Huazhong University of Science and Technology the 6th ...pi.math.cornell.edu/~fractals/6/slides/Yu.pdfThomas L. Warren et al., Wear 196, 1-15(1996) Huazhong University of Science and

$Page 1: Huazhong University of Science and Technology the 6th ...pi.math.cornell.edu/~fractals/6/slides/Yu.pdfThomas L. Warren et al., Wear 196, 1-15(1996) Huazhong University of Science and$
Huazhong University of Science and Technology

2017/6/16 12017/6/16

Boming YuSchool of Physics

Huazhong University of Sci. & [email protected]

A review on flow resistance in microchannels with rough surfaces by fractal geometry theory and technique

http://blog.sciencenet.cn/?398451Google Scholar:

https://scholar.google.com/citations?user=_NmWuUQAAAAJ&hl=en

the 6th Cornell conference on Fractalson June 13–17, 2017, Cornell University

http://blog.sciencenet.cn/?398451

https://scholar.google.com/citations?user=_NmWuUQAAAAJ&hl=en


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Outlines1. Introduction2. Rough surface by fractal description

7. Concluding remarks

3. Models for simulating rough surfaces4. Fractal geometry theory for rough surfaces5. Flow resistance in micro channels6. Other methodologies for flow resistance in

roughened channels


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1. IntroductionRough surfaces widely exist in natures such as roadsurface, airplane surface, metal surface, tube surface,channel surface, earth surface, etc.

Roughness of surfaces significantly influences the flowresistance when fluid flows through rough surfaces.

Absolutely smooth surface does not exists!


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2. Rough surface by fractal description

2.1 Description of typical rough surfaces


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A. Majumdar et al., Journal of Tribology, APRIL 1990, Vol. 112, p205

A. Majumdar et al., ASME J. Tribol. 1991, 113: 1–11


2017/6/16 6Suryaprakash Ganti, et al., Wear 180 (1995) 17-34

An NOP image at 4000 um scan length and an AFM image at 50 um scan length for a lapped steel surface.


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2.2 A self-affine fractal surface

Profile of a self-affine fractal surface

Weierstrass-Mandelbrot (W-M) function can be widely used to describe the profile of a rough surface ：

1;21;2cos)(1

)2()1(

DxGxznn

nD

nD



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I. G. Main, et al., Geological Society, London, Special Publications, 54: 81-96, 1990.

Natural surfaces,real fracturesin rock, such asdry hot rock.

Rough surfaces of Fracture networks


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Heilbronner R., Keulen N. Grain size and grain shape analysis of fault rocks. Tectonophysics, 2006, 427(1):199-216.

Characters of fractures:--- Irregular--- Random--- Different apertures--- Different lengths--- extremely rough surfaces

Fractured networks


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Consider oil/gas/water flowing in such fractures/tubes, the effects of roughness of surfaces on flow in channels/fractures should be taken into accounted.

(a) (b)

(a) Cross-section of a micro-channel tube(b) A profile of a rough surface of a micro-tube(c) Fluid distributor

(c)


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Artery or vena vessel

Antonets V.A.,et al. Fractal in the Fundamental and Applied Sciences. North-Holland: Elsevier, 1991. 59-71.

If fat is accumulated on the wall surface of artery, what will happen? High blood pressure happens!!!


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Afrin, N., et al. Int. J. Heat and Mass Transfer 54 (11): 2419-2426(2011).


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3. Models for simulating rough surfaces

Profile of a self-affine fractal surfaceWeierstrass-Mandelbrot (W-M) function can be used to describe the profile of a rough surface ：

1;21;2cos)(1

)2()1(

DxGxznn

nD

nD


3.1 Weierstrass-Mandelbrot (W-M) function

where G is a characteristic length scale, D is the fractal dimensionof the roughness profile, and is the scaling parameter.


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3.2 Cantor model for rough surfacesRough surfaces can be characterized by fractal Cantor structures

Cantor set


2017/6/16 15Yongping Chen et al., Int. J. Heat and Fluid Flow 31, 622(2010)

Thomas L. Warren et al., Wear 196, 1-15(1996)


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3.3 Random fractal spots for modeling rough surface

max( ) ( / )DN L d d d Typical morphology

J.-H. Li, et al., Chin. Phys. Lett. 26 (11): 116101(2009)


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As D=1.25 and G=9.4610-13m, a rough surface by simulation

3.4 A rough surface simulated by Fractal- Monte Carlo method

M.Q. Zou et al., Physica A 386, 176-186(2007).


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4. Fractal geometry theories for rough surfaces

Weierstrass-Mandelbrot (W-M) function can be used to describe the profile of a rough surface ：

1;21;2cos)(1

)2()1(

DxGxznn

nD

nD


4.1 Weierstrass-Mandelbrot (W-M) function

where G is a characteristic length scale, D is the fractal dimension of the roughness profile, and is the scaling parameter.


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/2~ fDN A a a

Mandelbrot in his book: The Fractal Geometry of Natureproposed that the cumulative size distribution of islands on earth follows the fractal scaling law:

where N is the total number of islands of area (A) greater than a, and Df is the fractal dimension of the surface.

B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, New York, 1983。

4.2 Model by extension of the fractal scaling law


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Marjumdar and Bhushan extended this power law to describe the contact spots on engineering surfaces, and the power-law relation is

/ 2max( ) / fDN A a a a

A. Majumdar et al., Journal of Tribology, April 1990, Vol. 112, p205

2maxmax ga

2ga where and , and g is a geometry factor.

,

is a spot diameter.


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Yu et al. again extended the above equation to describe the pore size distribution in porous media by

fDmax )()L(N

B.M. Yu, Analysis of flow in fractal porous media, Appl. Mech. Rev. 61, 050801(2008).

B.M. Yu and P. Cheng, Int. J. Heat Mass Transfer,V. 45, No. 14, 2983-2993(2002).


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5. Flow resistance in micro channels

Flow resistance is usually defined by

/P LPwhere is the pressure difference, and L

represents the straight length. or by Friction factor:

22 /( )w mf u where , and are respectively the wall shear, fluid density and mean velocity in a channel.

w mu


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5.1 Flow resistance for laminar flow in micro-channelswith smooth surfaces

For fully-developed, laminar, incompressible flow in a smooth rectangular microchannel with the height and width being respectively b and w, the equation of motion is


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2 2

2 2

1

u u dp

z y dxwhere u is the velocity in the x-direction, is the dynamic viscosity,

dp/dx is the pressure gradient along the flow direction,

x

Assume b<<W, then, Eq. (1) can be simplified as2

2

1

d u dpdz dx

(1)

(2)


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Due to the symmetry of the channel, the no-slip boundary condition on wall is

, 02

0, 0

bz u

duzdz

(3)

Solving Eq. (2) with the boundary condition Eq. (3) yields2

21 ( z )2 4

dp budx

(4)


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The mean velocity over the cross section can be obtained as2

20

1 1/ 2 12

b

mdp bu udz

b dx(5)

The volume flow (let w=1 and b<<w) rate is 32

- 2 12b

b

b dpQ udzdx

(6)


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The wall shear in smooth channel:

22

wbz

du b dpdz dx

(7a)

Substituting Eq. (6) into Eq. (7a) yields

2

6w

Qb (7b)


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From Eq. (6) we can obtain the pressure gradient across the length L as

3

12=（）SP QL b

(8)

Combining Eq. (5) and Eq. (7b) results in the fanning friction factor:

2

2 12w

m m

fu u b

(9)


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The Reynolds number is

Re

m hu D(10)

Since b<<w, the hydraulic diameter Dh can be simplified as

bDh 224 24 / Re

m h

fu D

(11)

(12)


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The Poiseuille number Po for a fully developed laminar flow in an infinite plate channel is

Re 24oP f (13)Similarly, we can obtain the friction factor f for a fully developed laminar flow through a smooth circular tube

64 / Ref (14)

(15)and the Poiseuille number Po:

is

Re 64Po f


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5.2 Flow resistance for laminar flow in micro-channelswith rough surfaces by fractal geometry

max( ) ( / )DN L d d d Typical morphology

J.-H. Li, et al., Chin. Phys. Lett. 26 (11): 116101(2009)


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The ratio of height to base diameter of conic peak is

h As shown in Fig. 1 (b), the base area for a conic peak/spot is

The effective average height of conic roughness elements can be found to be

2 / 4i iS

3max 2 1

3 3 1

Ds

eff

s

DhD

(16)

(17)

(18)


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The non-slip boundary on the walls of roughened microchannels is

( / 2 ), 0

0, 0

eff R

R

z b h uuzz

Solving Eq. (2) with the boundary condition Eq. (19) yields

(19)

2 21 [( ) ]2 2

effRdp bu h zdx

(20)


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The volume flow (when w=1 and b<<w) in roughened microchannel is

32

2

( 2 )12

eff

eff

b h effb Rh

b h dpQ u dzdx

(21)

The pressure gradient in roughened microchannel is

3

12=( -2 )

（）Reff

P QL b h

(22)


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Compared to the pressure gradient in smooth channels,Eq. (22) can be rewritten as

= )（）（R S RP P FL L (23)

where

3

1(1 )R

r

F

2 /r effh b and

where r is defined as the relative roughness in rectangular roughened microchannels.


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The relative increase of the pressure gradient is defined by

3

( ) ( ) 1 1 1(1 )( )

R S

R Rr

S

P PL L FP

L

(24)

The friction factor in rough channels can be obtained as 24ReR Rf F (25)

where FR>1, and friction factor is increased and similarresults for flow in rough cylindrical tube.

24R RPo F=and


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Reference

S.S Yang, et al., A fractal analysis of laminar flow resistance in roughened microchannels, Int. J. Heat Mass Transfer 77, 208-217(2014).


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6. Other methodologies for flow resistance in roughened channels

6.1 Numerical simulations

Y.P. Chen, et al., Int. J. Heat and Fluid Flow 31 (2010) 622–629

The Gauss–Seidal iterative technique, with successive over-relaxation to improve the convergence time.


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6.2 The Lattice Boltzmann method (LBM)

C.B. Zhang, et al., Int. J. Heat and Mass Transfer 70: 322 (2014)

Schematic of gas flow heat transfer in a rough micriochannel.


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7. Concluding remarksAnalytical solution for flow resistance in roughened channels can be obtained based on fractal geometry, but it was impossible based on Euclid geometry.

The flow resistance in roughened channels based on Weierstrass-Mandelbrot (W-M) function is open.The flow resistance in roughened channels based onCantor set model is also open.The flow resistance in roughened natural fracturesbased on fractal geometry is also open.


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You are welcome to submit your original manuscript to Fractals journal for publication at:

IF=1.22, 2014IF=1.412, 2015IF=1.540, 2016IF is higher in the subject of Mathematics.Publishing original papers in:Fractals in Sciences;Fractals in Engineering;Fractals in Mathematics.

http://www.worldscientific.com/worldscinet/fractals

http://www.worldscientific.com/worldscinet/fractals


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Documents

Huazhong University of Science and Technology the 6th ...pi.math.cornell.edu/~fractals/6/slides/Yu.pdfThomas L. Warren et al., Wear 196, 1-15(1996) Huazhong University of Science and