An Essential Need of Modern Civilization…

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Micro & Compressible Viscous Fluid Flows in Ducts

Engineering Solution for Hagen-Poiseuille Flow

2** 2

4

1Cru

• Conventional engineering flows: Kn < 0.001

0** ws

uu

• Micro Fluidic Devices : Kn < 0.1

02

*

***

w

r

uKnuu

ws

• Ultra Micro Fluidic Devices : Kn <1.0

02

22

*

*22

*

***

ww r

uKn

r

uKnuu

ws

The Wall Boundary Conditions.or

us

uwWall

Micro Engineering Mild Slip Hagen-Poiseuille Flow

The first order slip condition:

For a flow through an immobile pipe:

w

r

uKnuu

ws *

*** 2

w

r

uKnu

s *

** 2

**

*

2

1r

r

u

2

2* Knu

walls

2

2

4

12

KnC

2** 2

4

1Cru

The micro engineering pipe-flow solution is thus

2

2

4

12*

* Knru

Mean & Maximum Flow Velocities

The Wall Shear Stress

Friction Factor

Popular Creeping Flows

• Fully developed duct Flow.

• Flow about immersed bodies

• Flow in narrow but variable passages. First formulated by Reynolds (1886) and known as lubrication theory,

• Flow through porous media. This topic began with a famous treatise by Darcy (1856

• Civil engineers have long applied porous-media theory to groundwater movement.

• http://www.ae.metu.edu.tr/~ae244/docs/FluidMechanics-by-JamesFay/2003/Textbook/Nodes/chap06/node17.html

Further Use of Mean Velocity for High Speed Flows

Compressible (Average) Frictional Flow in A Constant Area Duct

0V

dVd

02

2

VhVd

w

Self similar compressible fully developed flow through ducts

Frictional Flow in A Constant Area Duct

AdpPdxdVm w

w

The shear stress is defined as an average viscous stress which is always opposite to the direction of flow for the entire length dx.

AdpPdxAVdV w Divide by V2

22 V

dpdx

A

P

VV

dV w

0V

dVd

002

2

VdVdTC

Vhd p

1D steady real flow through constant area duct : momentum equation

02 2

V

dpdx

A

Pf

V

dV

02 2

p

dp

V

pdx

A

Pf

V

dV

02 2

p

dp

V

pdx

A

Pf

V

dV

22 V

dpdx

A

P

VV

dV w

02 2

p

dp

V

pdx

A

Pf

V

dV

02 2

p

dp

V

p

dxA

Pf

V

dV

01

2 2

p

dp

Mdx

A

Pf

V

dV

One dimensional Frictional Flow of A Perfect Gas

0V

dVd

0VdVdTC p

T

dTd

p

dp

T

dT

V

dV

p

dp

01

2 2

p

dp

Mdx

A

Pf

V

dV

Sonic Equation

RT

V

c

VM

2

2

22

Differential form of above equation:

T

dT

V

dV

M

dM

2

T

dT

V

dV

p

dp

T

dT

M

dM

p

dp

2

2

222

RT

dTV

RT

VdVMdM

M

dM

M

M

T

dT

2

2

21

1

1

Energy equation can be modified as:

T

dT

M

dM

p

dp

2

M

dM

M

M

M

dM

p

dp

2

2

21

1

1

2

1

01

2 2

p

dp

Mdx

A

Pf

V

dV

M

dM

M

M

T

dT

2

2

21

1

1

M

dM

M

M

M

dM

p

dp

2

2

21

1

1

2

1

T

dT

V

dV

M

dM

2

Differential Equations for Frictional Flow Through Constant Area Duct

01

2 2

p

dp

Mdx

A

Pf

T

dT

M

dM

0

2

11

1

2

11

22

11

1

2

2

22

2

M

dM

M

M

M

dM

Mdx

A

Pf

M

dM

M

M

M

dM

dxA

Pf

M

MM

M

dM

212

11

2

22

Differential Equations for Frictional Flow Through Constant Area Duct