Fluid MechanicsChapter 1Fluids and their Properties

Faculty of Science, Technology, Engineering and Mathematics (FOSTEM)INTI International University

Fluid MechanicsBasic Text: Fluid Mechanics by J. F. Douglas, J. M. Gasiorek and J. A. Swaffield 6th edition, 5th edition, 4th edition, or 3rd edition.

Chapters 1, 2, 3, 4, 5, 6, 8, 9,10,14 + Handouts

References:J.F. Douglas and R.D. Mathews Solving Problems in Fluid Mechanics Volume 1 and Volume 2

Other text books on Fluid Mechanics

CVE3211 Fluid MechanicsBasic Text: Fluid Mechanics by J. F. Douglas, J. M. Gasiorek and J. A. Swaffield 6th edition, 5th edition, 4th edition, or 3rd edition.

Chapters 1, 2, 3, 4, 5, 6, 8, 9,10,14,15 + Handouts

References:J.F. Douglas and R.D. Mathews Solving Problems in Fluid Mechanics Volume 1 and Volume 2

Other text books on Fluid Mechanics

FluidsFluid Mechanics is that branch of applied mechanics that is concerned with the statics and dynamics of liquids and gases

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

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

FluidsIn everyday life, we recognize three states of matter: solid, liquid and gas.

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

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

FluidsFluids flow under the action of a force, deforming continuously for as long as the force is applied.

A fluid is unable to retain any unsupported shape; it flows under its own weight and takes the shape of any solid body with which it comes into contact.

Deformation is caused by shearing forces such as F (Fig. 1.1) and cause the material originally occupying the space ABCD to deform to ABCD.

Fig. 1.1 Deformation caused by shearing forces

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

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

Shear stress in a moving fluidThere can be no shear stress in a fluid at rest.Shear stresses are developed when the fluid is in motion.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 distorted.Considering successive layers parallel to the boundary (Fig 1.2), the velocity of the fluid will vary from layer to layer as y increases.

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

Shear stress in a moving fluidConsider a lubricating oil of viscosity undergoes steady shear between a fixed lower plate and an upper plate moving at a certain speed V

y moving plate y u = V

u = 0 u fixed plate Velocity profile

The fluid in contact with the upper plate sticks to the plate surface and moves with it at the same speed.The fluid in contact with the lower plate assumes the velocity of that plate, which is zero. This is referred to as the no-slip condition.

Shear stress in a moving fluid

Shear stress in a moving fluidFor a small angle, x = .yshear strain, = x/yrate of shear strain = (x/y).(1/t) = (x/t).(1/y) = u/ywhere u = velocity of the particle at E and y = distance from ED.

From experimental result, shear stress is proportional to the rate of shear strain, then shear stress, = constant . (u/y)where (u/y) = change of velocity with the distance y.

is known as Newtons law of viscositywhere is the dynamic viscosity of the fluid.Fig. 1.1 Deformation caused by shearing forces

Shear stress in a moving fluid y moving plate y u = V dy du u = 0 u fixed plate Velocity profile

Shear stress

where = dynamic viscosity and du/dy = velocity gradient

Newtons law of viscosity

Shear stress

where = dynamic viscosity (or) absolute viscosity du/dy = velocity gradient (or) rate of shear strain

The value of dynamic viscosity depends upon the fluid under consideration.

Differences between Solids and FluidsFor a solid, the strain is a function of the applied stress, providing that the elastic limit is not exceeded.

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

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 deformation disappears when the force is removed.

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

Newtonian FluidsFluids obeying the Newton's law of viscosity and for which has a constant value are called Newtonian fluids.

Newton's law of viscosity is given by Shear stress

where = dynamic viscosity (or) absolute viscosity of fluid du/dy = shear rate (or) rate of shear strain (or) velocity gradient

Non-Newtonian FluidsFluids which do not obey the Newtons law of viscosity are known as non-Newtonian fluids.

Bingham Plastic, for which the shear stress must reach a certain minimum value before flow commences. Thereafter, shear stress increases with the rate of shear (e.g. tooth paste, jellies, sewage sludge, etc.)

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

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

Newtonian and Non-Newtonian Fluids

Fig. 1.3: Variation of shear stress with velocity gradient

Liquids and GasesAlthough liquids and gases both share the common characteristics of fluids, they have many distinctive characteristics of their own.

A liquid is difficult to compress and, for many purposes, may be regarded as incompressible.A given mass of liquid occupies a fixed volume, and a free surface is formed.

A gas is comparatively easy to compress.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, does not form a free surface.liquidgasFree surface

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

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

Examples on SI unitsLength: metre (m) Area: square metre (m2)Volume: cubic metre (m3)

Volume rate of flow: cubic metres per second (m3/s)Volume rate of discharge: cubic metres per second (m3/s)Flow rate (or) Flow: (m3/s)Discharge: (m3/s)

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

Examples on SI unitsMass: (kg)

Mass density: kilogrammes per cubic metre (kg/m3)

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

Pressure = Force/Area (N/m2)

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

Power = Work/time (Nm/s) or (J/s) = (W)

Properties of FluidsDensity Mass density is defined as the mass of the substance per unit volume (mass/volume). Units: kilograms per cubic metre (kg/m3) water, 1000 kg/m3 air, 1.23 kg/m3

Specific weightSpecific weight w is defined as the weight per unit volume. w = gUnits: Newtons per cubic metre (N/m3)

Properties of FluidsRelative density Relative density (or) specific gravity is the ratio of density of a substance to density of water.

Specific gravity = substance /water No units: (dimensionless)

Specific volume Specific volume is defined as the reciprocal of mass density (m3/kg)

Properties of FluidsViscosity

Viscosity may be defined as the property of a fluid which determines its resistance to shearing stresses.

It is a measure of the internal fluid friction which causes resistance to flow.

Viscosity of fluids is due to cohesion and interaction between particles.

Viscosity may also be defined as the shear stress required to produce unit rate of shear strain.

Properties of FluidsViscosity Coefficient of dynamic viscosity (or absolute viscosity) can be defined as the shear force per unit area (or shear stress) required to drag one layer of fluid with unit velocity past another layer a unit distance away from it in the fluid.

Dynamic viscosity

Units: Newton seconds per square meter (Ns/m2) or (Nsm-2) For water, 1.14x10-3 Ns/m2 or (kg/ms) For air, 1.78x10-5 Ns/m2 or (kg/ms)

Properties of FluidsKinematic ViscosityKinematic viscosity is defined as the ratio of dynamic viscosity to mass density. Kinematic viscosity

Units: square meters per second (m2/s) For water, 1.14x10-6 m2/s For air, 1.46x10-5 m2/s

Surface tensionA molecule (I) within the body of the liquid is attracted equally in all directions by the other molecules surrounding it.

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

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

Surface tensionSurface tension is caused by the force of cohesion at the free surface.

Surface tension of a liquid is measured as the force acting across 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, and is the same at all points.

Surface tensionSurface tension causes drops of liquid to tend to take a spherical shape.

Surface tension

Pressure force inside the droplet = pressure x area =

Surface tension force around the circumference =

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

i.e., pressure,

Surface tension

CapillaritySurface tension causes the liquid to rise in a fine tube when its lower end is inverted in a liquid which wets the tube see Figure (a)

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

CapillarityIf = angle of contact between liquid and solid, = density of liquid, = surface tension (N/m), d = diameter of tube, d = perimeter of tube, H = height of liquid raised Upward pull due to surface tension = (d) cos Weight of liquid raised = g[(/4)d2] Hso that, d cos = g(/4) d2 HCapillary rise,

Capillary action is the source of error in reading gauge glasses

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

Rise or fall depends on the relative magnitude of cohesion of the liquid and the adhesion of the liquid to the walls of the containing vessel.

Cohesion: intermolecular attraction between molecules of the same liquidAdhesion: attraction between molecules of a liquid and molecules of solid surface in contact with the liquid

CapillarityLiquids rise in tubes if they wet (adhesion > cohesion)

Liquids fall in tubes that do not wet (cohesion > adhesion) adhesion > cohesioncohesion > adhesion

Vapour PressureA liquid in a closed container is subjected to partial vapour pressure due to the escaping molecules from the surface.

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

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

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

Free surfaceClosed ContainerLiquid

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

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

This phenomenon is known as cavitation and can cause serious problems since the flow of liquid can sweep the cloud of bubbles on into an area of higher pressure where the bubbles will collapse suddenly.

If this occur in contact with a solid surface, very serious damage can result due to the very large force with which the liquid hits the surface.

CavitationCavitation can affect the performance of pumps and turbines and the impact of collapsing bubbles can cause local erosion of metal surfaces.Vapour bubbles in a propeller

CavitationCavitation can affect the performance of pumps and turbines and the impact of collapsing bubbles can cause local erosion of metal surfaces.Cavitation damage in a PumpCavitation damage in a Turbine

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

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

minus sign indicates that the volume decreases as pressure increasesFor water, K = 2.05 x 109 N/m2For oil, K = 1.62 x 109 N/m2

The End