Chapter 6 Equations of Continuity and Motion

Session 6-2 Equation of motion

Contents

6.1 Continuity Equation

6.2 Stream Function in 2-D, Incompressible Flows 6.3 Rotational and Irrotational Motion

6.4 Equations of Motion

6.5 Examples of Laminar Motion 6.6 Irrotational Motion

6.7 Frictionless Flow 6.8 Vortex Motion

♦ Laminar motion

~ orderly state of flow in which macroscopic fluid particles move in layers

~ viscosity effect is dominant

~ no-slip condition @ boundary wall

~ apply concept of the Newtonian viscosity

~ low Re [Ex]

1. Laminar flow between two parallel plates → Couette flow

2. Laminar flow through a tube (pipe) of constant diameter → Poiseuelle flow

[Re] Reynolds number = inertial force / viscous force = destabilizing force / stabilizing force

• Viscous force

~ dissipative

~ have a stabilizing or damping effect on the motion

~ use Reynolds number [Cf] Turbulent flow

~ unstable flow

~ instantaneous velocity is no longer unidirectional

~ destabilizing force > stabilizing force

~ high Re

2-D flow (x, ^z) steady flow parallel flow z-axis coincides with h

6.5.1 Laminar flow between two parallel plates

Consider the two-dimensional, steady, laminar flow between parallel plates in which either of two surfaces is moving at constant velocity and there is also an external pressure gradient.

♦ Assumptions:

0 ; ( ) 0

v y

→ = ∂ =

( ) ∂ t 0

→ ∂ =

∂

0 ; ( ) w 0

w ∂

→ = =

∂

0 ; 1

h h h

x y z

∂ ∂ ∂

→ = = =

∂ ∂ ∂

a

y

z

a z/a

p 0 x

∂ <

0 ∂ p

x

∂ = 0 ∂

p x

∂ >

∂

p₁ p₂

♦ External pressure gradient

1 2

p > p p 0

x

∂ < →

i)

∂

pressure gradient assists the viscously induced motion to overcome the shear force at the lower surface

p 0 x

∂ > →

ii)

∂

pressure gradient resists the motion which is induced by the motion of the upper surface

1 2

p < p

Continuity eq. for two-dimensional, parallel flow:

u w 0

x z

∂ ∂

+ =

∂ ∂

( )

2 2

only

u 0 x

u f z

∂ =

→   ∂

 = 

N-S Eq.:

. : u x dir

t

− ∂

∂

u u x + ∂

∂

v u y + ∂

∂

w u z + ∂

∂

g h x

= − ∂

∂

2 2

1 p u

x x

µ

ρ ρ

∂ ∂

− +

∂ ∂

2 2

u y + ∂

∂

2 2

u z

 ∂ 

 + 

 ∂ 

 

2 2

0 1 p u

x z

µ

ρ ρ

∂  ∂ 

∴ = − ∂ +   ∂  

Steady flow

Continuity eq. for incompressible fluid 2D flow

parallel flow

(6.31a)

. : w z dir

t

− ∂

∂

u w x + ∂

∂

v w y + ∂

∂

w w z + ∂

∂

2 2

1

h p w

g z z x

µ

ρ ρ

∂ ∂ ∂

= − − +

∂ ∂ ∂

2 2

w y + ∂

∂

2 2

w z + ∂

∂

 

 

 

 

0 1 p

g ρ z

∴ = − − ∂

∂

^(6.31b)

(6.31b):

p

z ρ g γ

∂ = − = −

∂

( )

p γ z f x

∴ = − +

_(6.32)

→ hydrostatic pressure distribution normal to flow

→ For any orientation of z -axis. in case of a parallel flow, pressure is distributed hydrostatically in a direction normal to the flow.

(6.31a):

p dp

~ independent of z

x dx

∂ →

∂

(A)

2 2

dp u

dx µ ^∂ z

∴ =

∂

Pressure

drop Energy loss due to

viscosity

Integrate (A) twice w.r.t. z to derive u(z)

Use the boundary conditions,

2 2

dp u

dzdz dzdz

dx = µ ^∂ z

∫∫ ∫∫ ∂

1

dp u

z dz dz C dz

dx = µ ^∂ z +

∫ ∫ ∂ ∫

2

1 2

2

dp z u C z C

dx = µ + +

_(6.33)

i)

0, 0 dp 0 ( ) 0

_{2 2}

0

z u C C

dx µ

= = → × = + ∴ =

2 2

u dp z dx µ ^∂ =

∂

ii)

,

₁

2

z a u U dp a U C a

dx µ

= = → = +

2 1

1

2

C dp a U

a dx ^ µ ^

∴ =   −  

∴

(6.33) becomes

2

1

2

2 2

dp z dp a

u U z

dx ⁼ µ ⁺ a dx ^   − µ ^  

2

2 2

z dp az z

u U

a dx

µ µ ^{ }

∴ = −   −  

( ) ¹

2

U a dp z

u z u z z

a µ dx a

 

= = −  − 

 

^(6.34)

Velocity

driven Pressure

driven

i) If

dp 0

dx = →

Couette flow (plane Couette flow)

u U z

= a

^(6.35)

→ driving mechanism =

U

(velocity)

ii) If

U = 0 →

2-D Poiseuille flow (plane Poiseuille flow)

( ) ^parabolic

1 ~

2

u dp z a z µ dx

= −

_(6.36)

→ driving mechanism = external pressure gradient,

dp dx

max @

2 u z = a

2

max

8

u a dp

µ dx

= −

_(6.37)

V = average velocity

(6.38)

2 max

2

3 12

Q a dp

A u µ dx

= = = −

[Re] detail

(

²

)

³

0 0

1 1

2 12

a a

dp dp

Q u dz z az dz a

dx dx

µ µ

= ∫ = ∫ − = −

2

max

1 2

12 3

Q a dp

A a V u

A µ dx

= × ∴ = = − =

[Re] Dimensionless form

2

2

2 1

2

1

u z a dp z z

U a U dx a a

a dp

P U dx

u z z z

U a P a a

µ µ

 

= −   −  

= −

 

= +   −  

[Cf] Couette flow in the narrow gap of a journal bearing

Flow between closely spaced concentric cylinders in which one cylinder is fixed and the other cylinder rotates with a constant angular velocity, ω

i

o i

U r a r r

U a ω τ µ

=

= −

≈

→ Hagen-Poiseuille flow

→ Poiseuille flow: steady laminar flow due to pressure drop along a tube Assumptions:

– use cylindrical coordinates

6.5.2 Laminar flow in a circular tube of constant diameter

p 0 x

∂ <

∂

parallel flow →

Continuity eq. →

paraboloid →

steady flow →

0 0 v

r

v

_θ

=

=

z

0 v

z

∂ =

∂

z

0 v

θ

∂ =

∂

z

0 v

t

∂ =

∂

z 0 v ≠

z

0 v

r

∂ ≠

∂

Eq. (6.29c) becomes

By the way,

independent of r

0 p

_z

1 v

^z

g r

z ρ µ r r r

∂ ∂  ∂ 

= − ∂ + + ∂   ∂  

^(A)

( ) ( )

z

p d

g p h p h

z ρ z γ dz γ

∂ ∂

− + = − + = − +

∂ ∂

z

g g h

ρ ρ ^∂

z

 = − 

 ∂ 

 

( )

comp. Eq.

0

1

0

r

r

p g r

p r

r

ρ

γ

→

= −

−

∂ +

∂

→ ∂ + =

∂

Then (A) becomes

( ) ¹

^z

d v

p h r

dz + γ ⁼ µ r r ∂ ^{∂ }   ^∂ ∂ r ^  

( )

1 d v

_z

p h r r

dz γ r r

µ

∂  ∂ 

+ = ∂   ∂  

^(B)

Integrate (B) twice w.r.t. r to derive v_z(r)

( )

² ₁

1

2

d r v

z

r C

p h

dz γ r

µ

= ∂ +

+ ∂

^(C)

( )

¹

1 2

d v

z

C

p h r

dz γ r r

µ

= ∂ +

+ ∂

( )

² _{1 2}

1 ln

2 2

^z

d r

v C r C

p h

dz γ

µ ^{+ = + +}

^(D)

Using BCs

→ (C) :

→ (D) :

0 ,

_{z z}max

r = v = v

1

0

C =

0

,

_z

0

r = r v =

₂

¹ ( )

⁰²

2 2

r

C d p h

dz γ

= µ +

(D1)

Then, substitute (D1) into (D) to obtain ν_z piezometric pressure

( ) (

0^{2 2}

)

1

z

4

v d p h r r

dz γ

µ ^{ }

∴ =   − +   −

( )

^{02 2}

0

4 1

z

r

d r

v p h

dz γ r

µ

   

= − +    −       

(6.39)

→ equation of a paraboloid of revolution

(1) maximum velocity,

(2) mean velocity, V_z

zmax

v

(6.40)

max @

0

v

z

r =

( )

max

2 0 z

4

d r

v p h

dz γ

= − + µ

_(6.41)

dQ = v dA

z

( ) (

0^{2 2}

)

1 2

4

d p h r r rdr

dz γ π

µ ^{ }

=   − +   −

( ) ( )

0 2 2

0 0

1 2

4

r

d

Q p h r r rdr

dz γ π

µ ^{ }

= ∫   − +   −

( )

₀^{2 2 4 0}

2 2 4

r

r

d r r

p h r

dz

π γ

µ

 

 

=   − +     −  

( )

4 0

8

r d

p h

dz

π γ

µ ^{ }

=   − +  

( )

^max

2 0 2

0

8 2

z z

Q Q r d v

V p h

A r dz γ

π µ ^{ }

∴ = = =   − +   =

^(E)

[Cf] For 2 - D Poiseuille flow 2 _max V = 3u

(3) Head loss per unit length of pipe

Total head = piezometric head + velocity head Here, velocity head is constant.

Thus, total head loss = piezometric head change

( )

_{2 2}

0

1 8 32

f z z

h d V V

p h

L dz r D

µ µ

γ ^ γ ^ γ γ

≡   − +   = =

where

D = 2 r

₀

=

diameter (E)

(6.42) hf p

L

γ

≡ ∆

[Re] Consider Darcy-Weisbach Eq.

2 (F)

1 2

f z

h V

L = f D g

head loss due to friction

f

= h

friction factor

f =

Combine (6.42) and (F)

2 2

32 1

2

z z

V V

D f D g

µ

γ ⁼

^(6.43)

Differentiate (6.39) w.r.t. r

64 64 64

/ Re

z z

f V D V D ν

= = ν =

→ For laminar flow (6.44)

(4) Shear stress

r zr

v

τ = µ ^∂

z

∂

z vz

v

r

µ

r

 + ∂  = ∂

 ∂  ∂

 

^(G)

( ) ¹

2 v

z

d

p h r

r dz γ

µ

∂ = +

∂

^(H)

Combine (G) and (H)

( )

1

zr

2

d p h r τ = dz + γ

Linear profile

(6.45)

At center and walls

0 ,

_zr

0 r = τ =

( )

0 max 0

, 1

zr

2

r r d p h r

τ

dz

γ

= = +

max

0

zr zr

r

τ = τ r

31

*

0 0

Turbulent fl

1

ow:

1 ln

v

c

v y

v R

y r

R R

κ

τ τ = ^   − ^   = τ

− = −

2 2

0 0

Laminar flow : 1-

1

c

v v r

R y r

R R

τ τ τ

 

=  

 

 

=   −   =

32

( )

2.0 log 0 8

1 Re f .

f = −

2.0 log 1 1. 4

f = e +