# 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

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**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