ME 254
Chapter I
Integral Relations for a Control Volume
An engineering science like fluid dynamics rests on foundations
comprising both theory and experiment. With fluid dynamics,
progress has been especially dependent upon an intimate crossfertilization between the analytical and empirical branches; the
experimental results being most fruitfully interpreted in terms of
theoretical reasoning , and the analyses in turn suggesting
critical and illuminating experiments which further amplify and
strengthen the theory. All analyses concerning the motion of
compressible fluids must necessarily begin either directly or indirectly, with the statements of the four basic physical laws
governing such motions. These laws, which are independent of
the nature of the particular fluid are
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(i) The law of conservation of mass
(ii) Newton’s second law of motion
(iii) The first law of thermodynamics
(iv) The second law of thermodynamics
In addition to certain subsidiary laws such as equation of state
of a perfect gas, the proportionality law between shear stress and
rate of shear deformation in a Newtonian fluid, the Fourier law
of heat conduction, etc.
Definition of a Fluid. A substance which deforms continuously
under the action of shearing forces. “ a fluid cannot withstand
shearing stresses ” .
Liquids vs. Gases. Both of which are fluids. For most practical
purposes the words “liquid” and “gas” are of value insofar as
the former denotes a fluid which generally exhibits only small
percentage changes in density.
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The Concept of a Continuum
In most engineering problems our primary interest lies not in the
motions of molecules, but rather in the gross behavior of the
fluid thought of a continuous material. The treatment of fluids as
continua may be said to be valid whenever the smallest volume
of fluid of interest contains so many molecules as to make
statistical average meaningful. This course concerns the motion
of compressible fluids which may be treated as continua;
(continuum approach).
Properties of the Continuum
Density at a Point. The “density” as calculated from molecular
mass m within given volume  is plotted versus the size of the
unit volume. There is a limiting value   below which molecular variations may be important and above which aggregate
variations may be important. The density of a fluid is best
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defined as
m
  lim
   
  10 9 mm3
for all liquids and gases.
   10 9 mm 3 of air contains approximately 3x10 7 molecules at
standard conditions.
This definition illuminates the idea of a continuum and shows
the true nature of a continuum property “at a point” as fictitious
but highly useful concept.
Number of molecules per unit volume n = PT where P =pressure
T = temperature & σ = Boltzmann constant 1.38x10 23 J/kmol
Fluid Dynamic Description
Lagrangian Technique An identifiable mass is followed as it
moves through space ( microscopic formulation)
Eulerian Technique Flow through an identifiable region is
monitored ( macroscopic formulation). [flow has confined surface]
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Properties of the Velocity Field
The Velocity Field
Velocity is a vector function of position and time and thus has
three components u, v, and w, each of which is itself a scalar field:




V ( x, y, z, t )  i u ( x, y, z, t )  j v( x, y, z, t )  k w( x, y, z, t )

The acceleration of a particle: The acceleration ais fundamental
to fluid mechanics since it occurs in Newton’s law of dynamics





dV
v
v x
v y
v z

a 




dt
t
x t
y t
z t




(local acc.)
(convective
acceleration)
V
V
V
V

a 
 (u
 v
 w
)
t
x
y
z
The gradient operator 



del   i x  j y  k z




dV
V

a 

 (V .)V
dt
t
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Vector Product
Scalar Product
 
axb  absin   botha & b

a.b  ab cos 


  


 
 
V

a 
 (i u  j v  k w).(i
 j
k
)V
t
x
y
z




V
V
V
V

a 
u
v
w
t
x
y
z
The general expression for Eulerian time-derivative

D
d




(
V
. )
operator following a particle Dt dt t
This operator can be applied to any fluid property, scalar
or vector , e.g., the pressure.
dP P 
P
P P
P

 (V .) P 
u v w
dt t
t
x
y
z
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Volume and Mass Rate of Flow
Volume Rate of Flow
Consider volume rate of flow through an arbitrary surface. The
volume swept out of dA in time dt is the volume ofthe slanted
parallelepiped in the figure d  VdtdAcos   (V .n )dAdt
dV
The integral of dt is the total volume rate of flow Q through
the surfase S.
Q =  (V .n ) dA
s
Mass flux m
 is given by the following relation

    (V .n )dA   Vn dA
m
where V s is the normal component of velocity
For constant density m  Q
An average velocity passing through the surface is given by:
s
n
Vav 
Q

A

(
V
 .n )dA
s
 dA
s
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System versus Control Volume
Control Volume Terminology
A system is defined as an arbitrary quantity of mass of fixed
identity. Everything external to this system is denoted by the
term surroundings and the system is separated from its
surroundings by its boundaries.
The control volume is defined as an arbitrary volume, fixed in
space, and through which fluid flows. The surface which bounds
the control volume is called the control surface.
The Reynolds Transport Theorem
In order to convert a system analysis into a control volume
analysis we must convert our mathematics to apply to a specific
region rather than to individual masses. This conversion, called
the Reynolds transport theorem, can be applied to all basic laws.
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One-Dimensional Fixed Control Volume
 Example of inflow and outflow as three systems pass
through a control volume
 Now let B be any (extensive) property of the fluid (mass,
dB
energy, momentum, etc.) and let β = dm be the intensive
value or the amount of B per unit mass in any small
portion of the fluid. The total amount of B in the control
volume is thus Bcv   dv &   dB
dm
cv
where dv is the differential mass of the fluid
The time derivative of Bcv is defined by the calculus limit
d
1
1
( Bcv ) 
Bcv (t  dt )  Bcv (t )
dt
dt
dt
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d
1
1
( Bcv )  [ B2 (t  dt )  ( dv) out  ( dv) in ]  [ B2 (t )]
dt
dt
dt
d
1
1
1
( Bcv )  [ B2 (t  dt )  B2 (t )]  ( dv) out  ( v) in
dt
dt
dt
dt
d
d
1
1
( Bcv )  ( Bsystem )  ( dv) out  ( dv) in
dt
dt
dt
dt
d
d
( Bsystem )  (  dv)  ( AV ) out  ( AV ) in
dt
dt cv
For any Arbitrary Fixed Control Volume

d

( Bsystem )   dv    (V .n )dA
dt
t
cv
The rate of change of B for a given mass as it is moving around is
equal to the rate of change of B inside the control volume plus
the net efflux (flow out minus flow in) of B from the control
volume.
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 By letting the property B be the mass
dB dm


1
dm dm
dmsystem

 0  
dv   (  i AiVi ) in   (  i AiVi ) out
dt
t
i
i
cv
 For steady flow within the control volume

 0
t
Conservation of mass principle reduces to
 
  (V .n )dA = 0
cs
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