 The Reynolds Transport Theorem
 Correlation between System (Lagrangian) concept
↔ Control-volume (Eulerian) concept
for comprehensive understanding of fluid motion?
 Reynolds Transport Theorem
Let’s set a fundamental equation of physical parameters
B = mb
where B: Fluid property which is proportional to
amount of mass (Extensive property)
b: B per unit mass (Independent to the mass)
(Intensive property)


e.g. a) If B  mV (Linear momentum): Extensive property
 
then, b  V (Velocity): Intensive property
1
b) If B  mV 2 (Kinetic energy): Extensive property
2
1
then, b  V 2 : Intensive property
2
i. B of a system Bsys at a given instant,
 bi ( iVi ) = sys bdV
Bsys  lim
V  0 i
mi for ith fluid particle in the system
where Vi : Volume of ith fluid particle
And Time rate of change of Bsys,
dBsys
dt
=

d  sys bdV

dt
 in a control volume Bcv
ii. B of fluid
Bcv  lim
 bi ( iVi ) = cv bdV
V  0 i
and
d cv bdV 
dBcv
=
dt
dt

 Relationship
between
dBsys
dt
and
Only difference from
B of a system
dBcv
: Reynolds Transport Theorem
dt
 Derivation of the Reynolds Transport Theorem
Consider 1-D flow through a fixed control volume shown
Fixed control surface at t (coincide with a system boundary)
System boundary at t + t
a) At time t, Control volume (CV) & System (SYS): Coincide
b) At t  t (after t ),
CV: fixed
& SYS: Move slightly
 Fluid particles at section (1): Move a distance dl1  V1t
 Fluid particles at section (2): Move a distance dl2  V2t
 I : Volume of Inflow (entering CV)
 II : Volume of Outflow (leaving CV)
That is,
SYS (at time t)
= CV
SYS (at time t  t )
= CV – I + II
Or if B: Extensive fluid property, then
Bsys(t) = Bcv(t)
(at time t)
Bsys (t  t )  Bcv (t  t )  BI (t  t )  BII (t  t )
(at time t  t )
Then, Time rate of change in B can be;
Bsys Bsys (t  t )  Bsys (t )
= Bcv(t), at time t

t
t
Bcv (t  t )  BI (t  t )  BII (t  t )  Bsys (t )
=
t
B (t  t )  Bcv (t ) BI (t  t ) BII (t  t )


= cv
t
t
t
In the limit t  0 ,
Left-side:
Bsys DBsys
=
t
Dt
(according to Lagrangian Concept)


 bdV 
B
(
t


t
)

B
(
t
)

B


cv
1st term on Right-side: lim cv
= cv = cv
t  0
t
t
t
B (t  t )
 1A1V1b1
2nd term on Right-side: Bin  lim I
t  0
t
(4.13)
because BI (t  t )  ( 1V1 )b1  1 A1V1b1t
where A1 : Area at section (1)
V1 : Velocity at section (1)
B (t  t )
  2 A2V2b2
3rd term on Right-side: B out  lim II
t  0
t
(4.12)
because BII (t  t )  (  2V2 )b2  2 A2V2b2t
Relationship between the time rate of change of Bsys and Bcv

DBsys
Dt

Bcv 
B
 Bout  Bin  cv   2 A2V2b2  1 A1V1b1
t
t
: Special version of Reynolds transport theorem
- Fixed CV with one inlet and one outlet
- Velocity normal to Sec. (1) and (2)
 General expression of Reynolds Transport Theorem
Consider a general flow shown
At time t,
CV & SYS: Coincide
At time t  t , CV: Fixed & SYS: Move slightly
Bcv 
 Bout  Bin
Dt
t
 Still valid,
but Bout & Bin : Different
DBsys

 What are Bout & Bin ?
1) Bout : Net flowrate of B leaving CV (Outflow)
across the control surface between II and CV ( CS out )
B across the area element A on CS out
B  bV  b (V cost )A
where V (Fluid volume leaving CV across A
 lnA  l cos A  (Vt cos  )A
Then, the time rate of B across A
bV
( bV cos t )A
 lim
 bV cos A
t  0 t
t  0
t
B out  lim
By integrating over the entire CS out ,
Bout  CS
out
dBout  CS
out
bV cos  dA  CS
out

bV  nˆdA
2) Bout : Net flowrate of B entering CV (Inflow)
across the control surface between I and CV ( CS in )
By the similar manner,

Bin   CS bV cos  dA   CS bV  nˆdA
in
(because

2
 
3
)
2
in
Finally, Net flowrate of B across the entire CS (  CS in  CS out )
Bout  Bin  CS

out

bV  nˆdA  ( CS bV  nˆdA)
in

= CS bV  nˆdA
DBsys
Dt



Bcv

 CS bV  nˆdA  cv bdV  CS bV  nˆdA
t
t
: General expression of Reynolds Transport Theorem
 PHYSICAL INTERPRETATION

DBsys
Dt
: Time rate of change of an extensive B of a system
 Lagrangian concept

 bdV : Time rate of change of B within a control volume
t cv
 Eulerian concept

 CS bV  nˆdA : Net flowrate of B across the entire control surface

 Correlation term – Motion of a fluid
c.f. Comparison with the definition of Material Derivative
D    
 
 
    

u
v
w

 (V  )
Dt
t
x
y
z
t



D 
Dt
: Time rate of change of a property of fluid particle
 
t
: Time rate of change of a property at a local space

 (V  )
 Lagrangian concept
 Eulerian concept: Unsteady effect
: Change of a property due to the fluid motion
 Correlation term – Convective effect
 Reynolds Transport Theorem
 Transfer from Lagrangian viewpoint to Eulerian one (Finite size)
 Special cases
DBsys
1. Steady Effects.
Dt




bd
V


b
V
 nˆdA


cv
CS
t
 Any change in property B of a system
= Net difference in flowrates B entering CV and leaving CV
2. Unsteady Effects.

 bdV  0
t CV
 Any change in property B of a system
= Change in B within CV
+ Net difference in flowrates B entering and leaving CV
e.g. For 1-D flow

V  V0 (t )iˆ
  Constant
 



Choose B  mV (Momentum), and thus b  B / m  V  V0 (t )iˆ


CS bV  nˆdA  CS  (V0iˆ)V  nˆdA
 (1)  V0iˆ(V0 )dA  ( 2)  V0iˆ(V0 )dA  side  V0iˆ(V0 cos 90 )dA
  V02 Aiˆ  V02 Aiˆ  0 (Inflow of B = Outflow of B)

DBsys
Dt
=

 bdV
t CV
: No convective effect
 Reynolds Transport Theorem for a moving control volume
DBsys
Dt




bd
V


b
V
 nˆdA


cv
CS
t
: Valid for a stationary CV
In case of moving control volume as shown,

Consider a constant velocity of CV = Vcv
 Reynolds transport theorem
: Relation between a system and CV, (Neglect the surrounding)
 Velocity of a system: Defined w.r.t. the motion of CV
  
 Relative velocity of a system: W  V  VCV

where V : Absolute velocity of a system
Finally,
DBsys
Dt


 cv bdV  CS bW  nˆdA
t

: Valid for a stationary or moving CV with constant Vcv