FLUID MECHANICS

Dr. P. Jagadeesh

PROPERTIES OF FLUIDSUnit - IA

FLUID: Fluid may be defined as a substance which is capable of flowing. It hasno definite shape on its own, but conforms to the shape of the containing vessel. Further even a small amount of shear force exerted on a fluid will cause it to undergo deformation which continues as long as the force continues to be applied.

LIQUID: A liquid is a fluid which posses a definite volume, which varies only slightly with temperature and pressure. Since under ordinary conditions liquids are difficult to compress, they may be for all practical purposes regarded as incompressible.

GAS: A gas is a fluid, which is compressible and possesses no definite volume but it always expands until its volume is equal to that of the container. Even a slight change in the temperature of a gas has significant effect on its volume and pressure. However, if conditions are such that a gas undergoes a negligible change in its volume, it may be regarded as incompressible.

VAPOUR: A vapour is a gas whose temperature and pressure are such that it is very near the liquid state. Ex: Steam

IDEAL FLUIDS: Ideal fluids are those fluids which have no viscosity and surface tension and they are incompressible. As such for ideal fluids no resistance is encountered as the fluid moves. However, in nature the ideal fluids does not exist and therefore, these are only imaginary fluids. Fluids which have low viscosity such as Air, Water etc. may treated as ideal fluids.REAL/PRACTICAL FLUIDS: These fluids possess the properties such as viscosity, surface tension and compressibility and therefore a certain amount of resistance is always offered by these fluids when they are set in motion.DEVELOPMENT OF FLUID MECHANICSUNITS OF MEASUREMENTUnits may be defined as those standards in terms of which the various physical quantities like length, mass, time, force, area, volume, velocity, acceleration etc., are measured. The system of units used in mechanics are based upon Newton’s second law of motion, which states that force equals mass times acceleration or F=mxa, where F is the force, m is the mass and a acceleration. There are in general four systems of unit, two in metric (C.G.S or M.K.S.) system and two in English (F.P.S.) system.

The difference between the absolute or gravitational system is that in the former the standard is the unit of mass. The unit of force is then derived.In the gravitational system the standard is the unit of force and the unit of mass is derived by Newton’s law.

Quantity Metric Units English UnitsGravitational

Absolute Gravitational

Absolute

Length Metre (m) Metre (m) Foot (ft) Foot (ft)Time Seconds

(sec)Seconds(sec)

Seconds(sec)

Seconds(sec)

Mass Metric slug (msl)

Gram [gm(mass)]

Slug (sl) Pound [lb(mas)]

Force Kilogram [kg(f)]

Dyne Pound [lb(f)] Poundal(pdl)

Temperature C C F F

Table 1.1 Various units of measurement of some fundamental quantities

Quantity Unit Symbol

Length metre mMass kilogram kgTime second sElectric current ampere AThermodynamicTemperature

kelvin k

Luminous Intensity candela cdPlane angle radian radSolid angle steradian sr

Table 1.2 Various units of measurement of some fundamental quantities in SI system

MASS DENSITY (): Mass density (or specific mass) of a fluid is the mass which it possesses per unit volume. Metric gravitation system – metric slug/m3, SI-kg/m3

SPECIFIC WEIGHT (): Specific weight of a fluid is the weight it possesses per unit volume. SI- N/m3 , Metric gravitation system – kg(f)/m3

SPECIFIC VOLUME: Specific volume of a fluid is the volume of the fluid per unit weight. It is the reciprocal of specific weight.SPECIFIC GRAVITY (G): Specific gravity is the ratio of specific weight of a fluid to the specific weight of a standard fluid.

VISCOSITY: Viscosity is that property of a fluid by virtue of which it offersresistance to the movement of one layer of fluid over an adjacent layer. It isprimarily due to cohesion and molecular momentum exchange between fluidlayers, and as flow occurs, these effect appears as shearing stresses betweenthe moving layers of fluid. SI: N-s/m2; or kg/m-s.

g

. .. . tan

Sp wt of liquidGSp wt of s dard liquid

cos

AVFY

F V dvA Y dy

dv dy

Kinematic vi ity

NOTE: Common fluids such as air, water, glycerine, kerosene etc., follows Newtons Law of Viscosity.Units SI: m2/s; C.G.S= cm2/sCentistoke =1/100 stoke

Fig. Fluid motion between two parallel plates

Fig. Variation of shear stress with velocity gradient

Note:

Thixotropic liquid = Printers inkThe fluids with which engineers most often have to deal are Newtonian, that is,their viscosity is not dependent on the rate of angular deformation, and the term

‘fluid-mechanics’ generally refers only to Newtonian fluids.

The study of non-Newtonian fluids is however termed as ‘rheology’.

VAPOUR PRESSURE

When the liquid is confined in a closed vessel, the ejected vapour molecules getaccumulated in the space between the free liquid surface and the top of the vessel.

This accumulated vapour of the liquid exerts a partial pressure on the liquid surfaceWhich is known vapour pressure of the liquid. As the molecular activity increaseswith temperature, vapour pressure of the liquid also increases with temperature.

Ex: Mercury has a very low vapour pressure and hence it is an excellent fluid tobe used in barometer. On the other hand volatile liquids like benzene etc. have

high very vapour pressure.

Compressibility and Elasticity

All fluid may be compressed by the application of external force, and when the External force is removed the compressed volumes of fluids expand to their Original volumes. Thus fluid also possess elastic characteristics like elasticsolids. Compressibility of a fluid is quantitatively expressed as inverse of the

Bulk modulus of elasticity K of the fluid, which is defined as:

Note: The bulk modulus of elasticity of a fluid is not constant, but it increase withincrease in pressure. For example, the bulk modulus of water roughly doublesas the pressure is raised from 1 atmosphere to 3500 atmosphere.

Stress Change in pressure dpk dvStrain Change in volumeVOriginal volume

SURAFCE TENSION & CAPAILLARITY

Due to molecular attraction, liquid possess certain properties such as cohesion and adhesion.Cohesion means inter-molecular attraction between molecules of the same liquid. This means it is a tendency of the liquid to remain as one assemblage of particles.Adhesion means attraction between the molecules of a liquid and the molecules of a solid boundary surface in contact with the liquid.The property of cohesion enables a liquid to resist tensile stress, while adhesion enables it to stick to another body. Surface tension due to cohesion between liquid particles at the surface, whereas capillarity is due to both cohesion and adhesion.

SURFACE TENSIONi. Pressure intensity inside a dropletConsider a spherical droplet of radius ‘r’ having internal pressure intensity p in Excess of the outside pressure intensity. If the droplet is cut into two halves, then The forces acting on one half will be those due to pressure intensity p on the projected area (r2) and the tensile force due to surface tension acting around the Circumference (2 r). These two forces will be equal and opposite for equilibriumand hence we have,

2 2

2

p r r

pr

Fig. Pressure inside a droplet

ii. Pressure intensity inside a soap bubbleAs spherical soap bubble has two surfaces in contact with air, one inside and the other outside, each one of which contributes the same amount of tensile force due to surface tension. As such on a hemispherical section of a soap bubble of radius r the tensile force due to surface tension is equal to 2(2r). However, the pressure force acting on the hemispherical section of the soap bubble is same as in the case of a droplet and it is equal to p(r2). Thus equating these two forces of equilibrium, we have

2 2 2

4

p r r

pr

Fig. Pressure inside a soap bubble

iii. Pressure intensity inside a liquid jet

Consider a jet of liquid of radius r, length l and having internal pressure intensity p in excess of the outside pressure intensity. If the jet is cut into two halves, thenthe forces acting on one half will be those due to pressure intensity p on theprojected area (2l) and the tensile force due to surface tension acting along

the two sides (2l). These two forces will be equal and opposite for equilibrium and hence we have,

2 2p rl l

pr

Fig. Pressure inside a liquid jet

CAPILLARITY

A phenomenon of rise or fall of liquid surface relative to the adjacent general levelof liquid is known as capillarity. Accordingly the rise of liquid surface is designated as capillary rise and the lowering of liquid surface as capillary depression, and it is expressed in terms of m or mm of liquid in SI units.

Fig. Capillarity in circular glass tubes

The capillary rise (or depression) can be determined by considering theconditions of equilibrium in a circular tube of small diameter inserted in a liquid.It is supported that the level of liquid has risen (or fallen) by h above (or below)The general liquid surface when a tube of radius r is inserted in the liquid. Forthe equilibrium of vertical forces acting on the mass of liquid lying above (orbelow) the general liquid level, the weight of liquid column h (or the totalpressure in the case of capillary depression) must be balanced by the force, atsurface of the liquid, due to surface tension . Thus equating these two forceswe have,

where is the specific weight of water, s is specific gravity of liquid, and is the contact angle between the liquid and the tube.

2 2 cos2 cos

0 & 12

s r h r

hs r

For water s

hr

If a tube of radius r is inserted in mercury (sp. gr. s1) above which a liquid of sp. gr. S2 lies then by considering the conditions of equilibrium it can be shownthat the capillary depression h is given by,

1 2

2 coshr s s

Further if two vertical parallel plates t distance apart and each of width l are held partially immersed in a liquid of surface tension and sp. gr. s, then the capillary rise (or depression) h may be determined by equating the weight of the liquid column h (shlt) to the force due to surface tension (2lcos). Thus we have, 2 cos

2 coss hlt l

hs t

NOTE: Capillary effect is true only in the case of small diameters (r<2.5mm). However, for tubes of diameter 6mm or more the capillary rise (or depression) is negligible.