Heat transfer enhancement and drag reduction in transverse groove-bounded microchannels with offset

International Journal of Thermal Sciences 130 (2018) 240–255

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Heat transfer enhancement and drag reduction in transverse groovebounded microchannels with offset

T

Weiwei Rena, Yu Chenb,c,∗, Xiaojing Muc,∗∗, B.C. Khood, Feng Zhangc, Yi Xuc a

Singapore Institute of Manufacturing Technology (SIMTech), 2 Fusionopolis Way, 138634, Singapore Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis North, 138632, Singapore c Defense Key Disciplines Lab of Novel Micro-nano Devices and System Technology, International R&D Center of Micro-nano Systems and New Materials Technology, Key Laboratory of Optoelectronic Technology & Systems, Ministry of Education, Chongqing University, Chongqing 400044, PR China d Department of Mechanical Engineering, National University of Singapore, 117576, Singapore b

A R T I C LE I N FO

A B S T R A C T

Keywords: Lab-on-a-chip Heat transfer Superhydrophobic Transverse grooves

The shrinking physical scale of microfluidic chips brings significant benefits like less expense of chemicals, shorter diagnosing time and highly parallel testing, but at the same time presents big challenges in drag reduction and thermal management. The ridge/groove structure on the superhydrophobic surface inside microchannels, which entraps air so as to reduce the skin friction, is believed to affect the internal heat transfer. In the current study, the flow and heat transfer between two walls with eccentric transverse microgrooves was studied based on numerical simulations, to investigate its effect on overall drag reduction, heat transfer enhancement and possible mechanisms. The changes in drag reduction capability and convective heat transfer were respectively evaluated in terms of the effective slip length and Nusselt number. The overall thermal performance and efficiency were evaluated using the goodness factor. It was found that the eccentricity of transverse microgooves inside superhydrophobic channel increases the effective slip length but reduces heat transfer slightly, which is especially obvious at low Reynolds number, large pattern length and moderate shear free fraction. Besides drag reduction and heat transfer, the flow fields for different cases were investigated in details, which show that the increase in the effective slip length of superhydrophobic microchannel with offset can be attributed to the increase of average velocity on air-water interface.

1. Introduction Lab-on-a-chip based on microfluidics is a state-of-the-art technology with many different promising applications, such as point-of-care testing (POCT), fast molecular diagnosis etc. The physical dimension of micron-channel is reduced to accommodate more channels invariably such as in microfluidic-chip, leading to larger driving pressure needed to maintain the flow throughout to overcome the increased skin friction. It is challenging to ensure such a high value of pressure for the micron-scale system, where elasticity (i.e., compressibility) of the liquid has to be considered, which can take minutes or even longer to compress liquid in the chamber [1]. This motivates the utilization of other forms of driving force (such as electrical force, magnetic forces and Lorentz forces) and/or the search for drag reduction means to keep the pressure to a manageable level [1]. In the meanwhile, heat transfer within microchannels which play a significant role in many biochemical reactions, including polymerase chain reaction (PCR), DNA melting

∗

and detection of single base mismatches and so on, also needs to be managed carefully [2–5]. It is a common belief since the beginning of the modern fluid mechanics that the drag on a smooth surface is always lower than that on a rough surface [6–9]. This is actually not the case as documented through the development of special drag reducing surface topographies [see recent reviews in Ref. [10]. When the size of (micro) fluid system shrinks, surface forces such as capillary force become dominant. As such, it can be used as the driving force and employed to overcome skin friction by the so-called superhydrophobic effect [11]. This is combining the effect of hydrophobicity of the liquid w.r.t. Surface material and surface topography, with the research being inspired by the unique water-repellent properties of the lotus leaf [12]. When a super-hydrophobic surface is submerged inside a liquid, gas bubbles which are trapped in microstructures (grooves, post/pillars and pores/holes) on the solid substrate surface can effectively reduce the shear stress experienced by the liquid, as the shear

Corresponding author. Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis North, 138632, Singapore. Corresponding author. E-mail addresses: [email protected] (Y. Chen), [email protected] (X. Mu).

∗∗

https://doi.org/10.1016/j.ijthermalsci.2018.04.025 Received 27 May 2017; Received in revised form 17 April 2018; Accepted 18 April 2018 1290-0729/ © 2018 Elsevier Masson SAS. All rights reserved.

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on two walls, resulting in different drag reduction and heat transfer, especially when two opposing walls too close to each other. However, it is difficult to align the micro-structures on the upper and lower channel walls due to the low assembling accuracy for the industrial applications, making it important to understand how this unalignment influences the performance of superhydrophobic microchannels. According to authors' best knowledge, there is few research work investigating how the offset of microstructures on the channel walls affects the slip length and thermal performance. Thus, this motivates the current work to study the effect of offset of transverse microgrooves of two walls on overall drag reduction, effective slip length and heat transfer in microchannels. Note that the water-air interface can be assumed to be undeformed and flat in the current study as many researchers did [26,51,60] when capillary number Ca is sufficiently small (which implies that viscous forces are negligible compared to surface tension forces). In addition, as reported by Ou et al. [15], Ou and Rothstein [36] in their studies for water flow through superhydrophobic microchannels with microstructures, the experimentally measured pressure drop agrees well with the predictions based on such assumption. However, it shall be noted that the contact angle of gas-liquid interface and the shape of meniscus could influence the pressure drop and heat transfer obviously under several special situations [61,62]. The numerical methodology is introduced in §2, to be followed by the results on the effect of offset on slip length and heat transfer under different pattern length, Reynolds number and groove ratios in §3. At the end of §3, the flow field, such as velocity profile and skin friction distribution on channel walls, is discussed to reveal the mechanism of the effect of offset on slip length. The concluding remarks are given in §4.

between the liquid and solid is replaced by the shear between the liquid and gas. The overall effect of super-hydrophobic surface on the flow is that the original no-slip boundary condition is transformed into slip boundary condition, and the slip velocity defined as the mean velocity averaged over the ridge pitch of composite interface (e.g. solid-liquid interface and gas-liquid meniscus) is proportional to the shear rate at the wall

=λ

u w

∂u ∂y

w

where λ is the slip length [13,14]. For micro-fluidic or nano-fluidic devices with super-hydrophobic surface whose characteristic length scale Lc and slip length λ are both in the order of micrometers [15,16], slip has a significant effect on the flow and drag due to the large Knudsen number Kn = λ / Lc [17–19]. Thus, much works have been focused on the means to increase the slip length of super-hydrophobic surface, such as increasing hydrophobicity through changes in surface chemistry [20–24] and correctly shaping the surface pores/roughness [25]. Normally, the configuration of roughness on the surface influences the overall drag reduction significantly, where longitudinal microgrooves are believed to induce the greatest drag reduction [26–29]. It is shown that the drag reduction and effective slip length becomes greater for higher shear-free fraction and the reduced relative channel hydraulic diameter [16,30] found that the effective slip length is closely correlated to the local streamwise velocity profile and interfacial velocity, where higher interfacial velocity indicates greater drag reduction and thus higher slip length. At higher Reynolds number, less drag reduction is reported for laminar flow [30–32]. However, obvious drag reduction is still reported for turbulent flow over super-hydrophobic surface [33,34] which is even greater than the counterpart in the laminar flow [35]. While the laminar drag reduction due to the super-hydrophobic effects is well documented [15,36–38], techniques which rely on this effect to reduce the turbulent drag are still being developed [34,39,40]. Besides drag reduction capability and slip length, thermal transport and heat transfer in microchannels with superhydrophobic surfaces have attracted concurrent interest [41]. Inman [42] analytically investigated Nusselt numbers, and thermal entrance lengths for flow inside ducts where slip velocity and temperature jumps are prescribed at the walls. Enright et al. [43] predicted the hydrodynamic and thermal behavior for fully developed laminar flow bounded by two parallel plates with apparent slip length and uniform heat flux, and the validated it by numerical results for parallel or transverse ridges or pillar arrays. Specifically, thermal behavior for different surface structures such as parallel ridges [44–46], transverse ridges [47–50] and pillar/holes [51] have been investigated under isoflux and/or isothermal with the assumption of adiabatic water-air interface. Lam et al. [52] analyzed thermal resistance of galinstan—a liquid metal, for direct liquid cooling of on surface structures, and found that the resistance is dominated by its caloric component (related to friction) instead of convective resistance (related to Nusselt number). On further work, besides focusing on the surface roughness shape, the effective slip length is found to be sensitive on the curvature to the water-air interface, which has been reported in experimental [53], analytical [54,55] and computational studies [53,56–58]. In addition, Kashaninejad et al. [59] pointed out that microhole eccentricity affects the drag reduction efficiency of microchannel with super-hydrophobic surface, which is correlated to the contact angle hysteresis. Generally, microchannels with super-hydrophobic surface on both walls induces greater drag reduction and effective slip length than those employed on single wall, and it is not simply two times as perhaps expected due to the nonlinear interaction between the two walls [60]. This implies that the nonlinear interaction between the upper and lower channel walls would be modified (strengthened/weakened) by the offset of grooves

2. Problem formulation and methodology 2.1. Problem formulation and governing equations As shown in Fig. 1 on the cross section of microchannel, the incompressible fluid (water) flows inside the channel bounded by the walls with gas (air) entrapped inside grooves. The flow is assumed to be laminar and steady, and the water-gas interface is assumed flat. The xdirection is the flow direction where a pressure gradient is applied to drive the flow, the y-direction is the direction of the channel height, and the origin is at the center of channel. The transverse grooves are arranged in a periodic array along the x-direction, therefore the flow is also periodic and two-dimensional (2D). Generally, the dimensional form of governing equations are given as follows:

∂ui* =0 ∂x i*

(1)

* ∂ui*uj* ⎞ ∂ 2u * ∂p* ⎛ ∂u = − * + μ* * i * ρ* ⎜ *i + * ⎟ ∂t ∂x j ∂x j ∂x j ∂x j ⎠ ⎝

(2)

Fig. 1. Computation domain geometry. 241

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∂uj*T * ⎞ ∂ 2T * ⎛ ∂T * = k* * * ρ*cp* ⎜ * + * ⎟ ∂t ∂ ∂ x x j ∂x j j ⎠ ⎝

Dimensionless phase shift or offset (3)

ε=

where ρ* is the density, p* is the pressure, and T * is the temperature, and the subscript represents the direction (e.g. 1 represents the x-direction, 2d represents the y-direction). The term cp* is the specific heat capacity at constant pressure, k* is the thermal conductivity of the fluid. For the purpose of nondimensionalization, the channel's hydraulic diameter Dh* = 4H * is taken as the reference length scale, where H * is the half channel height. Besides, the reference velocity is the average

where e is the shear-free region length, E is the module length, Δ is the shift/offset length of the shear-free regions on the upper wall w.r.t. the lower wall, Dh is the channel's hydraulic diameter (Dh ≈ 4H ), and H is the half channel height. Following the approach introduced in other previous works (e.g. Ref. [30]), the effective slip length λ is defined to characterize the friction reduction arising from the superdydrophobic surfaces, by matching the flow resistance of the channel with slip boundary velocity

*

∫ H * u*dy

* = −H * . The pressure is decomposed into the mean bulk velocity uavg 2H and fluctuating components as follows:

p* = −β *x * + p′* ,

w

where is the dimensional mean pressure gradient in the streamwise direction, which is adjusted in the simulation process to ensure that the average bulk velocity uavg equals the prescribed value. Non-dimensional temperature is defined as

H*

,

∫−H * u*dy *

λ 1 1 = ln . E 2π cos (δπ /2)

which is a function of x varying in the streamwise direction. Please note that those variables with superscript ∗ are used to denote dimensional variables and those without ∗ are used for dimensionless variables. Considering the above reference quantities and the assumption of steady flow, the dimensionless continuity equation, momentum equation and energy equation take the following form:

∂ui = 0, ∂x i

∂x j

∂ (θuj ) ∂x j

=

∂p′ 1 ∂ 2u i + + βδi1, Re ∂x j ∂x j ∂x i

1 ∂ 2θ , RePr ∂x j ∂x j

(5)

from whose Taylor series expansion, it can be found that this nondimensionalized slip length λ / E is proportional to δ 2 as δ → 0 and goes to infinity as δ → 1, which makes it very difficult to compare the slip length of different configurations, especially at low δ. In order to evaluate the effect of offset on slip length in detail and to exclude the effect of δ on slip length as much as possible, a normalized effective slip length λ / Eδ 2 was introduced by Teo and Khoo [58] as

(6)

λ 1 ⎛ 8 1 ⎞ = − ⎜ ⎟. δe Lδ 2 ⎝ fRe 12 ⎠

* ρ*uavg Dh*

μ*

(7)

f=

cp*μ *

h=

.

˙ p (Tout − Tin ) mc E ΔTm

,

where the mass flow rate m˙ is a quantity defined in a unit width as H m˙ = ρ∫−H udy , Tin is the inlet bulk temperature, Tout is the outlet bulk temperature, and ΔTm is the logarithmic mean temperature difference

ΔTm =

(Tout − Tw ) − (Tin − Tw ) . ln(Tout − Tw ) − ln(Tin − Tw )

It shall be noted that periodic boundary conditions for temperature only applies to θ, not for T. Therefore, Tin ≠ Tout as Tb = Tb (x ) although θin = θout . The Nusselt number reads

E E = , Dh 4H

Nu =

Dimensionless shear free fraction

δ=

ΔPDh 1 2 ρuavg E 2

where ΔP is the pressure loss. The overall heat transfer coefficient is defined as

,

and the Prandtl number Pr = * is set as 7 for water herein. k Additionally, periodic boundary conditions for the velocity and nondimensional temperature θ are set at the inlet and outlet of the computational domain ( x = ± E/2 ): uin = uout ; vin = vout ; θin = θout [63]. Noslip boundary condition is set in the solid region along the upper and lower walls ( y = ± H ): u = v = 0 , T = Tw ; while shear-free boundary condition is set in the gas region ∂u/ ∂y = v = ∂T / ∂y = 0 . In order to systematically describe the phenomena in microchannels, other salient parameters are defined as follows: Channel's relative pattern length of surface structure

L=

(9)

Note that fRe = 96 for the completely no-slip channel, thus the slip length becomes zero, which is consistent with Eq. (9). The Darcy friction factor f is defined as

where δij is Kronecker delta and j is set as 1 to impose the pressure gradient in the streamwise direction. The Reynolds number is thus defined as

Re =

(8)

Lauga and Stone [26] have shown that for a parallel-plate channel in the limit regime Re → 0 , Kn → 0 with vanishing shear stress at the liquid-vapor cavity interfaces, the slip length may be expressed exactly as

H*

∫−H * u*T *dy *

=−

. w

λ 8 1 = − Dh fRe 12

where Tw* is the wall temperature, and the local bulk temperature Tb* (x ) is defined as

∂ (ui uj )

∂u ∂y

In other words, the slip length is computed from matching the flow rate of superhydrophobic channel with channel flow with slip walls and it can be related to the product of friction coefficient f and Reynolds number Re (e.g. Poiseuille number) as [30].

T * − Tw* , Tb* − Tw*

Tb* =

=λ

u

(4)

β*

θ=

Δ , E

hDh . k

Note that the water-air interface is assumed flat under infinite surface tension in the current study, as most of previous computational studies did [32,57,60]. This idealized assumption makes the

e , E 242

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Table 1 Mesh size used for simulation.

Table 2 Parameters of investigated microchannels at different offset ε = 0 , 0.1, 0.2, 0.3, 0.4 and 0.5.

Mesh size

Parameters

Δx Δymin number of cells in y

0.003125 0.000925 320

pattern length L shear free fraction δ Reynolds number Re

computations simplified and helps to understand the mechanism of the offset effect without loss of generality.

0.25, 0.5, 0.8, 1, 1.25, 1.6, 2, 2.5 0.05, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8, 0.9, 0.95 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000

depend on the channel's pattern length L, shear free fraction δ, and the working condition like Reynolds number Re. Thus, the modulation effect of offset ε on the frictional and thermal performance can be expected. So, the variation of slip length λ / δe and heat transfer rate Nu due to offset ε are investigated at different pattern lengths L, Reynolds number Re and shear free fraction δ as shown in Table 2. Note that the parameters set in Italic (at L = 2 , Re = 1, and δ = 0.5) are chosen the reference case to show how these parameters affect the performance of the microchannel with different offset. Furthermore, to reveal the underlying mechanisms responsible for changes in slip length and heat transfer coefficient due to offset ε, the flow field will be discussed in details at the end of this section.

2.2. Numerical validation Before the investigation propes, the accuracy of the assembled numerical code is tested vis-a-vis the smooth channels with no-slip surface. In the specific implementation process, a staggered grid is used for the finite volume discretization, while the central difference scheme is employed for spatial derivatives. The pressure field is updated by solving the Poisson equation to satisfy the continuity equation. A Cartesian mesh is utilized for the simulation, which is uniform in the streamwisedirection and is greatly refined near the walls in the wall-normal-direction. The mesh size in the streamwise- and wall-normal-direction are respectively shown in Table 1. This methodology and the code of flow solver has been employed and validated in the authors' previous publications [64–66]. For convenience of comparison, the product of friction coefficient f and Reynolds number Re ( fRe ) and Nusselt number (Nu ) are used to quantitatively validate the numerical code. As shown in Fig. 2(a), f Re approaches the analytical solution 96 when the shear free fraction goes to zero and match well with the published results [51,57]. On the other hand, the Nusselt number (Nu ) of solid flat channel without shear free section approaches the asymptotic limit 7.54 when Reynolds number Re becomes higher and higher as shown in Fig. 2(b), which has very good agreement with Cheng et al. [51]. It shall be noted that the variation of Nusselt number on Reynolds number is attributed to an axial conduction effect in the channel. These two comparisons attest to the accuracy of the numerical code and show its validity for simulation of flow inside the superhydrophobic microchannel.

3.1. Slip length Firstly, their capability of friction reduction is investigated in terms of nondimensional slip length λ / δe . 3.1.1. Pattern length The pattern length L dramatically influences the overall friction coefficient of microchannels, and hence affects the slip length λ / δe [51,60]. So, it is logical to study how the offset modulate the slip length at different pattern lengths. Fig. 3 demonstrates the variation of slip length λ / δe at different offsets ε with respect to pattern length L from 0.25 to 2.5. Herein, a typical shear free fraction δ = 0.5 is chosen and both low and high Reynolds numbers at Re = 1, 1000 are studied for simplicity without loss of generality. It can be seen from Fig. 3 that the effective slip length λ / δe increases along with offset ε at the same pattern length. In addition, the variation of the effective slip length λ / δe due to the offset ε becomes more significant at high pattern length. Similarly, comparison of the results at the two different Reynolds numbers Re = 1 and 1000 demonstrates that offset has much more obvious effect on slip length at relatively low Reynolds number (Re = 1), which is also the most pertinent working conditions for microchannel flows. It is observed that the effective slip length λ / δe with zero offset

3. Results and discussion In this section, the effect of phase-shift/offset ε between upper and lower super-hydrophobic sections on the effective slip length λ / δe and heat transfer coefficient Nu is studied and discussed. Generally, the nondimensional effective slip length λ / δe and heat transfer rate Nu

Fig. 2. Validation of numerical results in terms of Poiseuille number f Re and Nusselt number Nu : the solid line represents the results obtained by the present numerical tool, and the symbols represent the benchmark. 243

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Fig. 3. Variation of the slip length λ / δe due to offset ε at different pattern length L for Re = 1, 1000 and δ = 0.5.

Combining with the findings in the previous section, it can be concluded that the offset increases the slip length more significantly at high pattern length (L = 1 − 2.5) and low Reynolds number Re = 1 − 10 .

decreases with the pattern length L for both low and high Reynolds numbers. However, the effective slip length is dramatically increased by shifting ε at low Reynolds number (Re = 1) and high pattern length (see Fig. 3(a)). Such changes in slip length caused by offset (difference between ε = 0 and 0.5) gradually increases within L = 0.25–1.2 and then almost remains as a constant for L = 1.2 –2.5. So the highest slip length is achieved at L = 1.2 and ε = 0.5 for Re = 1, where the increase in slip length due to the change in offset overcomes the decrease of slip length due to the increase of pattern length. Thus, an interesting phenomenon found is that λ / δe increases then decreases with pattern length L at high enough offset (ε > 0.3). Conversely, the original decreasing trend of slip length against the pattern length for the reference case (with ε = 0 ) is maintained at high Reynolds number (Re = 1000 , see Fig. 3(b)) regardless of offset.

3.1.3. Shear free fraction Besides the pattern length and Reynolds number, the effect of offset on slip length at different shear free fraction is investigated and presented in Fig. 5. As may be expected, the variation of slip length with respect to free shear fraction caused by the offset ε is extremely small at low pattern length or high Reynolds number (see Fig. 5(a), (c) and (d)), while the variation of slip length is significant only when the pattern length is high and Reynolds number is low (see Fig. 5(b)). The increases of slip length induced by offset at different shear free fraction are of the same order for L = 2 and Re = 1, with wide shear free fraction ranging between 0.3 < δ < 0.95 (see Fig. 5(b)), thus the trend of slip length against the shear free fraction is broadly unchanged by the offset although the higher shear free fraction induces greater slip length.

3.1.2. Reynolds number Though microchannels are normally operating at relatively small Reynolds number (Re < 1), it is still worthwhile to investigate the effect of offset on the slip length at different Reynolds numbers ranging from Re = 1 to 1000. Fig. 4 demonstrates the variation of the effective slip length λ / δe with respect to Reynolds number at different offset ε for pattern length L = 0.25, 2 and a typical shear free fraction of δ = 0.5. It is shown that the effect of offset on slip length is quite negligible at all Reynolds numbers ranging from 1 to 1000 for small pattern length (L = 0.25, see Fig. 4(a)). It is also observed that the slip length λ / δe is significantly increased by shifting the shear free section for high pattern length (L = 2 , see Fig. 4(b)). Such increase of slip length is more obvious at relatively low Reynolds number (Re = 1−10), and gradually drops to a low level while Reynolds number increases up to 1000.

3.1.4. Summary From the above discussions on the variation of slip length due to offset at different pattern length, Reynolds number and shear free fraction, it is found that the offset significantly increases the slip length at relatively high pattern length (L > 1) and low Reynolds number (Re < 10 ) for very wide range of shear free fraction (0.3 < δ < 0.9). The effect of offsets on friction and slip length may be attributed to its influence on the boundary layer over the no-slip and shear-free portions of the channel walls. Thus, this nonlinear effect becomes more significant when two opposing walls are close enough to each other

Fig. 4. Reynolds number effect on the slip length λ / δe at L = 0.25, 2 and δ = 0.5. 244

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Fig. 5. The effect of the fraction of slip space δ on the slip length λ / δe at Re = 1, 1000 and L = 0.25, 2.

3.2.2. Reynolds number In Fig. 7, the variation of Nusselt number for Reynolds numbers ranging from 1 to 1000 is presented at L = 0.25 and 2, and δ = 0.5. It can be seen that at small pattern length of L = 0.25 (see Fig. 7(a)), the change of Nusselt number due to offset is fairly moderate regardless of the Reynolds number. Conversely, at the large pattern length of L = 2 (see Fig. 7(b)), the Nusselt number at different offset can be distinguished from each other readily. While the reduction of Nusselt number due to offset almost remains in the same order for relatively low Reynolds number (Re = 1−10), it gradually drops to an undistiguishable value when Reynolds number increases up to 1000.

(e.g. high pattern length L) such that boundary layer is thick enough to cover the full channel height. It is also reasonable to observe that the slip length is influenced by offset significantly at low Reynolds number, where the boundary layer develops faster in the streamwise direction since diffusion term is stronger than convection/inertial [30]. However, if the fraction ratio is too low or too high, the effect of shifting the boundary layers over no-slip and shear-free walls are depressed. All this hypothesis will be further discussed for detailed flow fields in §3.4. 3.2. Heat transfer characteristics Similar to the discussion on slip length, the offset also affect the heat transfer rate from the channel walls to the fluid, where the variation depends on the pattern length, Reynolds number and shear free fraction. It may be expected that similar to the slip length, the effect of offset on the heat transfer in super-hydrophobic microchannels is relatively significant for high pattern length, low Reynolds number and within a wide shear free fraction range (0.3 < δ < 0.9). However, this would require careful investigations of the heat transfer rate in terms of Nusselt number to ascertain this perspection.

3.2.3. Shear free fraction Fig. 8 shows the effect of offset on Nusselt number at different shear free fraction, where the Reynolds number Re = 1 and 1000, and L = 0.25 and 2. Consistent with the above findings, the heat transfers is minimally affected by the offset at high Reynolds number or small pattern length (see Fig. 8(a), (c) and (d)), and the effect of offset on heat transfer is much more obvious for low Reynolds number and high pattern length (see Fig. 8(b)). Slightly different from the variation of slip length due to offset at different shear free fractions, the variation of Nuseelt number due to offset remains as about the same order within a relatively narrower range of shear free fraction 0.4 < δ < 0.7 (see Fig. 8(b)).

3.2.1. Pattern length Firstly, Fig. 6 shows the variation of Nusselt number due to the offset at different pattern length from L = 0.25 to 2.5 for Re = 1 and 1000, and δ = 0.5. It can be observed that the offset induces a reduction of Nusselt number generally. At low Reynolds number (see Fig. 6(a)), it clearly shows that such reduction of Nusselt number becomes gradually apparent with pattern length. Conversely, such distribution has slight variation at the high Reynolds number of Re = 1000 with respect to pattern length (see Fig. 6(b)).

3.2.4. Summary Based on the above investigation and discussion on the heat transfer at different offset, it can be concluded that heat transfer is reduced by shifting the shear free section on the opposite wall. Obvious reduction in Nusselt number is observed at relatively high pattern length (L > 1) 245

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Fig. 6. The Nusselt number Nu against the pattern length L at Re = 1, 1000 and δ = 0.5.

Fig. 7. Reynolds number effect on the Nusselt number Nu at L = 0.25, 2 and δ = 0.5.

and low Reynolds number (Re < 10 ) within the shear free fraction range (0.4 < δ < 0.6). Similar to the slip length, the Nusselt number is also closely correlated to the thermal boundary layer thickness which is significantly affected by flow fields. And, considering the fact that Nusselt number is positively related to friction drag, it is natural to observe the offset reduces Nusselt while it increases slip length. More detailed discussion of the flow fields and the underlying mechanisms will be presented in §3.4.

Φc =

which is taken as the reference to evaluate the overall thermal performance of super-hydrophobic microchannels. A heat transfer enhancement is achieved in the superhydrophobic microchannel on comparing to the flat and solid channel if Φ/Φc > 1. 3.3.1. Pattern length In Fig. 9, the goodness factor with different offset are presented for Reynolds number Re = 1, 1000, δ = 0.5 and pattern length ranging from 0.25 to 2.5. It is observed that the goodness factor of superhydrophobic microchannel is lower at low Reynolds number (Re = 1) and higher at large Reynolds number (Re = 1000 ), compared to that of the solid flat channel. It is also shown that an increase in ε increases the goodness factor compared to the case without offset. In addition, such an enhancement due to offset is more obvious at large pattern length but negligible at small pattern length.

3.3. Goodness factor Based on the analysis in the above two sections, it is shown that both the friction (or slip length) and heat transfer are reduced by shifting the shear free sections on the opposite walls in super-hydrophobic microchannels. In order to evaluate the overall thermal performance for the super-hydrophobic microchannels, the goodness factor Φ proposed by London [67] and calculated in terms of the Colburn factor j and Darcy friction factor f is considered as follows:

Φ=

Nu 7.54 = , fRePr 1/3 96 × 71/3

3.3.2. Reynolds number Fig. 10 shows the variation of goodness factor due to offset at different Reynolds number for respectively small and large pattern length at L = 0.25 and 2. It is observed that the influence of offset on goodness factor is fairly marginal when the pattern length is small regardless of Reynolds number (see Fig. 10(a)). Conversely, the amplification of goodness factor due to offset at high pattern length (L = 2, see Fig. 10(b)) increases when Reynolds number first increases (1 < Re < 20 ) then decrease again for larger Re (i.e. Re > 20 ). The highest heat transfer enhancement is found to occur at about Re = 20 .

j Nu = . f fRePr 1/3

which evaluate the combination of hydrodynamic and thermal performance vis-a-vis the input pumping powers as measured by the hydrodynamic drag for a given pumping power and constant fluid thermomechanical properties, a higher goodness factor suggests more heat is taken away by the working fluid. For the fully developed laminar flow inside channel with constant wall temperature, the critical goodness factor is expressed as [51]: 246

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Fig. 8. The effect of the fraction of slip space δ on the Nusselt number Nu at Re = 1, 1000 and L = 0.25, 2.

factor for super-hydrophobic channel with various ε is studied for different pattern length, Reynolds number and shear free fraction. It can be concluded that increasing ε on the superhyrophobic channel walls induces heat transfer enhancement, and the enhancement gets amplified at larger pattern length and higher shear free fraction in a wide range of Reynolds number. It is generally consistent with the finding that frictional performance and heat transfer rate are significantly influenced by offset at large pattern length, relatively low Reynolds number and wide shear free fraction range, where the observed friction reduction is the determinative factor of heat transfer enhancement herein. Especially, it is consistent with the findings of [52] that the thermal resistance is dominated by its caloric component, as the heat

3.3.3. Shear free fraction Fig. 11 shows the variation of goodness factor due to offset at different shear free fraction δ for Reynolds number Re = 1 and 1000, and pattern length Ł = 0.25 and 2. It is found that the influence of offset on goodness factor is only observable when the pattern length is large (see Fig. 11(b) and (d)), and is barely visible for small pattern length (see Fig. 11(a) and (c)). It is also observed the goodness factor enhancement induced by the offset becomes larger at higher shear free fraction for large pattern length (see Fig. 11(b) and (d)). 3.3.4. Summary The overall heat transfer efficiency as indicated by the goodness

Fig. 9. The goodness factor Φ against the pattern length L at Re = 1, 1000 and δ = 0.5. 247

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Fig. 10. Reynolds number effect on the goodness factor Φ at L = 0.25, 2 and δ = 0.5.

transfer efficiency is enhanced by texturing microchannels due to the much larger contribution from hydrodynamics than the reduction in the Nusselt number (e.g./ caloric resistance is decreased more than the increase in convective resistance).

observed. (It may be noted that the temperature field is not presented since the goodness factor increment is mainly attributed to the reduction of drag and the heat transfer problem is also dominated by the flow field herein (as temperature is a passive scalar).)

3.4. Flow fields

3.4.1. Reference case For simplicity and without loss of generality, the super-hydrophobic channel with L = 2, Re = 1, and δ = 0.5 which induces an extremely obvious increment in the slip length with offset of shear free region is chosen as the reference case. Fig. 12 shows the effect of offset on the flow profile by presenting the respective stream-wise velocity contours. It is shown that inside the super-hydrophobic channel with no

It is found in the above sections that the influence of offset on frictional and thermal performance is clearly obvious at large pattern length, low Reynolds number and medium shear free fraction. So, it is deemed worthwhile to investigate the (underlying) flow field in the attempt to reveal the underlying mechanism(s) of the phenomena

Fig. 11. The effect of the fraction of slip space δ on the goodness factor Φ at Re = 1, 1000 and L = 0.25 , 2. 248

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Fig. 12. The contour of normalized velocity (u/ uavg ) for super-hydrophobic microchannel with different offset at L = 2 , Re = 1 and δ = 0.5.

increased from 1 to 1.4. It is observed from Fig. 13(b) that the shear stress on the lower wall is generally reduced as the offset ε increases then reaches its lowest limit when ε = 0.5, which can be associated with the movement of high speed flow region towards the shear-free boundary and the related increment of shear layer thickness (see Fig. 12). So, the overall shear drag and pressure drop is dramatically reduced for super-hydrophobic channel with offset at ε = 0.5, resulting in a significant increase in slip length (see Fig. 3(a)). The mechanism(s) for the obvious slip length increment for the reference case as discussed above are clearly associated with the increase of interfacial velocity and associated manipulation of shear layer over the no-slip sections. Next, several cases with different configurations are presented and discussed in the following sections to reveal how the pattern length, Reynolds number and shear free fraction affect the flow field thus the slip length.

offset, there is a high speed region at the centerline between the no-slip sections (see Fig. 12(a)). The fluid velocity on the shear free boundary (interfacial velocity) dramatically increases along the streamwise direction because the herein boundary condition suddenly changes into the shear free type and the fast flowing fluid at the centerline causes the fluid closer to the boundary to accelerate due to viscous effect. Meanwhile the velocity at the centerline is reduced to maintain the over flow rate at the shear free sections. Then the velocity on the wall is (suddenly) reduced back to zero again when it returns to the no-slip section. Thus a strong shear layer is formed when the fluid transits between the no-slip and shear free sections due to the sudden change of velocity on the walls. When the shear free section on the upper wall is shifted further downstream, the high speed flow shifts away from the (top and bottom) no-slip walls and moves towards the opposite side with the shear free boundary. Thus the shear on the lower wall is reduced due to the increase of shear layer thickness. If the shear free section on the upper wall is shifted even further downstream, the highest speed flow region moves closer to the shear free section on the upper wall (see Fig. 12(b)–(e)), and eventually totally disappears from the centerline of channel and rides on the shear free sections on the lower and upper walls (ε = 0.5, see Fig. 12(f)) when the no-slip section and shear free sections on opposite walls are directly facing each other. In other words, the interfacial velocity is dramatically increased via the shift of the upper wall (see Fig. 13(a)), and the maximum interfacial velocity is

3.4.2. Pattern length effect Firstly, the flow field for the case with L = 0.25, Re = 1, and δ = 0.5 is investigated for comparison with the reference case to evaluate the effect of pattern length. When the pattern length is small (channel length is small), the high speed flow region is concentrated at the centerline all through the streamwise direction (see Fig. 14(a)) due to the shorter shear free boundary barely affecting the velocity at the centerline. In addition, the 249

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Fig. 13. The shear distribution for super-hydrophobic microchannel with different offset at L = 2 , Re = 1 and δ = 0.5.

reduction of shear drag (see Fig. 15, the lines overlap with each other) and negligible increment of slip length (see Fig. 3(a)).

velocity on the shear free sections (interfacial velocity) for the case with smaller pattern length (L = 0.25, see Figs. 14(a) and 15(a)) is smaller than the counterpart for the case with a large pattern length (L = 2 , see Figs. 12(a) and 13(a)) since a long developing length is needed for velocity on the boundary to accelerate and gain speed. Based on the finding that the high speed flow region is barely affected by shear free boundary and distributed almost uniformly through the streamwise direction, it is therefore expected that the shifting of the shear free section on upper walls has minor influence on the flow field (see Fig. 14(b)–(f)). Hence the shift of the shear free section on the upper wall has minimal effect on the interfacial velocity and the thickness of shear layer over the no-slip section, resulting in a minor

3.4.3. Reynolds number effect Secondly, the flow field for the case with L = 0.25, Re = 1000 , δ = 0.5 is investigated to demonstrate the effect of Reynolds number via comparison with the reference case. Higher Reynolds number means higher inertial force becomes more dominant over the viscous force, thus the interfacial velocity herein (see Figs. 16(a) and 17(a)) is also smaller than the counterpart for low Reynolds number (see Fig. 12(a)). As such, the supposedly fast speed fluid is only minimally affected in the shear free section and distributed

Fig. 14. The contour of velocity for super-hydrophobic microchannel with different offset at L = 0.25, Re = 1 and δ = 0.5 250

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Fig. 15. The shear distribution for super-hydrophobic microchannel with different offset at L = 0.25, Re = 1 and δ = 0.5

It can be found that the effect of offset of grooves on flow shifting the boundary layer over non-slip and shear-free portions of channel walls, which is dominated by diffusion effect actually. Thus, at high ∂u Reynolds number, the nonlinear convective terms in N-S equations u ∂x becomes more important, and the diffusion term is vanishing meanwhile. In addition, the interfacial velocity is also skewed due to the

relatively more uniformly across the whole streamwise direction. The consequence is the shift of the upper wall has smaller influence on the interfacial velocity and the associated shear drag (see Fig. 17) than the reference case (see Fig. 13). It is not surprising to see the offset has minor effect on the effective slip length as discussed in the previous section (see Fig. 4(b)).

Fig. 16. The contour of velocity for super-hydrophobic microchannel with different offset at L = 2 , Re = 1000 and δ = 0.5 251

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Fig. 17. The shear distribution for super-hydrophobic microchannel with different offset at L = 2 , Re = 1000 and δ = 0.5

Fig. 18. The contour of velocity for super-hydrophobic microchannel with different offset at L = 2 , Re = 1 and δ = 0.05

significant inertial effect caused by the convective term [30], and it is rarely affected by the offset since inertial/convective momuntem transport is much stronger than diffusion at high Reynolds number.

3.4.4. Shear free fraction effect Lastly, the effect of shear free fraction is investigated by comparing the flow field for the cases with Re = 1, L = 2 , δ = 0.05 and 0.95 presented herein with the reference case. In Fig. 18, it is found that the flow field is very slightly affected by 252

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Fig. 19. The shear distribution for super-hydrophobic microchannel with different offset at L = 2 , Re = 1 and δ = 0.05

Fig. 20. The contour of velocity for super-hydrophobic microchannel with different offset at L = 2 , Re = 1 and δ = 0.95

the offset ε due to the small shear free fraction δ = 0.05, and thus the offset has little impact on the interfacial velocity and shear drag. Therefore, it is unsurprising to observe such a minor change in interfacial velocity and shear drag in Fig. 19.

For the case with very high shear free fraction (δ = 0.95), the velocity contour distribution is almost uniform in the spanwise direction (u/ uavg ≈ 1) except at a high speed region (umax / uavg ≈ 1.2 )in the shear free section (ε = 0 , see Fig. 20(a)). Thus, shifting the upper wall 253

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Fig. 21. The shear drag distribution for super-hydrophobic microchannel with different offset at L = 2 , Re = 1 and δ = 0.95

shifted shear free section moves the high speed fluid towards the shear free boundary, resulting in higher velocity on the water-gas interface, and thus lowering the overall shear drag and higher slip length. In conclusion, shifting the transverse grooves on the upper and lower walls of superhydrophobic microchannel can further increase the slip length (reduce the friction) and enhance the overall thermal efficiency. In the future, the experimental investigation is required to further confirm the accuracy and feasibility of the numerical findings in this study. The curvature of the air-water interface including its surface tension effect also needs to explored. Besides, superhydrophobic microchannel with non-aligned/shifted longitudinal grooves shall also be investigated similar to the transverse grooves herein.

manipulates the location of such high speed flow that it moves towards the shear free boundaries (see Fig. 20(b)–(f)), therefore resulting in larger interfacial velocity (see Fig. 21(a)) and thicker shear layer over the no-slip boundary. So the finding that the shear drag on the no-slip region is reduced as the offset increases can be explained (see Fig. 21(b)), which also results in an increment of slip length (see Fig. 5(b)). 3.4.5. Summary Generally, from the above discussion, it can be concluded that the flow field is manipulated via shifting the shear free section on the upper wall in a way that the high speed regions between the original opposite shear free sections moves towards the walls and then finally colocates on the shear free interface (ε = 0.5). In doing so, the interfacial velocity reaches its maximum value and the shear layer thickness over the noslip boundary becomes higher, hence resulting in lower overall shear drag and higher slip length. However, if the pattern length or shear free fraction is small, such influence of offset on the interfacial velocity is very minor because the shear free section is too short for flow development. Similarly, such effect becomes negligible when the Reynolds number becomes large due to the fact that the inertial force becomes dominant over the viscous force which accelerate the interfacial fluid. Conversely, such effect is still obvious at high pattern length (L = 2), low Reynolds number (Re = 1) and wide shear free fraction δ = 0.05 − 0.95).

Acknowledgements This project is supported by core funding of IHPC (A*STAR), National Key Research and Development Program of China (Grant No.2016YFB0402702), National Natural Science Foundation of China (Grant No. 51605060), the Fundamental Research Funds for the Central Universities (No.106112016CDJZR125504), Ministry of Education of the People's Republic of China, and the “thousands talents” program for the pioneer researcher and his innovation team, China. References [1] H.A. Stone, A.D. Stroock, A. Ajdari, Engineering flows in small devices: microfluidics toward a lab-on-a-chip, Annu Rev Fluid Mech 36 (2004) 381–411. [2] D.T. Chiu, D. Di Carlo, P.S. Doyle, C. Hansen, R.M. Maceiczyk, R.C. Wootton, et al., Small but perfectly formed? Successes, challenges, and opportunities for microfluidics in the chemical and biological sciences, Inside Chem 2 (2) (2017) 201–223. [3] L. Zhang, B. Ding, Q. Chen, Q. Feng, L. Lin, J. Sun, Point-of-care-testing of nucleic acids by microfluidics, Trac Trends Anal Chem 94 (2017) 106–116. [4] M.C. Giuffrida, G. Spoto, Integration of isothermal amplification methods in microfluidic devices: recent advances, Biosens Bioelectron 90 (2017) 174–186. [5] S. Hardt, D. Dadic, F. Doffing, K. Drese, G. Münchow, O. Sörensen, Development of a slug-flow PCR chip with minimum heating cycle times, Nanotechnology 1 (2004) 55–58. [6] Hagen G. Uber den Einfluss der Temperatur auf die Bewegung des Wasser in Röhren. Math. Abh. Akad. Wiss. 17. [7] H. Darcy, Recherches expérimentales relatives au mouvement de l'eau dans les tuyaux, Mallet-Bachelier, 1857. [8] J. Nikuradse, VDI-Forschungscheft 361; also NACA TM, Strömungsgesetze in Rauhen Rohren vol 1292, (1950). [9] L.F. Moody, N.J. Princeton, Friction factors for pipe flow, Trans. ASME 66 (8) (1944) 671–684. [10] R. García-Mayoral, J. Jiménez, Drag reduction by riblets, Philosophical Transactions of the royal society of London a: mathematical, Physical and Engineering Sciences 369 (1940) 1412–1427 (2011). [11] J.P. Rothstein, Slip on superhydrophobic surfaces, Annu Rev Fluid Mech 42 (2010) 89–109. [12] W. Barthlott, C. Neinhuis, Purity of the sacred lotus, or escape from contamination in biological surfaces, Planta 202 (1) (1997) 1–8. [13] C. Navier, Mémoire sur les lois du mouvement des fluides, Mémoires de L’Académie

4. Concluding summary In this paper, a numerical study on the flow inside microchannel with superhydrophobic walls with rib-groove structures was conduced, to investigate the effect of offset of grooves between the upper and lower walls on drag reduction and heat transfer for lab-on-a-chip applications. The numerical results showed that the shifted grooves on the upper and lower walls can increase the effective slip length by up to 100% especially for the high pattern length (L = 2 ), low Reynolds number (Re = 1) within a wide shear free fraction range (δ = 0.05 − 0.95). On the other hand, the shifted grooves only slightly decrease the heat transfer from the microchannel's walls into the flow, and such effect is maximized at up to 30% at high pattern length (L = 2 ), low Reynolds number (Re = 1) and medium shear free fraction (δ = 0.4 − 0.6 ). Considering heat transfer rate and pumping power at the same time in order to evaluate the overall cooling efficacy, the thermal goodness factor for microchannel with offset can become higher than those without offset. The increase in the goodness factor becomes obvious (up to 20%) at high pattern length (L = 2), and large shear free fraction (δ > 0.4 ) within a wide range of Reynolds number. Furthermore, the detailed analysis of the velocity field showed that the 254

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