IDEAL FLUID FLOW

The components of the Incompressible Flow Navier-Stokes

Equation ρ
∂V
∂t[ ]
+ ( V  ∇ ) V =−ρg ∇ h−∇ p+ μ ∇ V
2

( )
2 2 2

ρ( ∂u
∂t
+u
∂u
∂x
+v
∂u
∂y
+w
∂u
∂z
=−ρg )
∂h ∂ p
−
∂x ∂x
+μ
∂u ∂u ∂u
2+
∂x ∂ y ∂z
2+ 2

( )
2 2 2

ρ( ∂v
∂t
+u
∂v
∂x
+v
∂v
∂y
+w
∂v
∂z
=−ρg )
∂h ∂ p
−
∂y ∂y
+μ
∂ v ∂ v ∂v
2+
∂x ∂ y ∂z
2+ 2

( )
2 2 2

ρ( ∂w
∂t
+u
∂w
∂x
+v
∂w
∂y
+w
∂w
∂z
=−ρg − )
∂h ∂ p
∂z ∂z
+μ
∂w ∂ w ∂w
2 +
∂x ∂ y ∂z
2+ 2

would have either μ=0 or the laplacians zero to arrive at the

Euler Equation of Inviscid Flow.
∂()
In two dimensions (setting =0 and w=0):
∂z
1 ∂ ∂u ∂u ∂ u
− ( p+ γh )=u + v +
ρ ∂x ∂x ∂ y ∂t
1 ∂ ∂v ∂v ∂v
− ( p+γh )=u +v +
ρ ∂y ∂x ∂ y ∂t
By setting x=s: s= locus of points defining a curve tangent to
the velocity vector at all points on it, i.e. s is in the
direction of velocity; and setting y=n-axis always normal
to the velocity (natural coordinates):
1 ∂ ∂v ∂v vn= 0 at streamline
− ( p+γh )=v s s + s
ρ ∂s ∂s ∂t
Rate of change of vn
1 ∂v ∂v with respect to s and
− ∂ ( p+γh )=v s n + n
ρ ∂n ∂ s ∂t t not necessarily zero
Using r = radius of curvature of streamline at s: rδθ= δs
δθ vs2 δs δv n ∂ vn vs
n s
δθ δvn
r
=
v
⇒ ∂ s
=
r
vs1 s
r 2
1 v ∂v
δs vs2  − ∂ ( p+γh )= s + n
vs1 ρ ∂n r ∂t
Imposing steady flow: Variation
v
2
v
2 of pressure
1
− ∂ ( p+γh )= s ⇒ ∂ ( p+ γh )=−ρ s head across
ρ ∂n r ∂n r streamlines

()
2
1 vs Euler Equation of Motion
− ∂ ( p+γh )= ∂
ρ ∂s ∂s 2 Along a Streamline
Since s is the only independent variable:
1 dp dh dv dp
+ g + v s s =0⇒ + gdh+ v s dv s=0
ρ ds ds ds ρ
p v
2
Bernoulli Constant;
Integrating: + gh+ s =constant Bernoulli Equation
ρ 2
2
in energy/unit mass
p v
or, +h+ =constant Energy per unit weight
γ 2g
Example: Applying pressure head variation across
streamlines to the case of uniform rotation at ω of a fluid
about a vertical axis, dn=-dr; vs= ωr (tangential velocity):
2 2
d ωr 2
( p+γh )=ρ = ρω r
dr r
2 2
ωr
⇒ p+γh=ρ ∫ ω rdr= ρ
2
+c
2
2 2
ωr
When p=p0 at r=0 and h=0  p= p 0−γh+ ρ as derived
2
in fluid statics.

IRROTATIONAL FLOW:
∂ v ∂ u ∂ w ∂ v ∂u ∂ w
∇ ×v=0 ⇒ = ; = ; =
∂x ∂ y ∂ y ∂z ∂z ∂x
In two-dimensional flow in the x-y plane: udx + vdy is some
scalar function, say -d (meaning  decreases in the
direction of the velocity vector), i.e.
∂φ ∂φ
− dx− dy=−dφ=udx+vdy
∂x ∂y
In three dimensions:
∂φ ∂φ ∂φ
v=−∇ φ ⇒ u=− ; v=− ; w=−
∂x ∂y ∂z
 Irrotational Flow means: curl ( grad φ )=∇×(−∇ φ )=0

| |
i j k
∂ ∂ ∂
because ∇ ×∇ = ∂ x ∂y ∂ z =0
∂ ∂ ∂
∂x ∂y ∂z
∂u ∂ v ∂ w
Substituting into the continuity equation + + =0 :
∂x ∂ y ∂z
2 2 2
∂φ ∂φ ∂φ The Laplace
2+ 2+ 2 =0
∂x ∂ y ∂z Equation
2 2
or ∇ • v=∇  (−∇ φ )=−∇ φ=∇ φ=0
Linearity: If 1 is a solution to the Laplace Equation, and 2
2 2
is another solution, thus ∇ φ1 =0 and ∇ φ2 =0 , then
2 2 2 2
∇ ( φ 1 +φ 2 )=∇ φ1 + ∇ φ 2 =0 . Also, ∇ ( cφ 1 )=0 is true.
For any solution  that satisfies condition of irrotationality
in frictionless incompressible flow, the rate of change of 
in any direction is velocity. This means that  is a velocity
potential (e.g. pressure), and the flow is referred to as
Potential Flow.

()
2
∂u ∂v ∂ v
In the Euler equation, we substitute v =v = ,
∂y ∂x ∂x 2

( )
2

w
∂u
∂z
=w
∂w ∂ w
=
∂x ∂x 2
, and
∂u
∂t
=− ∂
∂φ
∂x ∂t ( ) to get

( )
2 2 2
∂ p + gh+ u + v + w − ∂ φ =0
∂x ρ 2 2 2 ∂t

( )
2
2 2 2
Using q = u + v + w : 2
 ∂ p + gh+ q − ∂ φ =0 .
∂x ρ 2 ∂t
 ( )
2
p q ∂φ
In the other directions, ∂ +gh+ − =0 and
∂y ρ 2 ∂t

( )
2
∂ p + gh+ q − ∂ φ =0 , meaning that the quantity in
∂z ρ 2 ∂t
parentheses does not depend on direction but on time only,
2
p q ∂φ
i.e., + gh+ − =f ( t ) .
ρ 2 ∂t
2
p q
In steady flow, + gh+ =constant (Bernoulli Equation).
ρ 2
Total pressure may be separated into static + dynamic
p p p p 1
pressure, i.e. = s + d . But s + gh= ( p s +γh ) is
ρ ρ ρ ρ ρ
hydrostatic pressure variation and may be included in the
pd q2
constant, say E, i.e. + =E .
ρ 2
 If pd and q are known at one location in the flow, then at
another point, p and q are related as:
2 2
p0 q0 p q
+ = + (subscript d omitted);
ρ 2 ρ 2

[ ( )]
2 2
ρq q
or, p= p 0 + 0 1−
2 q0

The Stream Function:

C P2
The rate of flow between
P1
points O and P1 is the B
same whether computed
as passing through line A
A or through line B. O

Flow between P1 and P2 is independent of the location of O.

If O is fixed and P1 and P2 are movable, flow rate across
any line connecting O to either point is a function of the
position of the respective point.
Convention: From O looking at any point P,  is positive if
flow is from right to left. Let 1 =(P1) and 2 =(P2),
then flow across any line C connecting P1 and P2 is 2 - 1.
Velocity components and can be obtained from :
δψ ∂ψ P1
u=− =−
δy ∂y u
δψ ∂ ψ δy v
v= = A P2
δx ∂ x δx
1 δψ
v r =− vr
r δθ rδθ
δθ r
δψ v0 δr
v θ=
δr θ
If P1 and P2 lie on the same streamline, 1 - 2 = 0 (no flow
across streamline),  =const. defines a streamline.
∂φ ∂ψ ∂φ ∂ψ The Cauchy-Riemann
= ; =−
∂x ∂ y ∂ y ∂x Conditions
x
Thus, the stream function can be obtained from the velocity
potential and vice versa. Both satisfy the Laplace
equation.
Two-Dimensional Flows:
Equipotential Line = line of constant 
Streamline = line of constant 
Flownet:
Δφ −Δc Δc =c+2Δc Δs
u s= =− =
Δs Δs Δs
Δn
'
Δψ Δ c
v s= = =c+Δc =c’
Δn Δn
=c’+Δc’
φ=φ ( x , y ) ; ψ=ψ ( x , y ) =c’+2Δc’
=c
 π π
φ= A • r α •cos( παθ ) ; ψ= A • r α • sin( παθ ) ; = 0 at θ = 0 and θ = α

2 2
φ= A ( x − y ) Vortex: φ=− μθ ; ψ=−μ ln (r)
2
ψ=2 Axy= A •r •sin (2 θ ) or φ= μ θ ; ψ= μ ln (r)
ψ=0 at θ =0 and θ = π Source: φ=− μ ln (r) ; ψ =−μ θ
2
Sink: φ= μ ln (r) ; ψ =μ θ
Strength = 2πμ in these 3 cases.
Let a = distance between Source and Sink of equal strength:
A Doublet is the limit condition as a becomes zero;
μ = strength of a doublet

cos(θ )
φ= μ
r
x
μ 2 2
x +y

sin(θ )
ψ=−μ
r

y
−μ 2 2
x +y

For uniform flow in the -x direction (u = -U):

φ=Ux=Ur cos(θ ) ; ψ=Uy=Ur sin (θ )
Doublet + Uniform Flow: Doublet + Uniform Flow
+ Vortex:

( ) ( )
2 2
cosθ a a
φ=Ur cos θ + μ =U r+ cos θ φ=U r + cos θ − Γ θ
r r r 2π

( ) ( )
2 2
sin θ a a
ψ=Ur sin θ − μ =U r− sin θ ψ=U r− sin θ + Γ θ
r r r 2π