Fluid Mechanics Introduction - Properties of Fluid - Fluid Mechanics

Fluid Mechanics

 INTRODUCTION

• Very strong intermolecular attractive forces exist in solids which give them the property of rigidity.

These forces are weaker in liquids and very weak in gases. A solid has volume and shape, a liquid

has volume but no shape and a gas has neither.

• A fluid is a substance which deforms continuously under the action of shear stress, regardless of its

magnitude. Hence, both liquid and gas are fluids.

• In liquids the molecules are so closely spaced that strong molecular cohesive force compels the

fluid to behave as a continuous mass.

• Density is mas per unit volume.

r = =

• Specific gravity =

 

• Specific weight = dg

The specific weight of water under normal condition = 9.81 × 10

3 N/m3

.

• Specific volume: It is the volume occupied by a unit mass of fluid. It is commonly applied to gases.

Specific volume = .

• Viscosity is the property by virtue of which a fluid offers resistance to the movement of one layer

over another adjacent layer under the influence of a shear force.

Newton’s law of viscosity is

t = m

 

where t = shear stress

m = coefficient of viscosity or absolute viscosity

= rate of shear strain and is known as velocity gradient.

Thus, for a given shear stress acting on a fluid element, the rate at which the fluid deforms is

inversely proportional to the viscosity.

• A fluid which obeys Newton’s law of viscosity is known as a Newtonian fluid. Air, water light oils

and gasolines are Newtonian fluids.

• Non-Newtonian fluids are those in which shear stress is not linearly dependent upon the velocity

gradient. Examples of such fluids are: human blood, lubricating oils, printers ink, molten rubber,

and sewage sludge. They have the relation of the form

t = A + B.

If n > 1, Dilatant = ea

: printing ink, butter, quick sand

n = 1, Bingham plastic = ex

: Sewage sludge

n < 1, Pseudoplastic = eu

: Paper pulp, suspension paints, polymeric solutions

• Some fluids are time dependent and some are time independent. In time dependent fluids the rate of

deformation and the viscosity depend upon both the shear stress and the duration of its application.

• Ideal fluid is non-viscous (frictionless) and incompressible (inelastic). It is a creation of

mathematicians, just to simplify the analysis.

• The dimension of dynamic viscocity is ML

–1 T

–1

.

• The ratio of the dynamic viscosity and mass density is called the kinematic viscosity. It can be

defined by only length and time dimension (L

2 T

–1

). Its unit in S.I. system is stoke which is equal to

1 = 10

–4 m2

/s.

• With increase in temperature, the intermolecular cohesive force decreases rapidly, resulting in the

decrease of viscocity.

Kinematic viscocity of water

=

is the equation given by Poiseuille.

• In case of gases, the viscocity is mainly due to transfer of molecular momentum in the transverse

direction brought about by the molecular agitation. As the molecular agitation increases with the

rise of temperature, the viscocity of the gases also increases with temperature. Thus, the viscocity

of fluid is due to:

1. intermolecular cohesion, and

2. transfer of molecular momentum.

Surface Tension

• Surface tension is a force which exists on the surface of a liquid when it is in contact with another

fluid or a solid boundary. It acts normal to a line of unit length drawn imaginarily on the surface. It

is a line force. Its unit in SI system is N/m. It depends directly upon intermolecular cohesion.

Capillary Action

• Most of the liquids completely wet the surface of solids because the molecules of a solid surface

attract liquid molecules with a greater force than that exists between the liquid molecules. Mercury

is exception to it.

• If the adhesive force is greater than that of cohesive force, liquid tends to spread out and wets the

surface. But in case of mercury cohesive force is more and hence it does not wet the surface of

solids. Hence, when a glass tube is dipped in water, the level increases in the tube while if it is

dipped in mercury, the level of mercury in the tube is lower.

• The phenomenon of rise or fall of liquid level in the tube is said to be due to capillary action.

• If q is the angle for contact between the liquid and solid surfaces, water in the glass tube will

continue to rise until the vertical component of the surface tension is equal to the weight of the

water column.

• For pure water and clean glass q is zero. But for ordinary water, Gibson found q = 25°.32¢. For

mercury q = 128°52¢.

Vapour Pressure

• Liquid molecules escaping from the free surface to air is known as vaporisation.

• After some time when air contains enough liquid molecules, it starts exerting pressure on liquid

molecules and forces them to rejoin the liquid surface. When equilibrium is established,

vaporisation stops. The pressure exerted by saturated air or liquid is called vapour pressure.

• Liquid starts boiling when the pressure on it is slightly below vapour pressure. Hence, boiling can

be achieved by either raising the temperature or by lowering the pressure of overlaying air below

the vapour pressure.

• Vapour pressure increases with temperature. For water, vapour pressure at 0°C is 0.063 m while its

value is 10.790 m when temperature is 100°C.

• At 20°C water has vapour pressure of 2.345 × 10

3 N/m2

, i.e., 2.345 × 10

3 Pascal i.e., 0.239 m

while mercury has 0.160 N/m2 only.

Incompressible and Compressible Fluids

• The compressibility is the measure of change of volume when a substance is subjected to pressure.

• The reciprocal of coefficient of compressibility is known as bulk modulus of elasticity.

• The compressibility of liquid is so small that, to simplify analysis it is many times assumed

incompressible.

• The values of bulk modulus Ev

for water at standard atmospheric conditions is Ev = 2.11 10

3 N/m2

.

• The ratio of the velocity of flow V to the velocity of sound in the fluid medium is known as Mach

number and denoted as M.

This is a measure of compressibility. If:

M < 1, compressibility effect is negligible

M > 1, compressibility of fluid is to be considered

• The flow is,

subsonic if M < 1

supersonic if M > 1

hypersonic if M > 5.

Fluid Statics

• In static fluid velocity gradient = 0

• Hence, viscosity of a fluid has no effect.

• If the distance h is measured from the free surface, pressure p is given by

p = g h where g = is unit weight of fluid.

• The atmospheric pressure p at a height h from sea level is

p = po – gah where po = pressure at mean sea level.

and ga = specific weight of air.

• The atmospheric pressure at sea level is 760 mm of mercury.

• Absolute pressure is the pressure measured above complete vacuum (i.e., the absolute zero).

• Gauge pressure is the difference between its absolute value and the local atmospheric pressure.

Thus

Gauge pressure = Absolute pressure – Local atmospheric pressure.

• If gauge pressure is negative, then vacuum or negative pressure

= Local atmospheric pressure – Absolute pressure.

• Mechanical pressure gauges like Bourdon gauge, measure gauge pressure.

• Local atmospheric is measured by a mercury barometer or by an aneroid barometer.

• The height of liquid that rises in a piezometer tube above the point is absolute pressure at that point.

The piezometers when employed for pressure measurements are called manometers. Types of

manometers are:

1. Simple manometer

2. Differential manometer

3. Micro-manometer.

• Simple manometer can only be used for measuring small and moderate pressures because as the

pressure gets higher, the length of the piezometer becomes larger.

• Simple manometer may be used of measuring negative pressure also by providing U-shaped turn to

the tube.

• Differential manometer is used to measure the difference in pressures at two points.

• Use of mercury is advantageous where the pressure difference is large.

• The micro-manometers are used for measuring small difference of pressure.

• Inclined manometer is used for precise measurement of small pressure in low velocity gas flow. In

this amplification of h is l/h where l is inclined length measured. Thus, amplification = = , if

q is the inclination of inclined manometer.

• The pressure on horizontal surface h below the free surface is g h and on vertical surface it varies

from zero at free surface to maximum of g h at bottom of vertical surface, linearly.

• When a body is submerged in liquid, it is subjected to vertical force by liquid and this is known as

buoyancy. The Archimedes’ principle states that a submerged body is subjected to a buoyancy

equal to the weight of the fluid displaced by it.

• The point through which the buoyancy force acts is known as the centre of buoyancy (CB) and is at

the centre of gravity of the displaced fluid.

• In gravity dam, the overturning moment due to hydrostatic pressure is resisted by stabilizing moment

by weight of the dam and weight of the wedge-shaped water body supported by upstream side

face.

• In arch dams the overturning moment by hydrostatic pressure is resisted by end thrust.

• Pipes subjected to internal pressure p are subjected to hoop tension s = = .

• If B is the centre of buoyancy and G is the centre of gravity of a submarine, the submerged body is

in:

1. Neutral equilibrium when B and G coincide and B lies above G.

2. Stable equilibrium, if slight rotational displacement generates the forces which oppose the

change of position and tend to bring the body to its original position.

3. Unstable equilibrium if G lies above B.

• The intersection of line of action of buoyancy force with the axis of symmetry of floating body is

known as metacentre.

• If meracentre lies between centre of gravity G and centre of buoyancy B tilting of vessel progresses.

• The distance between centre f gravity G and metacentre M is called metacentric height. If

metacentric height is positive (M above G) it is the condition of stable equilibrium.

• The initial metacentric height for a passenger steamer varies from 300 to 600 mm while for naval

vessel it varies from 900 to 1200 mm.

• For a floating cylinder metacentre is zero.

• A vessel with sides diverging upwards has better stability.