of Equations Change for Isothermal Systems
The equation of continuity (or mass balance equation)
Mass influx
(inward arrows)
Mass balance over a fixed control volume
xy z
y z u x x u x x x t
y
z
x
t
xz u y u y
y
y y
y
x y u z z u z z z t
ux u y uz
. u
x
t
y
z
For incompressible fluids: .u 0
Mass outflux
(outward arrows)
x
z
Equations of Change for Isothermal Systems
The equation of motion
Derivation of the equation of motion using momentum principle
The balance for x-momentum over a fixed control volume
xyz
with
ux
yz xx x xx x x xz yx yx
y
t
xy zx z zx z z xyz g x
ix ui u x p ix ix
y y
i x, y , z
xx yx zx
u x
gx
t
y
z
x
1
Equations of Change for Isothermal Systems
The equation of motion in a general form
u j
, z i uiu j p ij ij g j
t
i x, y
ij p
gj
ui u j
j
i x , y , z i
i x , y , z i
For Newtonian flow
u j u
2
i
ij .u
ij
xi x j
3
Equations of Change for Isothermal Systems
Simplification of the equation of motion
1. Constant density and viscosity : Navier-Stokes equation
For incompressible fluids: .u 0
i x, y, z
ij
i
2u j
i
i x, y, z
2u j
2
u
, z ii 2u j
j i x , y
.u 0
u
uiu j u j i ui ui ij
i x , y , z i
i x, y , z
u j
u j i ui ui i
i x, y , z
i x, y,z
u j
u j
u j i ui ui i ui i
i x, y , z
i x, y,z
i x, y , z
2
Equations of Change for Isothermal Systems
Simplification of the equation of motion
1. Constant density and viscosity : Navier-Stokes equation
For incompressible fluids: .u 0
i x, y , z
ij
i
2u j
u
uiu j ui ij
i x , y , z i
i x, y, z
Equations of Change for Isothermal Systems
Simplification of the equation of motion
1. Constant density and viscosity : Navier-Stokes equation
The general equation of motion is reduced to
u j
t
u j p
2
u j g j
i j
u j
p
2
u j g j
i
j
u
i x, y , z
i
u j
ui
t i x , y , z
3
Equations of Change for Isothermal Systems
Simplification of the equation of motion
2. Navier-Stokes equation with negligible acceleration term
Stokes flow equation: for very slow flow
u j
u j
p
2
ui
u j g j
j
i
t x , y , z i
j x, y , z
0
p
2u j g j 0
j
j x, y , z
This equation is important in lubrication theory, particle motion in suspension,
and flow in porous media
Equations of Change for Isothermal Systems
Simplification of the equation of motion
3. Negligible viscous forces in general equation of motion
Euler equation:
u j
t
uiu j iij p g j
j
i x , y , z i
i x, y , z
0
With the help of the equation of continuity, Euler equation may be rewritten as
u j
t
u j
p
ui
gj
i j
i x, y , z
4
Equations of Change for Isothermal Systems
Simplification of the equation of motion
3. Negligible viscous forces in general equation of motion (cont.)
Euler equation with flow under steady-state conditions
u j
p
ui
gj
i j
i x, y , z
From this equation, one can derive Bernoulli equation for steady-state
streamline flow
B
12
1
u dp gh
2
C
D
C
D
B is constant along the streamline in general
For example:
1 2 pC 1 2 p D
uC
uD
2
2
BC
BD
Equations of Change for Isothermal Systems
Simplification of the equation of motion
4. Small variation of density and constant viscosity
Boussinesq approximation for temperature-dependent density = (T)
- For small variation of with respect to T, the continuity equation has the
small variation of
respect to T, the continuity equation has the
incompressible form
.u 0
- and the equation of motion is slightly different from Navier-Stokes equation
u j
t
u j
p
, z ui i j 2u j T g j
i x, y
is the density evaluated at average temperature
5
Equations of Change for Isothermal Systems
Three components of the equation of
motion
Momentum influx
(inward arrows)
X-component
u x
ui u x p ix ix g x
t
i x , y , z i
ix
y
z
Y-component
u u p g
i y
iy
iy
y
t
i x , y , z i
iy
x
u y
Momentum outflux
(outward arrows)
Z-component
x
y
z
u z
u u p g
iz
iz
z
iz
t
i x , y , z i
iy
Equations of Change for Isothermal Systems
A methodology for solving flow problems
Statement of flow problems
1. Define the domain boundary.
2. Choose a coordinate system such that there are more vanishing terms (i.e.
coordinate system such that there are more vanishing terms (i
components of velocity, velocity gradients, and forces).
3. State assumptions to simplify the problem (i.e. set zero more terms).
4. List non-zero components of velocity, velocity gradient, and forces.
5. Apply directly the equations of continuity and motions for each phase.
6. Identify initial and boundary conditions, and then solve the velocity
equations.
7. Determine other quantities related to velocity profiles.
6
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Flow of falling film
The equation of continuity (see Table B.4)
u u u u
y
x
z
z
0
x
t
y
z
z
uz is uniform in z-direction
The equation of motion (see Table B.5)
u x
u
u
u
ux x u y x uz x
t
x
y
z
yx zx
p
xx
x x
y
z
gx
u y
u y
u y
u y
ux
uy
uz
t
x
y
z
yy
zy
p xy
gy
y
z
y x
yz
u z
u
u
u
p
u x z u y z u z z xz
zz g z
t
x
y
z
z
z x
y
Equations Change of for Isothermal Systems
Applications of the equations of continuity and motion
Flow of falling film
The equation of continuity (see Table B.4)
uz
0
z
The equation of motion (see Table B.5)
p
p
g x g sin
x
x
p
0
y
p
0 xz g cos
z x
0
7
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
The x-component equation gives
p gx sin f ( y , z )
S ince
p / y 0 and p x 0 f ( y , z ) patm
We have
p ( x ) gx sin patm
and
d xz
g cos
cos
dx
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Flow in tube at steady state
The equation of continuity (see Table B.4)
1
1
rur
u u z 0
t
r r
r
z
The equation of motion (see Table B.6)
ur
1
u u
p
u
u u2
1 2u
2u
2 u
ur r r uz r
rur 2 2r 2r 2 gr
t
r
r
r
z
z
r
r
r
r r r
u
1
u
u u
u u u
1 p
1 2u 2u
2 u
ur uz r
ru 2 2 2 2 r g
t
r
r
r
z
r
z
r
r
r r r
2
1
uz
u u
p
u
u
2u
1 u
ur z z uz z
ruz 2 2z 2z gz
z
r
z
z
r
r
r r r
t
8
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Flow in tube at steady state
The equation of continuity (see Table B.4)
uz 0
z
The equation of motion (see Table B.6)
p
r
1 p
0
r
p
1
0
ruz gz
z
r r r
0
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Rotational vortex at steady state
The equation of continuity (see Table B.4)
1
1
ru r
u u z 0
r r
r
t
z
z0
z
R
The equation of motion (see Table B.6)
r
2
1
ur
u u
2u
2 u
p
u
u u2
1 u
ur r r uz r
rur 2 2r 2r 2 gr
r
r
r
z
r
r
z
r
r r r
t
u
1
u
u u
u u u
2u
1 p
1 2u
2 u
ur uz r
ru 2 2 2 2 r g
t
r
r
r
r
z
r
z
r
r r r
uz
u u
u
u
ur z z uz z
t
r
z
r
1
p
1 2u
2u
ruz 2 2z 2z
z
z
r
r r r
gz
9
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Rotational vortex at steady state
The equation of continuity (see Table B.4)
1
u 0
r
The equation of motion (see Table B.6)
u 2
p
r
r
1
0
ru
r r r
p
0
gz
z
z0
z
R
r
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Rotational vortex at steady state
Integrating the -component equation gives
u
C
1
C1r 2
r
2
Applying boundary conditions
u
rR
R
and u
r 0
C1 2
finite
C2 0
u r
10
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Rotational vortex at steady state
Integrating r- and z-component equations
u 2
p
1
2 r p 2 r 2 f1 ( , z )
r
r
2
p
and
g p gz f 2 ( r , )
z
Since p ( r , z ) , choosing
f1 gz C and f 2 2 r 2 C
g ives p gz
1
2r 2 C
2
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Rotational vortex at steady state
Applying boundary conditions
zz
p r 00 patm
1
p patm g z z0 2 r 2
2
The shape of the air-liquid interface
2 2
z z0 r
2g
11
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Pitot tube
p
1 2 p1 1 2 p2
u1
u2
2
2
2
0
2 p2 p1
u1
p2 gh2
with p1 gh1
1/ 2
u1 2 g h2 h1
1/ 2
p2
1
u12 p1 is called 'stagnation presure'
2
1
u12 is called 'dynamic presure'
2
Equations of Change for Isothermal Systems
Applications of the equations of continuity and motion
Orifice flow
Distribution of p-patm
at the orifice
p
1 2 p atm
u
atm gh
2
u
2 gh
loss of potential energy = gain of kinetic energy
w Ac u Ac 2 gh
Ac Aorifice
for rounded orifice, otherwise 60% Aorifice for sharp-edged orifice
12
Equations of Change for Isothermal Systems
Dimensional analysis of the equations of change
Non-dimensionalizing Navier-Stokes equation
Consider the x-component of the Navier-Stokes equation
u x
u
u
u
p
u x x u y x u z x 2u x g x
x
y
z
x
t
Introducing the following dimensionless variables using characteristic length (l0),
characteristic velocity (u0), and characteristic pressure (P0)
x x / l0 ;
t u0t / l0 ;
y y / l0 ;
z z / l0 ;
P P0
u x u x / u0 ; P
2
u0
Equations of Change for Isothermal Systems
Dimensional analysis of the equations of change
Non-dimensionalizing Navier-Stokes equation
Inserting the dimensionless variables into the x-component Navier-Stokes equation gives
2
2
2
u0 ux u0 ux u0 ux u0 ux
ux u y uz
l0 / u0 t l0
x l0
y l0
z
2
u0 p
u0 2
2 ux g x
l0 x
l0
ux ux ux ux
p 2 l0 g x
u
ux u y uz
2
x
u0
t
x
y
z
x u0 l0
13
Equations of Change for Isothermal Systems
Dimensional analysis of the equations of change
Non-dimensionalizing Navier-Stokes equation (cont.)
Or
ux ux ux ux
p 1 2 1
u x u y u z ux
Fr
t
x
y
z
x Re
1
1 l0 g x
2
Re u0 l0
Fr u0
Re and Fr are called Reynolds and Froude numbers, respectively. The dimensionless
equation shows that
- Its solutions are independent on the dimensions of flow domain, projection of gravity
force on the x-direction, and fluid properties (i.e. viscosity and density).
- Its solutions are dependent on dimensionless numbers (i.e. Re and Fr)
- It can be simplified for very high or low values of Re and Fr. For example
Re ;
ux ux ux ux
p 1
ux u y uz
t
x
y
z
x Fr
14