FLUID

Characteristics of Fluid Flow (1)

Steady flow (lamina flow, streamline flow)The fluid velocity (both magnitude and direction) at any given point is constant in time

The flow pattern does not change with time

Non-steady flow (turbulent flow)Velocities vary irregularly with time

e.g. rapids, waterfall

Rotational and irrotational flowThe element of fluid at each point has a net angular velocity about that pointOtherwise it is irrotationalExample: whirlpools

Compressible and incompressible fluidLiquids are usually considered as incompressibleGas are usually considered as highly compressible

Characteristics of Fluid Flow (2)

Viscous and non-viscous fluidViscosity in fluid motion is the analog of friction in the motion of solids

It introduces tangential forces between layers of fluid in relative motion and results in dissipation of mechanical energy

Characteristics of Fluid Flow (3)

Streamline

A streamline is a curve whose tangent at any point is along the velocity of the fluid particle at that point It is parallel to the velocity of the fluid particles at every point No two streamlines can cross one another In steady flow the pattern of streamlines in a flow is stationary with time

Change of speed of flow with cross-sectional area

If the same mass of fluid is to pass through every section at any time, the fluid speed must be higher in the narrower region

Therefore, within a constriction the streamlines must get closer together

Kinematics (1)

Mass of fluid flowing past area Aa =avatA

a

Mass of the fluid flowing past area Ab = bv

btAb

In a steady flow, the total mass in the bundle must be the same

avaAa t= bvbAb ti.e. avaAa = bvbAb or vA = constant

The above equation is called the continuity equation For incompressible fluids vA = constant

Kinematics (2)

Further reading

Static liquid pressure

The pressure at a point within a liquid acts in all directions

The pressure depends on the density of the liquid and the depth below the surfaceP = gh

Further reading

Bernoulli’s equation

Bernoulli’s equationThis states that for an incompressible, non-viscous fluid undergoing steady lamina flow, the pressure plus the kinetic energy per unit volume plus the potential energy per unit volume is constant at all points on a streamline

i.e. constant 221 ghvp

Derivation of Bernoulli’s equation (1)

The pressure is the same at all points on the same horizontal level in a fluid at rest

In a flowing fluid, a decrease of pressure accompanies an increase of velocity

In a small time interval t, fluid XY has moved to a position X’Y’

At X, work done on the fluid XY by the pushing pressure= force distance moved

= force velocity time

= p1A1 v1 t

Derivation of Bernoulli’s equation (2)

figure

At Y, work done by the fluid XY emerging from the tube against the pressure= p2A2 v2 t

Net work done on the fluidW = (p1A1 v1 - p2A2 v2)t

For incompressible fluid, A1v1= A2v2

W = (p1 - p2)A1 v1 t

Derivation of Bernoulli’s equation (3)

figure

Gain of p.e. when XY moves to X’Y’= p.e. of X’Y’ - p.e. of XY

= p.e. of X’Y + p.e. of YY’ - p.e. of XX’ - p.e. of X’Y

= p.e. of YY’ - p.e. of XX’

= (A2 v2 t)gh2 - (A1 v1 t)gh1

= A1 v1 tg(h2 - h1)

Derivation of Bernoulli’s equation (4)

figure

Gain of k.e. when XY moves to X’Y’= k.e. of YY’ - k.e. of XX’

=

=

Derivation of Bernoulli’s equation (5)

1

2

1

22 2 22

1 1 12A v t v A v t v

1

2 1 1 22

12A v t v v

figure

For non-viscous fluidnet work done on fluid = gain of p.e. + gain of

k.e.

(p1 - p2)A1 v1 t = A1 v1 tg(h2 - h1) +

Derivation of Bernoulli’s equation (6)

1

2 1 1 22

12A v t v v

p p g h h v v1 2 2 1 22

121

2 ( ) ( )

figure

or

Derivation of Bernoulli’s equation (7)

p h g1

2v p h g

1

2v1 1 1

22 2 2

2

p h g1

2v constant2

figure

Assumptions made in deriving the equationNegligible viscous force

The flow is steady

The fluid is incompressible

There is no source of energy

The pressure and velocity are uniform over any cross-section of the tube

Derivation of Bernoulli’s equation (8)

Further reading

Applications of Bernoulli principle (1)

Jets and nozzlesBernoulli’s equation suggests that for fluid flow where the potential energy change hg is very small or zero, as in a horizontal pipe, the pressure falls when the velocity rises

The velocity increases at a constriction and this creates a pressure drop. The following devices make use of this effect in their action

Bunsen burnerThe coal gas is made to pass a constriction before entering the burner The decrease in cross-sectional area causes a sudden increase in flow speed The reduction in pressure causes air to be sucked in from the air hole The coal gas is well mixed with air before leaving the barrel and this enables complete combustion

Applications of Bernoulli principle (2)

Carburettor of a car engine The air first flows through a filter which removes dust and particles It then enters a narrow region where the flow velocity increases The reduced pressure sucks the fuel vapour from the fuel reservoir, and so the proper air-fuel mixture is produced for the internal combustion engine

Applications of Bernoulli principle (3)

Filter pumpThe velocity of the running water increases at the constriction

The surrounding air is dragged along by the water jet and this causes a drop in pressure

Air is then sucked in from the vessel to be evacuated

Applications of Bernoulli principle (4)

Spinning ball

If a tennis ball is `cut’ it spins as it travels through the air and experiences a sideways force which causes it to curve in flight This is due to air being dragged round by the spinning ball, thereby increasing the air flow on one side and decreasing it on the other A pressure difference is thus created

Further readingfigure

Aerofoil

A device which is shaped so that the relative motion between it and a fluid produces a force perpendicular to the flowFluid flows faster over the top surface than over the bottom. It follows that the pressure underneath is increased and that above reduced. A resultant upwards force is thus created, normal to the flow e.g. aircraft wings, turbine blades, sails of a yacht

Pitot tube (1)

a device for measuring flow velocity and in essence is a manometer with one limb parallel to the flow and open to the oncoming fluid

The pressure within a flowing fluid is measured at two points, A and B. At A, the fluid is flowing freely with velocity va. At B where the Pitot tube is pl

aced, the flow has been stopped

By Bernoulli’s equation:

Pitot tube (2)

P v Pa a b 1

202

P gh v P gho a a o b 1

22

where P0 = atmospheric pressure

Note:• In real cases, v varies across the diameter of the pipe

carrying the fluid (because of the viscosity) but if the open end of the Pitot tube is offset from the axis by 0.7 radius of the pipe, then v is the average flow velocity

• The total pressure can be considered as the sum of two components: the static and dynamic pressures

Pitot tube (3)

v 2g h ha b a

A moving fluid exerts its total pressure in the direction of flow. In directions at right angles to the flow, the fluid exerts its static pressure only figures

P p gh vT ( ) 12

2

Total Static Dynamic

pressure pressure pressure

Pitot tube (4)

Further reading: paragraph of ‘Pitot Static System’ near the bottom of the page

Venturi meter (1)

This consists of a horizontal tube with a constriction. Two vertical tubes serving as manometers are placed perpendicular to the direction of flow, one in the normal part and the other in the constriction

In steady flow the liquid level in the manometer connected to the wider part of the tube is higher than that in the narrower part

figure

Venturi meter (2)

From Bernoulli’s principle2

222

11 2

1

2

1vPvP (h1 = h2)

)1)((2

1

2

1

2

1 22

1

222

12

12

221 v

vvvvPP

For an incompressible fluid,

A1v1 = A2v2

2

1

1

2

A

A

v

v

Hence

Venturi meter (3)

]1)[(2

1 2

2

12121 A

AvPP

v1 can be deduced

Streamline

PQ

vPvQ

Change of speed in a constriction

Streamlines are closer when the fluid flows faster

Derivation of Bernoulli’s equation

X

Y

X’

Y’v1

p1A1

p2A2

v1t

v2t

Area A1

Area A2

v2

h1

h2

Bunsen burner

Carburettor

fuel

air

filter

to engine cylinder

Filter pump

Spinning ball

Aerofoil

Pitot tube (1)

Pitot tube (2)

Pitot is here

Pitot tube: fluid velocity measurement (1)

Flow of air

Stagnant air,

higher pressure

inside tube

Fast moving air,

lower pressure

inside chamber

Total tube

Static tube

Static pressure

holes

P1= total pressure

P2= static pressure

P2 – P1 = ½(v2)

Pitot tube: fluid velocity measurement (2)

Ventri meter (1)

Venturi meter (2)

Venturi meter (3)

A1A2

v1 v2

Density of

liquid =