Natural Fluid Flow to Engineering Fluid Flow…

P M V SubbaraoProfessor

Mechanical Engineering DepartmentI I T Delhi

Poiseuille (pressure-driven) steady duct flows

J. L. M. Poiseuille

• Poiseuille possess extraordinary sense of experimental precision.

• He carried out his doctoral research on, ”The force of the aortic heart” in 1828.

• Poiseuille invented the U-tube mercury manometer (called the hemodynamometer) and used it to measure pressures in the arteries of horses and dogs.

• To this day blood pressures are reported in mm Hg due to Poiseuille's invention.

Stud of Blood Flow in Capillaries Using Liquid flow in Glass Tubes

• Poiseuille set out to find a functional relationship among four variables: the volumetric efflux rate of distilled water from a tube Q, the driving pressure differential P, the tube length L, and the tube diameter D.

• Poiseuille summarized his findings at first stage by the equation Q = KP.

• The coefficient K was a function, to be determined, of tube length, diameter, and temperature.

Law of Lengths

• By investigating the influence of tube length Poiseuille was able to show that the flow time was proportional to tube length (the "law of lengths").

• At this point Poiseuille could state that K = K' / L.• Therefore, Q = K' P / L, where K' was a function of tube

diameter and temperature.

Origin of Hydraulic Diameter

• To assign a diameter to one of his noncircula':, noncylindrical tubes, Poiseuille first calculated a geometrical average diameter for each end.

• This was defined as the diameter of the circle having the same area as an ellipse with the maximum and minimum diameters of the tube section.

• The arithmetic average of the geometrical means at the two ends was taken as the average diameter of the tube.

• To determine the effect of tube diameter on flow, Poiseuille analyzed the data of seven of his previous experiments from which he was able to discern that the efflux volumes (in 500 s) varied directly as the fourth power of the average diameter.

Poiseuille Law of Flow through Tubes

• K" being simply a function of temperature and the type of liquid flowing.

• For l0C, average value of K" = 2495.224 for distilled water expressed in mixed units of (mg/s)/(mm Hg) mm3•

LPdKQ

4''

Poiseuille Flow through Ducts• Whereas Couette flow is driven by moving walls, Poiseuille

flows are generated by pressure gradients, with application primarily to ducts.

• For liquid flows:

i

j

j

i

jii

j

ij

i

xv

xv

xxpg

xvv

tv

0 v

Flow That follows Poiseuille’s Laws

Regardless of duct shape, the entrance length can be correlatedfor laminar flow in the form

hDeL Re05.05.0

Fully Developed Duct Flow• For x > Le, the velocity becomes purely axial and varies

only with the lateral coordinates.• v = w = 0 and u = u(y,z). • The flow is then called fully developed flow.

For fully developed flow, the continuity and momentum equations for incompressible flow are simplified as:

xp

zu

yu

2

2

2

2

With 0&0

zp

yp

xu

These indicate that the pressure p is a function of x only for this fully developed flow. Further, since u does not vary with x, it follows from the x-momentum equation that the gradient dp/dx must only be a (negative) constant. Then the basic equation of fully developed duct flow is

xp

zu

yu

2

2

2

2

subject only to the slip/no-slip condition everywhere on the duct surface

This is the classic Poisson equation and is exactly equivalent to the torsional stress problem in elasticity

Characteristics of Poiseuille Flow

• Like the Couette flow problems, the acceleration terms vanish here, taking the density with them.

• These flows are true creeping flows in the sense that they are independent of density.

• The Reynolds number is not even a required parameter • There is no characteristic velocity U and no axial length

scale L either, since we are supposedly far from the entrance or exit.

• The proper scaling of Poiseuille Equation should include , dp/dx, and some characteristic duct width h.

Dimensionless variables for Poiseuille Flow

hyy *

hzz *

dxdph

uu2

*

Dimensionless Poiseuille Equation

1*2

*2

*2

*2

zu

yu

The Circular Pipe: Hagen-Poiseuille Flow

• The circular pipe is perhaps our most celebrated viscous flow, first studied by Hagen (1839) and Poiseuille (1840).

• The single variable is r* = r/R, where R, is the pipe radius. • The equation reduces to an ODE:

11*

**

**

drdur

drd

r

The solution of above Equation is: 2*

1** ln

41 2

CrCru

•Engineering Conditions: •The velocity cannot be infinite at the centerline.• On engineering grounds, the logarithm term must be rejected and set C1 = 0.

Engineering Solution for Hagen-Poiseuille Flow

2** 2

41 Cru

• Conventional engineering flows: Kn < 0.001

0** ws

uu• Micro Fluidic Devices : Kn < 0.1

02*

***

wr

uKnuuws

• Ultra Micro Fluidic Devices : Kn <1.0

02

22

*

*22

*

***

ww r

uKnruKnuu

ws

The Wall Boundary Conditions

Macro Engineering No-Slip Hagen-Poiseuille Flow

The no-slip condition:

2** 141 ru The macro engineering pipe-flow solution is thus

0* w

u

041

2** 2

CRuw

1 as 0 *** ruuws

For a flow through an immobile pipe:

41

2 C

Dimensional Solution to Macro Engineering No-Slip Hagen-Poiseuille Flow

dxdpR

uu2

*

Rrr *

2** 141 ru

2

21

41

Rr

dxdpR

u

2

2

141

Rrdx

dpRu

22

4rRdx

dpu

The Capacity of A Pipe

• Thus the velocity distribution in fully developed laminar pipe flow is a paraboloid of revolution about the centerline.

• This is called as the Poiseuille paraboloid. • The total volume rate of flow Q is of interest, as defined for

any duct by

tioncross

udAQsec

Rr

rrudrQ

02

Rr

rdrrRrdx

dpQ

0

22

2

dxdpRQ

8

4

Mean & Maximum Flow Velocities

• The mean velocity is defined by

AQu

• The maximum velocity occurs at the center, r=0.

4

2

max

Rdxdp

u

dxdpR

RdxdpR

u

88 2

2

4

21

max

u

u

The Wall Shear Stress

• The wall shear stress is given bywall

wall drdu

22

4rRdx

dpu

2

rdxdp

drdu

dxdpR

drdu

wallwall 2

Ru

wall 4

Friction Factor

w is proportional to mean velocity.• It is customary, to nondimensionalize wall shear with the

pipe dynamic pressure.

DuuC wall

f

16

22

Two different friction factor definitions are in common use in the literature:

Darcy Friction Factor

This is called as standard Fanning friction factor, or skin-friction coefficient.

Re16

fC

Re6464

2

82

Duuwall

Hagen’s Pipe Flow Experiments

• Hagen was born in Köni gsberg, East Prussia, and studied there, having among his teachers the famous mathematician Bessel.

• He became an engineer, teacher, and writer and published a handbook on hydraulic engineering in 1841.

• He is best known for his study in 1839 of pipe-flow resistance, for water flow.

• At heads of 0.7 to 40 cm, diameters of 2.5 to 6 mm, and lengths of 47 to 110 cm.

• The measurements indicated that the pressure drop was proportional to Q at low heads.

Hagen’s Paradox

Engineering Solution for Hagen-Poiseuille Flow

2** 2

41 Cru

• Conventional engineering flows: Kn < 0.001

0** ws

uu• Micro Fluidic Devices : Kn < 0.1

02*

***

wr

uKnuuws

• Ultra Micro Fluidic Devices : Kn <1.0

02

22

*

*22

*

***

ww r

uKnruKnuu

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:

wr

uKnuuws *

*** 2

wr

uKnus *

** 2

**

*

21 r

ru

22* Knu

walls

22

41

2KnC

2** 2

41 Cru

The micro engineering pipe-flow solution is thus

2

24

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