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Momentum Conservation. Newton’s Second Law Force = Mass * Acceleration Alternate method: From Reynold’s theorem Fluid Flow Force = Momentum flux + Momentum Accumulation rate. Flux. Accumulation Rate. 0. Example. Straight Pipe. Steady State. A 2 V 2. A 1 V 1. Assumption: No friction.

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momentum conservation
Momentum Conservation
  • Newton’s Second Law
    • Force = Mass * Acceleration
    • Alternate method: From Reynold’s theorem
  • Fluid Flow
    • Force = Momentum flux + Momentum Accumulation rate

Flux

Accumulation Rate

example

0

Example
  • Straight Pipe
  • Steady State

A2

V2

A1

V1

  • Assumption: No friction
example1

1

2

Example
  • Flow in a straight pipe. Realistic case
example2

0

r1=1g/cm3

d1=8cm

  • Steady State

V1=5m/s

d2=5cm

Example
  • Problem: 5.32

d2,V2

d1,V1

1

2

h=58 cm

example3

y

x

Weight of Fluid

Example
  • Example 1 (bent pipe), page 47

r1,V1

By

Area=A1

Area=A2=A1

Q

Bx

r2,V2

example4

out

in

in

out

Example
  • Steady State
    • d/dt =0
  • From Eqn of Conservation of Mass
example5
Example

V1

By

Area=A1

Area=A2=A1

Q

Bx

V2

example6
Example
  • Pipe with U turn

P1,V1

F

P2,V2

Use Gage Pressure!

In case of gas, use absolute pressure to calculate density

example7
Example

N

  • “L” bend

P1

E

F

Assume the force by the pipe on the fluid is in the positive direction

P2

What will the force be, if the flow is reversed (a) in a straight pipe? (b) in a L bend?

example8

Conservation of Mass

X

Y

Example

Pbm. 5.13 & 5.19

V2

r1

P2

r2

Vw

P1

V1

Stationary CV

example9

X

Y

Example

Pressure difference?

Pbm. 5.19

V1=-3 m/s

V2=0 m/s

Vw=Velocity of Sound in water

V2

r1

r2

Vw

P2

P1

V1

Stationary CV

~ 41 atm

example10

y

x

Example

V2=?

Pbm. 5.1,5,3

V1=6 m/s

0.6 m

1.2 m

momentum conservation1
Momentum Conservation
  • Angular Momentum
  • In a moving system
    • Torque = Angular Momentum flux + Angular Momentum Accumulation rate

Flux

Accumulation Rate

example11

Q

Example
  • Example 4 in book
  • Find the torque on the shaft
  • In a moving system
    • Torque = Angular Momentum flux + Angular Momentum Accumulation rate
example12

Q

Example
  • Approach-1. Find effective Force in X direction
  • Find the moment of Force
  • Assume: No frictional loss, ignore gravity, steady state, atmospheric pressure everywhere
example13
Example
  • Approach-2. Using conservation of angular momentum
  • Stationary CV
example14
Example
  • Consider a jet hitting a moving plate
  • After 1 second
  • Vnoz water has entered into the CV
  • Plate has moved by Vplate
  • In a control volume which moves with the plate, Vnoz-Vplate water has entered the CV (and exited at the bottom)
example17
Example
  • Pbm 5.24
  • Thickness of slit =t, vol flow rate =Q, dia of pipe=d, density given
  • Ignore gravity effects

3ft

6ft

Flux

Accumulation Rate

example18
Example

1

3ft

  • Pbm 5.31
  • P1, P2, density, dia, vol flow rate given

2

  • Calculate velocity at 1 (=2)
energy conservation

Friction Loss (Viscous)

Mechanical

Work

done by

the system

Heat

Work done

by pressure force

Energy Conservation
energy conservation1
Energy Conservation
  • No Frictional losses
  • Incompressible
  • Steady
  • No heat, work
  • No internal energy change
example19
Example
  • Flow from a tank

Dia = d1

1

h1

0

2

Dia = d2

  • Pressure = atm at the top and at the outlet

h3

3

  • Velocity at 1 ~ 0
  • Toricelli’s Law
  • Sections 2 and 3
example20
Example
  • How long does it take to empty the tank?
  • What if you had a pipe all the way upto level 3?

Dia = d1

1

h1

0

2

Dia = d2

h3

3

  • Pressure @ section 2 != atm
  • Pressure @ section 3 = atm
example21
Example
  • What if you had a pipe all the way upto level 3?

Dia = d1

1

h1

0

2

Dia = d2

h3

3

  • More flow with the pipe
  • Turbulence, friction
  • Unsteady flow
  • Vortex formation
example22
Example

Height is known

  • Moving reference; Aircraft

60 km/h

  • Find P and r (eg from tables)

150 km/h

  • Flight as Reference

1

2

3

example23
Example
  • Pbm. 6.4
  • Steady flow through pipe , with friction
  • Friction loss head = 10 psi
  • Area, vol flow rate given
  • Find temp increase
  • Assume no heat transfer
example24
Example

D2

  • Pbm. 6.10
  • Fluid entering from bottom,
  • exiting at radial direction
  • Steady, no friction

t

P2=atm

h2

D1

  • Find Q, F on the top plate

P1=10 psig

example25
Example

F

y

D1

P1=10 psig

  • If the velocity distribution just below the top plate is known, then P can be found using Bernoulli’s eqn
modifications to eqn
Modifications to Eqn
  • Unsteady state, for points 1 and 2 along a stream line
draining of a tank

R

1

H

L

2

D

Draining of a tank
  • We can obtain the time it takes to drain a tank
    • (i) Assume no friction in the drain pipe
    • (ii) Assume you know the relationship between friction and velocity
  • Le us take that the bottom location is 2 and the top fluid surface is 1
  • Incompressible fluid
draining of a tank quasi steady state
Draining of a tank: Quasi steady state
  • Quasi steady state assumption
    • Velocity at fluid surface at 1 is very small
    • i.e. R >> D
  • No friction : L is negligible
  • P1 = P2 = Patm
draining of a tank quasi steady state1
Draining of a tank: Quasi steady state
  • Original level of liquid is at H = H0
  • Integrating above equation from t=0, H=H0 to t=tfinal, H =0, we can find the efflux time
draining of a tank unsteady state
Draining of a tank: Unsteady state
  • BSL eg.7.7.1
  • At any point of time, the kinetic + potential energy of the fluid in tank is converted into kinetic energy of the outgoing fluid
  • We still neglect friction
  • Potential Energy of a disk at height z and thickness dz
draining of a tank unsteady state1
Draining of a tank: Unsteady state
  • Also, using continuity equation
  • Substituting, you get a 2nd order non linear ODE with two initial conditions. Please refer to BSL for solution
draining of a tank accounting for friction

R

1

H

L

2

D

Draining of a tank (accounting for friction)
  • What if the flow in the tube is laminar and you want to account for friction?
  • Bernoulli’s eqn is not used (friction present)
  • Continuity
  • Hagen-Poiseuille’s eqn
draining of a tank accounting for friction1
Draining of a tank (accounting for friction)
  • Substituting and re arranging,
  • Integrating with limits
  • Note: The answer is given in terms of diameter of tube, so that it is easier to compare with the answer given in the book
example26
Example
  • A1,A2, initial height h1 known
  • A1 >> A2

1

L

  • Consider section 3 and 2

h1

3

2

  • Pseudo Steady state ==> Toricelli’s law
example27
Example
  • Rearranging and solving, we get
  • As t increases, the solution approaches the Toricelli’s equation
appendix example
Appendix:Example
  • Oscillating fluid in a U-tube

1

  • Let h=h1-h2

h1

2

h2

L3

appendix example1
Appendix:Example
  • Blood Flow in vessels
  • Minimization of ‘work’
  • Murray’s Law:
  • Laminar Flow, negligible friction loss (other than that due to viscous loss in laminar flow) , steady
  • Turbulent, pulsating flow
  • Assume
appendix example2
Appendix:Example
  • If the ratio of ‘smaller’ to larger capillary is constant
  • And Metabolic requirement =m= power/volume
  • Work for maintaining blood vessel
  • Total work
  • Optimum radius
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