kg)/kg)

Properties of FluidsProperties of Fluids

ViscosityViscosity Coefficient of dynamic viscosityCoefficient of dynamic viscosity can be defined as the can be defined as the

shear force per unit area (or shear stress) required to drag shear force per unit area (or shear stress) required to drag one layer of fluid with unit velocity past another layer a unit one layer of fluid with unit velocity past another layer a unit distance away from it in the fluiddistance away from it in the fluid

= /(du/dy)du/dy)

Units: Newton seconds per square meter (Ns/mUnits: Newton seconds per square meter (Ns/m22) or (Nsm) or (Nsm-2-2))For water, 1.14x10For water, 1.14x10-3 -3 Ns/mNs/m22 or (kg/ms) or (kg/ms)For air, 1.78x10For air, 1.78x10-5-5 Ns/m Ns/m22 or (kg/ms) or (kg/ms)

Properties of FluidsProperties of Fluids

Kinematic viscosityKinematic viscosity is defined as the ratio of is defined as the ratio of dynamic viscosity to mass density dynamic viscosity to mass density

= = //

Units: square meters per second (mUnits: square meters per second (m22/s)/s)For water, 1.14x10For water, 1.14x10-6-6 m m22/s/sFor air, 1.46x10For air, 1.46x10-5-5 m m22/s/s

Surface tensionSurface tension

Surface tensionSurface tension A molecule (I) within the body of the liquid is attracted equally in all A molecule (I) within the body of the liquid is attracted equally in all

directions by the other molecules surrounding itdirections by the other molecules surrounding it

But at the surface between liquid and air, the upward and downward But at the surface between liquid and air, the upward and downward attractions are unbalanced, the surface molecules (S) being pulled attractions are unbalanced, the surface molecules (S) being pulled inward towards the bulk of the liquidinward towards the bulk of the liquid

This effect causes the liquid surface to behave as if it were an This effect causes the liquid surface to behave as if it were an elastic membrane under tensionelastic membrane under tension

Surface tensionSurface tension

Surface tension of a liquid is measured as the force acting across Surface tension of a liquid is measured as the force acting across the unit length of a line drawn in the surface (N/m)the unit length of a line drawn in the surface (N/m)

It acts in the plane of the surface, normal to any line in the surface, It acts in the plane of the surface, normal to any line in the surface, and is the same at all pointsand is the same at all points

Surface tensionSurface tension

Surface tension causes drops of liquid to tend to take a spherical Surface tension causes drops of liquid to tend to take a spherical shapeshape

Surface tensionSurface tension

Surface tension causes drops of liquid to tend to take a spherical Surface tension causes drops of liquid to tend to take a spherical shapeshape

Surface tensionSurface tension

Pressure force inside the droplet =Pressure force inside the droplet =

Surface tension force around the circumference = Surface tension force around the circumference =

Under equilibrium condition the two forces will be equal and opposite,Under equilibrium condition the two forces will be equal and opposite,

i.e.,i.e.,

2d4

p

dd4

p 2

d4p

d

Surface tensionSurface tension Surface tension causes the liquid to rise in a fine tube when its Surface tension causes the liquid to rise in a fine tube when its

lower end is inverted in a liquid which wets the tube lower end is inverted in a liquid which wets the tube see Figure see Figure (a)(a)

If the liquid does not wet the tube, it will be depressed in the fine If the liquid does not wet the tube, it will be depressed in the fine tube below the surface outside tube below the surface outside see Figure (b) see Figure (b)

Surface tensionSurface tension If If θθ = angle of contact between liquid and solid, = angle of contact between liquid and solid, = density of liquid, = density of liquid,

d = diameter of tube, h = height of liquid raised and d = diameter of tube, h = height of liquid raised and σσ = surface tension (N/m) = surface tension (N/m)

Upward pull due to surface tension = Upward pull due to surface tension = σσd cos d cos θθ

Weight of liquid raised = Weight of liquid raised = g(g(/4) d/4) d22 h h

So that So that σσd cos d cos θθ = = g(g(/4) d/4) d22 h h

Capillary action is the source of error in reading gauge glassesCapillary action is the source of error in reading gauge glasses

gdcos4h

Surface tensionSurface tension

Surface tensionSurface tension

CapillarityCapillarity

Rise or fall of a liquid in a capillary tube is caused by surface tension Rise or fall of a liquid in a capillary tube is caused by surface tension

Rise or fall depends on the relative magnitude of Rise or fall depends on the relative magnitude of cohesioncohesion of the liquid of the liquid and the and the adhesionadhesion of the liquid to the walls of the containing vessel of the liquid to the walls of the containing vessel

Cohesion – intermolecular attraction between molecules of the same liquidCohesion – intermolecular attraction between molecules of the same liquid Adhesion – attraction between molecules of a liquid and molecules of solid Adhesion – attraction between molecules of a liquid and molecules of solid

surface in contact with the liquidsurface in contact with the liquid

CapillarityCapillarity

Liquids rise in tubes if they wet (Liquids rise in tubes if they wet (adhesion > cohesionadhesion > cohesion) ) Liquids fall in tubes that do not wet (Liquids fall in tubes that do not wet (cohesion > adhesioncohesion > adhesion) )

adhesion > cohesionadhesion > cohesion cohesion > adhesioncohesion > adhesion

Vapour PressureVapour Pressure

A liquid in a closed container is subjected to partial A liquid in a closed container is subjected to partial vapour pressurevapour pressure due to the escaping molecules from the surfacedue to the escaping molecules from the surface

It reaches a stage of equilibrium when this pressure reaches It reaches a stage of equilibrium when this pressure reaches saturated saturated vapour pressurevapour pressure

Since this depends upon molecular activity, the Since this depends upon molecular activity, the vapour pressurevapour pressure of a of a fluid depends upon its temperature and increases with itfluid depends upon its temperature and increases with it

Boiling will occur when the Boiling will occur when the vapour pressurevapour pressure is equal to the pressure is equal to the pressure above the liquid. above the liquid.

Free surface

liquid

Vapour PressureVapour Pressure

CavitationCavitation

Under certain conditions, areas of low pressure can occur locally in a Under certain conditions, areas of low pressure can occur locally in a flowing fluid flowing fluid

If the pressure in such areas falls below the If the pressure in such areas falls below the vapour pressurevapour pressure, there , there will be local boiling and a cloud of vapour bubbles will formwill be local boiling and a cloud of vapour bubbles will form

This phenomenon is known as This phenomenon is known as cavitationcavitation and can cause serious and can cause serious problems since the flow of liquid can sweep the cloud of bubbles into problems since the flow of liquid can sweep the cloud of bubbles into an area of high pressure where the bubbles will collapse suddenlyan area of high pressure where the bubbles will collapse suddenly

Serious damage can result due to the very large force with which the Serious damage can result due to the very large force with which the liquid hits the surfaceliquid hits the surface

CavitationCavitation can affect the performance of pumps and turbines can affect the performance of pumps and turbines

Cavitation cause by collapse of vapour bubblesCavitation cause by collapse of vapour bubbles

Effect of Cavitation in PumpsEffect of Cavitation in Pumps

Effect of Cavitation in Turbines

Cavitation damage in Francis TurbineCavitation damage in Francis Turbine

Compressibility and Bulk Modulus Compressibility and Bulk Modulus

All fluids are compressible under the application of an All fluids are compressible under the application of an external force and when the force is removed they expand external force and when the force is removed they expand back to their original volume exhibiting the property that back to their original volume exhibiting the property that stress is proportional to volumetric strain stress is proportional to volumetric strain

Bulk modulus K = pressure change/volumetric strainBulk modulus K = pressure change/volumetric strain = -dp/(dV/V) = -dp/(dV/V)

For water, K = 2.05 x 10For water, K = 2.05 x 1099 N/m N/m22

For oil, K = 1.62 x 10For oil, K = 1.62 x 1099 N/m N/m22

The End