Basics of Fluid Mechanics, 2014a

326 CHAPTER 10. POTENTIAL FLOW
These aspects are very important in certain regions which can be evaluated using dimensional
analysis. The determination of what regions or their boundaries is a question
of experience or results of a sophisticated dimensional analysis which will be discussed
later.
The inviscid flow is applied to incompressible flow as well to compressible flow.
However, the main emphasis here is on incompressible flow because the simplicity. The
expansion will be suggested when possible.
10.1.1 Inviscid Momentum Equations
The Naiver–Stokes equations (equation (8.112), (8.113) and (8.114)) under the discussion
above reduced to
( ∂Ux
ρ
∂t
( ∂Uy
ρ
∂t
( ∂Uz
ρ
∂t
Euler Equations in Cartesian Coordinates
)
∂U x
+ U x
∂x + U ∂U x
y
∂y + U z
∂U y
+ U x
∂x + U ∂U y
y
∂y + U z
∂U z
+ U x
∂x + U ∂U z
y
∂y + U z
∂U x
∂z
∂U y
∂z
∂U z
∂z
)
)
= − ∂P
∂x + ρg x
= − ∂P
∂y + ρg y
= − ∂P
∂z + ρg z
(10.1)
These equations (10.1) are known as Euler’s equations in Cartesian Coordinates. Euler
equations can be written in a vector form as
ρ DU = −∇P − ∇ ρ g l (10.2)
Dt
where l represents the distance from a reference point. Where the D U/ Dt is the
material derivative or the substantial derivative. The substantial derivative, in Cartesian
Coordinates, is
( )
DU
Dt = i ∂Ux ∂U x
+ U x
∂t ∂x + U ∂U x
y
∂y + U ∂U x
z
∂z
+ j
( ∂Uy
∂t
)
∂U y
+ U x
∂x + U ∂U y
y
∂y + U ∂U y
z
∂z
+ k
( ∂Uz
∂t
)
∂U z
+ U x
∂x + U ∂U z
y
∂y + U ∂U z
z
∂z
In the following derivations, the identity of the partial derivative is used
(10.3)
U i
∂U i
∂i
= 1 2
∂ (U i ) 2
∂i
(10.4)