Basics of Fluid Mechanics, 2014a

8.6. BOUNDARY CONDITIONS AND DRIVING FORCES 255
For in index form as
ρ DU i
Dt
= − ∂
∂x i
(
P +
( ) )
2
3 μ − λ ∇·U + ∂ ( ( ∂Ui
μ + ∂U ))
j
+ f Bi
∂x j ∂x j ∂x i
(8.109)
For incompressible flow the term ∇·U vanishes, thus equation (8.108) is reduced to
Momentum for Incompressible Flow
ρ DU
Dt = −∇P + μ ∇2 U + f B
(8.110)
or in the index notation it is written
ρ DU i
Dt
= − ∂P + μ
∂2 U
+ f Bi (8.111)
∂x i ∂x i ∂x j
The momentum equation in Cartesian coordinate can be written explicitly for x coordinate
as
( )
∂Ux
ρ
∂t + U ∂U x
x
∂x + U ∂U x
y
∂y + U ∂U x
z =
∂z
− ∂P ( ∂ 2 )
(8.112)
∂x + μ U x
∂x 2 + ∂2 U x
∂y 2 + ∂2 U x
∂z 2 + ρg x
Where g x is the the body force in the x direction (î · g). In the y coordinate the
momentum equation is
( )
∂Uy
ρ
∂t + U ∂U y
x
∂x + U ∂U y
y
∂y + U ∂U y
z =
∂z
− ∂P ( ∂ 2 )
(8.113)
∂y + μ v
∂x 2 + ∂2 v
∂y 2 + ∂2 v
∂z 2 + ρg y
in z coordinate the momentum equation is
( ∂Uz
ρ
∂t +
U ∂U z
x
∂x + U ∂U z
y
∂y + U z
)
∂U z
=
∂z
− ∂P ( ∂ 2 )
(8.114)
∂z + μ U z
∂x 2 + ∂2 U z
∂y 2 + ∂2 U z
∂z 2 + ρg z
8.6 Boundary Conditions and Driving Forces
8.6.1 Boundary Conditions Categories
The governing equations that were developed earlier requires some boundary conditions
and initial conditions. These conditions described physical situations that are believed
or should exist or approximated. These conditions can be categorized by the velocity,