Fluid mechanics spring 2021 chapter 2 u lei 李雨

Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei ( 李雨 )
Specific heat ratio as function of temperature
(from White, “viscous fluid flows,” 1991)
p
v
C
C
γ =
(from White’s book)
gases

Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei ( 李雨 )
Non-Newtonian fluid
2
1 pseudoplastic
1 newtonian ( )
1 dilatant
n
xy xy
C
n
n C
n
τ ε
μ
≈
<
= =
>
(from White’s book)
(nonlinear)
(for simple shear flow)
Equations on page 75
need to be modified
(different constitutive equations)

Derivation of the Continuity using control volume approach (1)
x Δ
y Δ
z Δ
→
→
↑
↑


( )
2
u
x
u y z
x
ρ
ρ
∂
 
Δ
+ Δ Δ
 
∂
 
( )
2
w
z
w x y
z
ρ
ρ
∂  
Δ
+ Δ Δ
 
∂
 
( )
2
v
y
v x z
y
ρ
ρ
∂  
Δ
+ Δ Δ
 
∂
 
( )
2
u
x
u y z
x
ρ
ρ
∂  
Δ
− Δ Δ
 
∂
 
( )
2
v
y
v x z
y
ρ
ρ
∂  
Δ
− Δ Δ
 
∂
 
( )
2
w
z
w x y
z
ρ
ρ
∂
 
Δ
− Δ Δ
 
∂
 
Velocity at the center of the cube: ( , , ) u v w
(Fluxes at the surfaces of an infinitesimal cubic control volume)
(Control volume: a fixed volume
in space, mass may cross the
surface of the control volume).

Derivation of the Continuity using control
volume approach (2)
( ) ( )
( ) ( )
( )
2 2
2 2
Mass Balance: Rate of mass increase = net influx of mass
u u
x x
x y z u y z u y z
t x x
v v
y y
v x z v x z
y y
w
w
ρ ρ
ρ
ρ ρ
ρ ρ
ρ ρ
ρ
ρ
∂ ∂    
∂ Δ Δ
Δ Δ Δ = − Δ Δ − + Δ Δ
   
∂ ∂ ∂
   
∂ ∂    
Δ Δ
+ − Δ Δ − + Δ Δ
   
∂ ∂
   
∂
+ −
∂
( )
( ) ( ) ( )
( ) ( ) ( )
2 2
0
0
w
z z
x y w x y
z z
u v w
x y z x y z x y z
x y z
u v w
t x y z
t
ρ
ρ
ρ ρ ρ
ρ ρ ρ
ρ
ρ
ρ
∂    
Δ Δ
Δ Δ − + Δ Δ
   
∂
   
∂ ∂ ∂
= − Δ Δ Δ − Δ Δ Δ − Δ Δ Δ
∂ ∂ ∂
∂ ∂ ∂
∂
→ + + + =
∂ ∂ ∂ ∂
∂
→ + ∇ ⋅ =
∂
u
(similar approach for deriving
momentum and energy equations)

The following 8 pages are for the detailed
equations written in Cartesian, cylindrical
and spherical coordinates, which are adopted
from the appendix of the book by R. L.
Panton, “Incompressible flow,” John Wiley
and Sons, 1984
.
Electronic version of Panton’s book (2013)
is available at NTU library.

(in the lecture notes)
ij ij
S D =

Stokes hypothesis has been assumed
2
In general, replacing
3
in the normal stress components.
μ λ − ∇ ⋅ = ∇ ⋅ v v

Incompressible Newtonian fluid

Incompressible Newtonian fluid

Some useful vector identities
(from the appendix of the book “Fundamental mechanics
of fluids,” by I. G. Currie, McGraw-Hill, 1974
.
__________
___________________
_______________________________
_________

Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei ( 李雨 )
Boundary and Initial Conditions
Since there exist spatial derivative terms in the
governing equations, it is required to specify certain
values of the flow quantities on the boundaries of the
fluid domain of interest.
 Boundary conditions
If the flow is not steady, then it is necessary to specify
the flow quantities having time derivatives at a specified time.
 Initial conditions

Fluid Mechanics (Spring 2021) – Chapter 2 - U. Lei ( 李雨 )
Boundary conditions (1)
(for continuum fluid mechanics)
 Experimentally, it is found that the fluid adheres to the boundary surface.
 B.C. for Navier-Stokes equation
At solid boundaries
,
B
u u =
(no-slip boundary condition)
n u n u ⋅ = ⋅
B
(slip boundary condition)
 Inviscid fluids, including perfect and ideal fluid, slip on the wall.