Ice cube in a glass of water

water

ice

water

icesub

mggmV

After the piece of ice melts:

Water level, h?

AVVh subwater

0subV subwater

ice

water

waterwater VmmV

Barge with steel beams:

water

steelsub

mV

steel

steelsteeldisp

mVV

Steady flow in a river.

Velocity in each point is shown by a vector with the length proportional to the velocity. Velocity gets higher, where the river gets narrower.

Flow represented by streamlines, that are everywhere tangent to flow direction. Higher density of the streamlines corresponds to higher flow velocity.

In a steady flow there are no variations in velocity and pattern of flow in time. Nevertheless, the actual fluid elements flowing past any particular point at different times are always different. The fluid elements also get accelerated and decelerated as they move along the streamlines.

Motion of fluids obeys the standard laws of mechanics.

Newton’s second law: amF

Becomes Navier-Stokes equation:

VPVVtV

/)(

Newton’s second law is actually a complicated differential equation! 2

2

dtrdmF

Any way to make our life easier?!

Let’s try to use the laws of conservation!!

Motion of fluids obeys the standard laws of mechanics.

Conservation of mass: constm

Conservation of momentum: constvmvm 21

Conservation of energy:

constmghmvPEKE 2

2

Using the laws of conservation means doing appropriate bookkeeping and doing algebra instead of solving differential equations!

Flow tube - A small tubelike region bounded on its sides by a continuous set of streamlinesand on its ends by small areas at right angles to the streamlines.

Cross-section areas on the left and right ends are:A1 and A2.Densities and velocities are:1, 2 and v1, v2

Steady flow

Mass of fluid entering the tube from the left over the time interval t

tvAVm 11111 By mass conservation, over the time interval t, the same mass is exiting the tube from the right

tvAm 222

Therefore tvAtvA 222111

constvA everywhere along a flow tube

If the fluid is incompressible and its density, , is constant, we have

constvA

tvx 11 tvAV 111

222111 vAvA

Steady flow

constvA Does it work for traffic?

Once you pass the spot of accident there are more lanes available (larger A) and the traffic speeds up (higher v).What is the matter?

Traffic is highly compressible.You have got to use

constvA

How does the total energy of a small fluid element change, as it moves inside the flow tube from cross-section 1 to cross-section 2?

Kinetic energy: )(21 2

122 vvmKE

Potential energy: )( 12 hhmgPE

How does this change in the total energy become possible?

There are external forces originating from pressure of the liquid outside the tube, which do work on the fluid element!

The total energy balance 21 WWPEKE

Positive work as it enters from the left 111111 xAPxFW

Negative work as it exits from the right 222222 xAPxFW

2221111221

22 )()(

21 xAPxAPhhmgvvm

The total energy balance

2221111221

22 )()(

21 xAPxAPhhmgvvm

Incompressible fluids – constant density and volume 2211 xAxAV

22221

211 2

121 ghvPghvP

constghvP 2

21

Bernoulli’s equation

V

constghvP 2

21Bernoulli’s equation

Inertial jet

The fire truck pump is generating a pressure P.

What is the maximal velocity at the nozzle?

What is the maximal height the jet can reach?

The venturi flowmeter is a practical instrument which makes use of the Bernoulli effect and a manometer pressure gauge.

Venturi Flowmeter