# Effective Boundary Slip Induced by Surface Roughness and Their Coupled Effect on Convective Heat Transfer of Liquid Flow

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Model

_{p}of the reference surface position above the bottom can be calculated as [44,45]:

^{5}Pa), and room temperature is 293.15 K. Then the density of water ρ = 0.9982071 g/cm

^{3}, and dynamic viscosity μ = 1.0050 × 10

^{−3}Pa⋅s [46]. Based on the setup, the flow field can be assumed to remains laminar. p

_{in}is set as a constant at the inlet of the channel while p

_{out}is set as 0 Pa at the outlet of the channel. Moreover, the lateral sides of the fluid flow field are defined by symmetric plane. Based on the simulation by COMSOL 5.3, the flow rate Q can be obtained for each model.

_{eff}, then the relationship between effective slip b

_{eff}and flow rate Q can be described as [47]:

_{eff}can be obtained with Equation (3). The height H of the channel need to be calculated for each model by H = H

^{*}− m

_{p}considering the surface roughness.

## 3. Results and Discussion

^{5}. The results are shown as following.

#### 3.1. Effective Boundary Slip

_{AFM}. Then the results were corrected by simply subtract the distance between the two different reference surface position which is referred to the R

_{pm}of the surface. The corrected effective boundary slip is marked as b

_{AFM-C}. Then the error of the corrected slip are obtained and shown in Figure 4. In Figure 3, it is shown that with smaller Rsm, the errors are always smaller than ±5% which is acceptable for slip measurement. However, when the Rsm reaches 1000 nm or even 2000 nm, the error can be as large as 15%. Error as 15% for boundary slip measurement seems to be too large, however, it should be noticed that the error of the uncorrected experimentally measured slip is in range of 60–600%, as a result, the correction is still necessary and the error is acceptable, especially when Rsm is smaller than 200 nm.

#### 3.2. Convective Heat Transfer

_{0}. The variation of Nusselt number with different Ra is shown in Figure 5. From Figure 5, it can be found that when the Ra increases from 0, the Nusselt number increases faster when Ra is lower than 25 nm, followed by a slow and unobvious increase. The increasing Nusselt number with the increasing Ra when Ra is small is in well agreement with the previous theoretical and experimental studies [48,49,50], this is because that the increase of Ra extends the real contact area of solid and liquid, which can enhance the convective heat transfer. In the meanwhile, the effective boundary slip decreases continually. However, when Ra is larger enough, the decreased effective boundary slip induced by surface roughness will lead to a significant reduction of the liquid flow velocity, which will weaken the convective heat transfer. The enhancement of Ra on the heat transfer by increasing the real contact area of solid and liquid and the reduction of Ra on the heat transfer by decreasing the effective slip counteract each other.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Stokes, V.K. Couple stresses in fluids. Phys. Fluids
**1996**, 9, 1709–1715. [Google Scholar] [CrossRef] - Vinogradova, O.I. Drainage of a Thin Liquid-Film Confined between Hydrophobic Surfaces. Langmuir
**1995**, 11, 2213–2220. [Google Scholar] [CrossRef] - Ou, J.; Perot, B.; Rothstein, J.P. Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids
**2004**, 16, 4635–4643. [Google Scholar] [CrossRef] - Choi, C.-H.; Kim, C.-J. Large Slip of Aqueous Liquid Flow over a Nanoengineered Superhydrophobic Surface. Phys. Rev. Lett.
**2006**, 96, 066001. [Google Scholar] [CrossRef] [PubMed] - Steinberger, A.; Cottin-Bizonne, C.; Kleimann, P.; Charlaix, E. High friction on a bubble mattress. Nat. Mater.
**2007**, 6, 665–668. [Google Scholar] [CrossRef] [PubMed] - Lee, C.; Kim, C.J. Maximizing the Giant Liquid Slip on Superhydrophobic Microstructures by Nanostructuring Their Sidewalls. Langmuir
**2009**, 25, 12812–12818. [Google Scholar] [CrossRef] [PubMed] - Wang, Y.; Bhushan, B. Boundary slip and nanobubble study in micro/nanofluidics using atomic force microscopy. Soft Matter
**2010**, 6, 29–66. [Google Scholar] [CrossRef] - Pan, Y.; Bhushan, B.; Zhao, X. The study of surface wetting, nanobubbles and boundary slip with an applied voltage: A review. Beilstein J. Nanotech.
**2014**, 5, 1042–1065. [Google Scholar] [CrossRef] [PubMed] - Di Federicao, V.; Longo, S.; King, S.E.; Chiapponi, L.; Petrolo, D.; Ciriello, V. Gravity-driven flow of Herschel–Bulkley fluid in a fracture and in a 2D porous medium. J. Fluid Mech.
**2017**, 821, 59–84. [Google Scholar] [CrossRef] - Longo, S.; Chiapponi, L.; Di Federicao, V. On the propagation of viscous gravity currents of non-Newtonian fluids in channels with varying cross section and inclination. J. Non-Newton Fluid
**2016**, 235, 95–108. [Google Scholar] [CrossRef] - Choi, W.; Tuteja, A.; Chhatre, S.; Mabry, J.M.; Cohen, R.E.; McKinley, G.H. Fabrics with tunable oleophobicity. Adv. Mater.
**2009**, 21, 2190–2195. [Google Scholar] [CrossRef] - Brown, P.S.; Bhushan, B. Bioinspired, roughness-induced, water and oil super-philic and super-phobic coatings prepared by adaptable layer-by-layer technique. Sci. Rep.
**2015**, 5, 14030. [Google Scholar] [CrossRef] [PubMed] - Li, F.; Wang, Z.; Huang, S.; Pan, Y.; Zhao, X. Flexible, Durable, and Unconditioned Superoleophobic/Superhydrophilic Surfaces for Controllable Transport and Oil–Water Separation. Adv. Func. Mater.
**2018**, 28. [Google Scholar] [CrossRef] - Bakli, C.; Chakraborty, S. Slippery to Sticky Transition of Hydrophobic Nanochannels. Nano Lett.
**2015**, 15, 7497–7502. [Google Scholar] [CrossRef] [PubMed] - Vo, T.Q.; Park, B.; Park, C.; Kim, B. Nano-scale liquid film sheared between strong wetting surfaces: Effects of interface region on the flow. J. Mech. Sci. Technol.
**2015**, 29, 1681–1688. [Google Scholar] [CrossRef] - Bao, L.; Priezjev, N.V.; Hu, H.; Luo, K. Effects of viscous heating and wall-fluid interaction energy on rate-dependent slip behavior of simple fluids. Phys. Rev. E
**2017**, 96. [Google Scholar] [CrossRef] [PubMed] - Ghorbanian, J.; Beskok, A. Scale effects in Nano-channel liquid flows. Microfluid Nanofluid
**2016**, 20, 121. [Google Scholar] [CrossRef] - Jeong, M.; Kim, Y.; Zhou, W.; Tao, W.; Ha, M.Y. Effects of surface wettability, roughness and moving wall velocity on the Couette flow in nano-channel using multi-scale hybrid method. Comput. Fluid.
**2017**, 147, 1–11. [Google Scholar] [CrossRef] - Noorian, H.; Toghraie, D.; Azimian, A.R. Molecular dynamics simulation of Poiseuille flow in a rough nano channel with checker surface roughnesses geometry. Heat Mass Transf.
**2014**, 50, 105–113. [Google Scholar] [CrossRef] - Cao, B.; Chen, M.; Guo, Z. Effect of surface roughness on gas flow in microchannels by molecular dynamics simulation. Int. J. Eng. Sci.
**2006**, 44, 927–937. [Google Scholar] [CrossRef] - Nosonovsky, M.; Bhushan, B. Multiscale Dissipative Mechanisms and Hierarchical Surfaces: Friction, Superhydrophobicity, and Biomimetics; Springer: Heidelberg, Germany, 2008. [Google Scholar]
- Baldoni, F. On slippage induced by surface diffusion. J. Eng. Math.
**1996**, 30, 647–659. [Google Scholar] [CrossRef] - Jabbarzadeh, A.; Atkinson, J.D.; Tanner, R.I. Effect of the wall roughness on slip and rheological properties of hexadecane in molecular dynamics simulation of Couette shear flow between two sinusoidal walls. Phys. Rev. E
**2000**, 61, 690–699. [Google Scholar] [CrossRef] - Pit, R.; Hervet, H.; Leger, L. Direct experimental evidence of slip in hexadecane: Solid interfaces. Phys. Rev. Lett.
**2000**, 85, 980–983. [Google Scholar] [CrossRef] [PubMed] - Ponomarev, I.V.; Meyerovich, A.E. Surface roughness and effective stick-slip motion. Phys. Rev. E
**2003**, 67, 026302. [Google Scholar] [CrossRef] [PubMed] - Bonaccurso, E.; Butt, H.-J.; Craig, V.S.J. Surface Roughness and Hydrodynamic Boundary Slip of a Newtonian Fluid in a Completely Wetting System. Phys. Rev. E
**2003**, 90, 144501. [Google Scholar] [CrossRef] [PubMed] - Zhu, Y.; Granick, S. Limits of the Hydrodynamic No-Slip Boundary Condition. Phys. Rev. Lett.
**2002**, 88, 106102. [Google Scholar] [CrossRef] [PubMed] - Schmatko, T.; Hervet, H.; Léger, L. Effect of nanometric-scale roughness on slip at the wall of simple fluids. Langmuir
**2006**, 22, 6843–6850. [Google Scholar] [CrossRef] [PubMed] - Guriyanova, S.; Semin, B.; Rodrigues, T.S.; Butt, H.-J.; Bonaccurso, E. Hydrodynamic drainage force in a highly confined geometry: Role of surface roughness on different length scales. Microfluid Nanofluid
**2010**, 8, 653–663. [Google Scholar] [CrossRef] - Pan, Y.; Jing, D.; Zhao, X. Effect of Surface Roughness on the Measurement of Boundary Slip Based on Atomic Force Microscope. Sci. Adv. Mater.
**2017**, 9, 122–127. [Google Scholar] [CrossRef] - Vinogradova, O.I.; Yakubov, G.E. Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E
**2006**, 73, 045302(R). [Google Scholar] [CrossRef] [PubMed] - Cottin-Bizonne, C.; Cross, B.; Steinberger, A.; Charlaix, E. Boundary slip on smooth hydrophobic surfaces: Intrinsic effects and possible artifacts. Phys. Rev. Lett.
**2005**, 94, 056102. [Google Scholar] [CrossRef] [PubMed] - Bhushan, B.; Wang, Y.; Maali, A. Boundary Slip Study on Hydrophilic, Hydrophobic, and Superhydrophobic Surfaces with Dynamic Atomic Force Microscopy. Langmuir
**2009**, 25, 8117–8121. [Google Scholar] [CrossRef] [PubMed] - Jing, D.; Bhushan, B. The coupling of surface charge and boundary slip at the solid–liquid interface and their combined effect on fluid drag: A review. J. Colloid Interface Sci.
**2015**, 454, 152–179. [Google Scholar] [CrossRef] [PubMed] - Kandlikar, S.G.; Garimella, S.; Li, D.Q.; Colin, S.; King, M.R. Heat Transfer and Fluid Flow in Minichannels and Microchannels; Elsevier: Oxford, UK, 2006. [Google Scholar]
- Li, D. Encyclopedia of Microfluidics and Nanofluidics; Springer Science & Business Media: Berlin, Germany, 2008. [Google Scholar]
- Lin, B. Microfluidics: Technologies and Applications; Springer: Berlin, Germany, 2011. [Google Scholar]
- Ngoma, G.D.; Erchiqui, F. Heat flux and slip effects on liquid flow in a microchannel. Int. J. Therm. Sci.
**2007**, 46, 1076–1083. [Google Scholar] [CrossRef] - Yavari, H.; Sadeghi, A.; Saidi, M.H.; Chakraborty, S. Combined influences of viscous dissipation, non-uniform Joule heating and variable thermophysical properties on convective heat transfer in microtubes. Int. J. Heat Mass Transf.
**2012**, 55, 762–772. [Google Scholar] [CrossRef] - Tan, D.K.; Liu, Y. Combined effects of streaming potential and wall slip on flow and heat transfer in microchannels. Int. Commun. Heat Mass
**2014**, 53, 39–42. [Google Scholar] [CrossRef] - Keramati, H.; Sadeghi, A.; Saidi, M.H.; Chakraborty, S. Analytical solutions for thermo-fluidic transport in electroosmotic flow through rough microtubes. Int. J. Heat Mass Transf.
**2016**, 92, 244–251. [Google Scholar] [CrossRef] - Jing, D.; Pan, Y. Electroviscous effect and convective heat transfer of pressure-driven flow through microtubes with surface charge-dependent slip. Int. J. Heat Mass Transf.
**2016**, 101, 648–655. [Google Scholar] [CrossRef] - Jing, D.; Pan, Y.; Wang, X. Joule heating, viscous dissipation and convective heat transfer of pressure-driven flow in a microchannel with surface charge-dependent slip. Int. J. Heat Mass Transf.
**2017**, 108, 1305–1313. [Google Scholar] [CrossRef] - Anonymous. Geometrical Product Specifications—Surface Texture: Profile Method-Nominal Characteristics of Contact (Stylus) Instruments. ISO3274. International Standardization Organization: Geneva, Switzerland, 1 December 1996. [Google Scholar]
- Anonymous. Surface Texture (Surface Roughness, Waviness, and Lay). ANSI/ASME B46.1-2009. ASME: New York, NY, USA, 20 August 2010. [Google Scholar]
- Haynes, W.M. Handbook of Chemistry and Physics, 96th ed.; Academic Press: New York, NY, USA, 2015. [Google Scholar]
- Huang, Y.; Zhao, X.; Pan, Y.; Ahmad, K. Simulation of effective slip and drag in pressure-driven flow on superhydrophobic surfaces. J. Nanomater.
**2016**, 2016. [Google Scholar] [CrossRef] - Li, Z.; He, Y.L.; Tang, G.H.; Tao, W.Q. Experimental and numerical studies of liquid flow and heat transfer in microtubes. Int. J. Heat. Mass Transf.
**2007**, 50, 3447–3460. [Google Scholar] [CrossRef] - Wu, H.Y.; Cheng, P. An experimental study of convective heat transfer in silicon microchannels with different surface conditions. Int. J. Heat. Mass Transf.
**2003**, 46, 2547–2556. [Google Scholar] [CrossRef] - Guo, L.; Xu, H.; Gong, L. Influence of wall roughness models on fluid flow and heat transfer in microchannels. Appl. Therm. Eng.
**2015**, 84, 399–408. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the roughness texture in (

**a**) the cone model and (

**b**) the groove model, and the schematic of the flow on the (

**c**) Cone model, (

**d**) Groove-P model and (

**e**) Groove-V model.

**Figure 2.**Effective boundary slip (reversed) vs. Ra on the cone model with Rsm = 10 nm (Green); 20 nm (Black); 100 nm (Dark blue); 200 nm (Pink); 1000 nm (Red); 2000 nm (Light blue). The −b

_{eff}and Ra are in log scale.

**Figure 3.**Effective boundary slip (reversed) vs. Ra on three different surface models: Cone model; Groove-V model and Groove-P model. The Ra is in log scale.

**Figure 4.**Error of corrected effective boundary slip (obtained experimentally from AFM) on cone model with different R

_{a}and R

_{sm}.

**Figure 5.**Increasing rate of Nusselt number for a pressure-driven flow in a micro channel with one-side rough surface in Groove-V model. The Rsm is fixed at 100 nm. The effective boundary slip are also shown.

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**MDPI and ACS Style**

Pan, Y.; Jing, D.; Zhang, H.; Zhao, X.
Effective Boundary Slip Induced by Surface Roughness and Their Coupled Effect on Convective Heat Transfer of Liquid Flow. *Entropy* **2018**, *20*, 334.
https://doi.org/10.3390/e20050334

**AMA Style**

Pan Y, Jing D, Zhang H, Zhao X.
Effective Boundary Slip Induced by Surface Roughness and Their Coupled Effect on Convective Heat Transfer of Liquid Flow. *Entropy*. 2018; 20(5):334.
https://doi.org/10.3390/e20050334

**Chicago/Turabian Style**

Pan, Yunlu, Dalei Jing, He Zhang, and Xuezeng Zhao.
2018. "Effective Boundary Slip Induced by Surface Roughness and Their Coupled Effect on Convective Heat Transfer of Liquid Flow" *Entropy* 20, no. 5: 334.
https://doi.org/10.3390/e20050334