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the 6th Cornell conference on Fractals Huazhong University of Science and Technology on June 13–17, 2017, Cornell University A review on flow resistance in microchannels with rough surfaces by fractal geometry theory and technique Boming Yu School of Physics Huazhong University of Sci. & Tech. yubm_2012@hust.edu.cn http://blog.sciencenet.cn/?398451 Google Scholar: https://scholar.google.com/citations?user=_NmWuUQAAAAJ&hl=en 1 2017/6/16 2017/6/16

Huazhong University of Science and Technology Outlines 1. Introduction 2. Rough surface by fractal description 3. Models for simulating rough surfaces 4. Fractal geometry theory for rough surfaces 5. Flow resistance in micro channels 6. Other methodologies for flow resistance in roughened channels 7. Concluding remarks 2 2017/6/16 2017/6/16

Huazhong University of Science and Technology 1. Introduction Rough surfaces widely exist in natures such as road surface, airplane surface, metal surface, tube surface, channel surface, earth surface, etc. Absolutely smooth surface does not exists! Roughness of surfaces significantly influences the flow resistance when fluid flows through rough surfaces. 3 2017/6/16

Huazhong University of Science and Technology 2. Rough surface by fractal description 2.1 Description of typical rough surfaces 4 2017/6/16

Huazhong University of Science and Technology A. Majumdar et al., Journal of Tribology, APRIL 1990, Vol. 112, p205 A. Majumdar et al., ASME J. Tribol . 1991, 113: 1–11 5 2017/6/16

Huazhong University of Science and Technology An NOP image at 4000 um scan length and an AFM image at 50 um scan length for a lapped steel surface. Suryaprakash Ganti, et al., Wear 180 (1995) 17-34 6 2017/6/16

Huazhong University of Science and Technology 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 ：   n x cos 2        ( D 1 ) z x G D ( ) ; 1 2 ; 1   D n ( 2 )  n n 1 A. Majumdar et al., ASME J. Tribol . 1991, 113: 1–11 7 2017/6/16

Huazhong University of Science and Technology Rough surfaces of Fracture networks Natural surfaces, real fractures in rock, such as dry hot rock. I. G. Main, et al., Geological Society, London, Special Publications, 54: 81-96, 1990. 8 8 2017/6/16 2017/6/16

Fractured networks Huazhong University of Science and Technology Characters of fractures: --- Irregular --- Random --- Different apertures --- Different lengths --- extremely rough surfaces Heilbronner R., Keulen N. Grain size and grain shape analysis of fault rocks. Tectonophysics, 2006, 427(1):199-216. 9 9 2017/6/16 2017/6/16

Huazhong University of Science and Technology 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. (b) (c) (a) (a) Cross-section of a micro-channel tube (b) A profile of a rough surface of a micro-tube (c) Fluid distributor 10 2017/6/16

Huazhong University of Science and Technology Artery or vena vessel If fat is accumulated on the wall surface of artery, what will happen? High blood pressure happens!!! Antonets V.A.,et al. Fractal in the Fundamental and Applied Sciences. North-Holland: Elsevier, 1991. 59-71. 11 2017/6/16

Huazhong University of Science and Technology Afrin, N., et al. Int. J. Heat and Mass Transfer 54 (11): 2419-2426( 2011) . 12 2017/6/16

Huazhong University of Science and Technology 3. Models for simulating rough surfaces 3.1 Weierstrass-Mandelbrot (W-M) function Profile of a self-affine fractal surface Weierstrass-Mandelbrot (W-M) function can be used to describe the profile of a rough surface ：   n cos 2 x        ( D 1 ) z ( x ) G ; 1 D 2 ; 1   ( 2 D ) n  n n 1 where G is a characteristic length scale, D is the fractal dimension  of the roughness profile, and is the scaling parameter . A. Majumdar et al., ASME J. Tribol . 1991, 113: 1–11 13 2017/6/16

Huazhong University of Science and Technology 3.2 Cantor model for rough surfaces Rough surfaces can be characterized by fractal Cantor structures Cantor set 14 2017/6/16

Huazhong University of Science and Technology Thomas L. Warren et al., Wear 196, 1-15(1996) 15 Yongping Chen et al., Int. J. Heat and Fluid Flow 31, 622(2010) 2017/6/16

Huazhong University of Science and Technology 3.3 Random fractal spots for modeling rough surface   / ) D N L ( d ) ( d d Typical morphology max J.-H. Li, et al., Chin. Phys. Lett. 26 (11): 116101(2009) 16 2017/6/16

Huazhong University of Science and Technology 3.4 A rough surface simulated by Fractal- Monte Carlo method As D =1.25 and G =9.46  10 -13 m , a rough surface by simulation M.Q. Zou et al., Physica A 386, 176-186(2007). 17 2017/6/16

Huazhong University of Science and Technology 4. Fractal geometry theories for rough surfaces 4.1 Weierstrass-Mandelbrot (W-M) function Weierstrass-Mandelbrot (W-M) function can be used to describe the profile of a rough surface ：   n x cos 2        D ( 1 ) z x G D ( ) ; 1 2 ; 1   D n ( 2 )  n n 1 where G is a characteristic length scale, D is the fractal  dimension of the roughness profile, and is the scaling parameter . A. Majumdar et al., ASME J. Tribol . 1991, 113: 1–11 18 2017/6/16

Huazhong University of Science and Technology 4.2 Model by extension of the fractal scaling law Mandelbrot in his book: The Fractal Geometry of Nature proposed that the cumulative size distribution of islands on earth follows the fractal scaling law:     D /2 N A a ~ a f where N is the total number of islands of area ( A ) greater than a , and D f is the fractal dimension of the surface. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, New York, 1983 。 19 2017/6/16

Huazhong University of Science and Technology Marjumdar and Bhushan extended this power law to describe the contact spots on engineering surfaces, and the power-law relation is     D /2 f N A ( a ) a / a max a   2   , 2 g a g where and , and g is a max max  geometry factor. is a spot diameter. A. Majumdar et al., Journal of Tribology, April 1990, Vol. 112, p205 20 2017/6/16

Huazhong University of Science and Technology Yu et al. again extended the above equation to describe the pore size distribution in porous media by  max ) D    f N ( L ) (  B.M. Yu and P. Cheng, Int. J. Heat Mass Transfer, V. 45, No. 14, 2983-2993(2002). B.M. Yu, Analysis of flow in fractal porous media, Appl. Mech. Rev . 61, 050801(2008). 21 2017/6/16

Huazhong University of Science and Technology 5. Flow resistance in micro channels Flow resistance is usually defined by  P L /  P where is the pressure difference, and L represents the straight length. or by Friction factor:    2 f 2 /( u ) w m   u where , and are respectively the m w wall shear , fluid density and mean velocity in a channel. 22 2017/6/16

Huazhong University of Science and Technology 5.1 Flow resistance for laminar flow in micro-channels with 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 23 2017/6/16

Huazhong University of Science and Technology   2 2 u u 1 dp   (1)    2 2 z y dx where u is the velocity in the x -direction,  is the dynamic viscosity, x dp/dx is the pressure gradient along the flow direction, Assume b<<W, then, Eq. (1) can be simplified as 2 d u 1 dp  (2)  2 dz dx 24 2017/6/16

Huazhong University of Science and Technology Due to the symmetry of the channel, the no-slip boundary condition on wall is    b  z , u 0   2  (3) du    z 0, 0   dz Solving Eq. (2) with the boundary condition Eq. (3) yields 2 1 dp b   2 (4) u ( z )  2 dx 4 25 2017/6/16

Huazhong University of Science and Technology The mean velocity over the cross section can be obtained as b 2 1 1 dp b    2 u udz (5)  m b / 2 dx 12 0 The volume flow (let w=1 and b<<w ) rate is 3 b dp  b 2   Q udz (6)  12 dx - b 2 26 2017/6/16

Huazhong University of Science and Technology The wall shear in smooth channel: du b dp     (7a) w dz 2 dx b  z 2 Substituting Eq. (6) into Eq. (7a) yields  6 Q   (7b) w 2 b 27 2017/6/16

Huazhong University of Science and Technology From Eq. (6) we can obtain the pressure gradient across the length L as   P 12 Q （ ） (8) = S 3 L b Combining Eq. (5) and Eq. (7b) results in the fanning friction factor:   2 12   w f (9)   2 u u b m m 28 2017/6/16