Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T11:02:50.581Z Has data issue: false hasContentIssue false
  • English
  • Français

Universality in statistics of Stokes flow over a no-slip wall with random roughness

Published online by Cambridge University Press:  17 January 2019

Vladimir Parfenyev Open the ORCID record for Vladimir Parfenyev [Opens in a new window] ,
Sergey Belan Open the ORCID record for Sergey Belan [Opens in a new window]  and
Vladimir Lebedev
Vladimir Parfenyev*
Affiliation:
Landau Institute for Theoretical Physics, Ak. Semenova 1-A, 142432, Chernogolovka, Russia National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000, Moscow, Russia
Sergey Belan
Affiliation:
Massachusetts Institute of Technology, Department of Physics, Cambridge, MA 02139, USA
Vladimir Lebedev
Affiliation:
Landau Institute for Theoretical Physics, Ak. Semenova 1-A, 142432, Chernogolovka, Russia National Research University Higher School of Economics, Faculty of Physics, Myasnitskaya 20, 101000, Moscow, Russia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Stochastic roughness is a widespread feature of natural surfaces and is an inherent byproduct of most fabrication techniques. In view of the rapid development of microfluidics, the important question is how this inevitable problem affects the low-Reynolds-number flows that are common for micro-devices. Moreover, one could potentially turn the flaw into a virtue and control the flow properties by means of specially ‘tuned’ random roughness. In this paper we investigate theoretically the statistics of fluctuations in fluid velocity produced by the waviness irregularities at the surface of a no-slip wall. Particular emphasis is laid on the issue of the universality of our findings.

JFM classification

Micro-/Nano-fluid dynamics: Microfluidics Waves/Free-surface Flows: Channel flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 1084 - 1104
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdo, B., El-Tamimi, A., Anwar, S., Umer, U., Alahmari, A. & Ghaleb, M. 2018 Experimental investigation and multi-objective optimization of Nd:Yag laser micro-channeling process of zirconia dental ceramic. Intl J. Adv. Manuf. Technol. 98 (5–8), 2213–2230.Google Scholar
Bhushan, B. 2001 Modern Tribology Handbook. vol. 1. CRC.Google Scholar
Bigerelle, M., Gautier, A. & Iost, A. 2007 Roughness characteristic length scales of micro-machined surfaces: a multi-scale modelling. Sensors Actuators B 126 (1), 126–137.Google Scholar
Chen, Y., Zhang, C., Shi, M. & Peterson, G. P. 2009 Role of surface roughness characterized by fractal geometry on laminar flow in microchannels. Phys. Rev. E 80 (2), 026301.Google Scholar
Darcy, H. 1857 Recherches Expérimentales Relatives au Mouvement de l’eau dans les tuyaux. Mallet-Bachelier.Google Scholar
Duparre, A., Ferre-Borrull, J., Gliech, S., Notni, G., Steinert, J. & Bennett, J. M. 2002 Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components. Appl. Opt. 41 (1), 154–171.Google Scholar
Gale, B., Jafek, A., Lambert, C., Goenner, B., Moghimifam, H., Nze, U. & Kamarapu, S. 2018 A review of current methods in microfluidic device fabrication and future commercialization prospects. Inventions 3 (3), 60.Google Scholar
Grzesik, W., Rech, J. & Żak, K. 2015 High-precision finishing hard steel surfaces using cutting, abrasive and burnishing operations. Procedia Manuf. 1, 619–627.Google Scholar
Hao, P.-F., Yao, Z.-H., He, F. & Zhu, K.-Q. 2006 Experimental investigation of water flow in smooth and rough silicon microchannels. J. Micromech. Microengng. 16 (7), 1397.Google Scholar
Jaeger, R., Ren, J., Xie, Y., Sundararajan, S., Olsen, M. G. & Ganapathysubramanian, B. 2012 Nanoscale surface roughness affects low Reynolds number flow: experiments and modeling. Appl. Phys. Lett. 101 (18), 184102.Google Scholar
Jansons, K. M. & Lister, J. R. 1988 The general solution of Stokes flow in a half-space as an integral of the velocity on the boundary. Phys. Fluids 31 (6), 1321–1323.Google Scholar
Jia, J., Song, Q., Liu, Zh. & Wang, B. 2018 Effect of wall roughness on performance of microchannel applied in microfluidic device. Microsystem Technologies. pp. 1–13. Springer.Google Scholar
Kandlikar, S. G., Joshi, S. & Tian, S. 2003 Effect of surface roughness on heat transfer and fluid flow characteristics at low Reynolds numbers in small diameter tubes. Heat Transfer Engng 24 (3), 4–16.Google Scholar
Kunert, C. & Harting, J. 2007 Roughness induced boundary slip in microchannel flows. Phys. Rev. Lett. 99 (17), 176001.Google Scholar
Kuo, C.-C. & Chao, C.-S. 2010 Rapid optical measurement of surface roughness of polycrystalline thin films. Opt. Lasers Engng 48 (12), 1166–1169.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd English edn (Course of Theoretical Physics) , vol. 6. Pergamon Press.Google Scholar
Lessen, M. & Huang, P.-S. 1976 Poiseuille flow in a pipe with axially symmetric wavy walls. Phys. Fluids 19 (7), 945–950.Google Scholar
MacDonald, M., Chung, D., Hutchins, N., Chan, L., Ooi, A. & García-Mayoral, R. 2017 The minimal-span channel for rough-wall turbulent flows. J. Fluid Mech. 816, 5–42.Google Scholar
Moody, L. F. 1944 Friction factors for pipe flow. Trans. ASME 66, 671–684.Google Scholar
Nikuradse, J.1950 Laws of flow in rough pipes. National Advisory Committee for Aeronautics, NACA TM 1292 (translation from VDI-Forschungsheft 361, 1933).Google Scholar
Panzer, P., Liu, M. & Einzel, D. 1992 The effects of boundary curvature on hydrodynamic fluid flow: calculation of slip lengths. Intl J. Mod. Phys. B 6 (20), 3251–3278.Google Scholar
Phan-Thien, N. 1980 On Stokes flow between parallel plates with stationary random surface roughness. Z. Angew. Math. Mech.-J. Appl. Math. Mech. 60 (12), 675–679.Google Scholar
Phan-Thien, N. 1981a On Stokes flow of a Newtonian fluid through a pipe with stationary random surface roughness. Phys. Fluids 24 (4), 579–582.Google Scholar
Phan-Thien, N. 1981b On Stokes flows in channels and pipes with parallel stationary random surface roughness. Z. Angew. Math. Mech.-J. Appl. Maths Mech. 61 (3–5), 193–199.Google Scholar
Pozrikidis, C. 1987 Creeping flow in two-dimensional channels. J. Fluid Mech. 180, 495–514.Google Scholar
Prakash, S. & Kumar, S. 2015 Fabrication of microchannels: a review. Proc. Inst. Mech. Engrs B 229 (8), 1273–1288.Google Scholar
Ren, J.2013 Micro/nano scale surface roughness tailoring and its effect on microfluidic flow. PhD thesis, Iowa State University.Google Scholar
Ren, J., Ganapathysubramanian, B. & Sundararajan, S. 2011 Experimental analysis of the surface roughness evolution of etched glass for micro/nanofluidic devices. J. Micromech. Microengng 21 (2), 025012.Google Scholar
Ren, J. & Sundararajan, S. 2012 Microfluidic channel fabrication with tailored wall roughness. In ASME 2012 International Manufacturing Science and Engineering Conference Collocated with the 40th North American Manufacturing Research Conference and in Participation with the International Conference on Tribology Materials and Processing, pp. 527–532. American Society of Mechanical Engineers.Google Scholar
Richardson, S. 1973 On the no-slip boundary condition. J. Fluid Mech. 59 (4), 707–719.Google Scholar
Silva, G., Leal, N. & Semiao, V. 2008 Micro-PIV and CFD characterization of flows in a microchannel: velocity profiles, surface roughness and Poiseuille numbers. Intl J. Heat Fluid Flow 29 (4), 1211–1220.Google Scholar
Taylor, J. B., Carrano, A. L. & Kandlikar, S. G. 2006 Characterization of the effect of surface roughness and texture on fluid flow past, present, and future. Intl J. Thermal Sci. 45 (10), 962–968.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar
Vinogradova, O. I. & Belyaev, A. V. 2011 Wetting, roughness and flow boundary conditions. J. Phys.: Condens. Matter 23 (18), 184104.Google Scholar
Wang, C.-Y. 1976 Parallel flow between corrugated plates. J. Engng Mech. Div. 102 (6), 1088–1090.Google Scholar
Wang, C.-Y. 1978 Drag due to a striated boundary in slow Couette flow. Phys. Fluids 21 (4), 697–698.Google Scholar
Wang, C.-Y. 1979 On Stokes flow between corrugated plates. J. Appl. Mech. 46 (2), 462–464.Google Scholar
Wang, C.-Y. 2006 Effect of helical corrugations on the low Reynolds number flow in a tube. AIChE J. 52 (6), 2008–2012.Google Scholar
Wang, H. & Wang, Y. 2007 Flow in microchannels with rough walls: flow pattern and pressure drop. J. Micromech. Microengng 17 (3), 586.Google Scholar
Wang, H., Wang, Y. & Zhang, J. 2005 Influence of ribbon structure rough wall on the microscale Poiseuille flow. J. Fluids Engng 127 (6), 1140–1145.Google Scholar
Wang, Z., Zheng, H., Lim, R., Wang, Z. & Lam, Y. 2011 Improving surface smoothness of laser-fabricated microchannels for microfluidic application. J. Micromech. Microengng 21 (9), 095008.Google Scholar
Yan, H., Zhang, W.-M., Peng, Z.-K. & Meng, G. 2015 Effect of random surface topography on the gaseous flow in microtubes with an extended slip model. Microfluid. Nanofluid. 18 (5–6), 897–910.Google Scholar
Yilbas, Z. & Hasmi, M. S. J. 1999 Surface roughness measurement using an optical system. J. Mater. Process. Technol. 88 (1), 10–22.Google Scholar
Young, P. L., Brackbill, T. P. & Kandlikar, S. G. 2009 Comparison of roughness parameters for various microchannel surfaces in single-phase flow applications. Heat Transfer Engng 30 (1–2), 78–90.Google Scholar
Yu, J., Cao, J. L., Namba, Y. & Ma, Y. Y. 1996 Surface roughness characterization of soft X-ray multilayer films on the nanometer scale. J. Vacuum Sci. Technol. B 14 (1), 42–47.Google Scholar
Zayernouri, M., Park, S.-W., Tartakovsky, D. M. & Karniadakis, G. E. 2013 Stochastic smoothed profile method for modeling random roughness in flow problems. Comput. Meth. Appl. Mech. Engng 263, 99–112.Google Scholar
Zhou, Z., Chen, D., Wang, X. & Jiang, J. 2017 Milling positive master for polydimethylsiloxane microfluidic devices: the microfabrication and roughness issues. Micromachines 8 (10), 287.Google Scholar
Zhu, Q. Z. & Zhang, Z. M. 2004 Anisotropic slope distribution and bidirectional reflectance of a rough silicon surface. J. Heat Transfer 126 (6), 985–993.Google Scholar