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Research Papers: Forced Convection

Solution of the Graetz–Nusselt Problem for Liquid Flow Over Isothermal Parallel Ridges OPEN ACCESS

[+] Author and Article Information
Georgios Karamanis

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: georgios.karamanis@tufts.edu

Marc Hodes

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: marc.hodes@tufts.edu

Toby Kirk

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: toby.kirk12@imperial.ac.uk

Demetrios T. Papageorgiou

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: d.papageorgiou@imperial.ac.uk

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 13, 2016; final manuscript received March 7, 2017; published online May 2, 2017. Assoc. Editor: Jim A. Liburdy.

J. Heat Transfer 139(9), 091702 (May 02, 2017) (12 pages) Paper No: HT-16-1508; doi: 10.1115/1.4036281 History: Received August 13, 2016; Revised March 07, 2017

We consider convective heat transfer for laminar flow of liquid between parallel plates that are textured with isothermal ridges oriented parallel to the flow. Three different flow configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned; and both plates textured, but the ridges staggered by half a pitch. The liquid is assumed to be in the Cassie state on the textured surface(s), to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. Heat is exchanged with the liquid either through the ridges of one plate with the other plate adiabatic, or through the ridges of both plates. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). Axial conduction is neglected and the inlet temperature profile is arbitrary. We solve for the three-dimensional developing temperature profile assuming a hydrodynamically developed flow, i.e., we consider the Graetz–Nusselt problem. Using the method of separation of variables, the thermal problem is essentially reduced to a two-dimensional eigenvalue problem in the transverse coordinates, which is solved numerically. Expressions for the local Nusselt number and those averaged over the period of the ridges in the developing and fully developed regions are provided. Nusselt numbers averaged over the period and length of the domain are also provided. Our approach enables the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics (CFD) solver.

Background.

Superhydrophobic surfaces, i.e., those with hydrophobic micro- and/or nanoscale protrusions, are of interest in the context of liquid flow through microchannels, especially in direct liquid cooling applications as a means to reduce flow and thus caloric resistance [1]. When criteria are met [1,2], the solid–liquid interfaces are confined to the tips of the structures, forming a composite interface along with the liquid–gas interfaces (menisci), as per Fig. 1, and the liquid is said to be in the unwetted or Cassie state [3,4]. Then, the solid–liquid interfaces are subjected to the no-slip [5,6] boundary condition, whereas the menisci are subjected to a low-shear boundary condition. Thus, a lubrication effect is provided which reduces caloric resistance. However, the reduction in the solid–liquid interface area reduces the Nusselt number (Nu) and thus increases the convective component of thermal resistance. A net reduction of the total, i.e., caloric plus convective, thermal resistance can be achieved with proper sizing of the structures [1] and it requires knowledge of Nusselt numbers as a function of the geometry of the channel and the structures. The surfaces can be textured with a variety of periodic structures such as pillars, transverse ridges, or parallel ridges [2].1 The latter configuration for the ridges is examined here and it is the most favorable from a heat transfer perspective [1,7].

The hydrodynamic effects of structured surfaces with parallel ridges have been studied for flat and curved menisci [812]. However, there is a relatively limited body of work on heat transfer effects. Enright et al. [7] derived an expression for the Nusselt number for fully developed flow through a microchannel with isoflux structured surfaces as a function of the (apparent) hydrodynamic and thermal slip lengths. Moreover, Enright et al. [7] developed analytical expressions for slip lengths for structured surfaces with parallel or transverse ridges or pillar arrays assuming flat and adiabatic menisci. Ng and Wang [13] derived semi-analytical expressions for the thermal slip length for isothermal parallel ridges while accounting for conduction through the gas phase. Lam et al. [14] derived expressions for the thermal slip length for isoflux and isothermal parallel ridges accounting for small meniscus curvature. Hodes et al. [15] captured the effects of evaporation and condensation along menisci on the thermal slip length for isoflux ridges. Lam et al. [16] developed expressions for the Nusselt number for Couette flow as a function of the slip lengths for various boundary conditions. Also, Lam et al. [16] discussed when Nu results from the molecular slip literature can be used to capture the effects of apparent slip. Maynes et al. [17] numerically investigated the thermal transport in microchannels with isothermal transverse ridges and flat menisci taking into account the heat transfer through the gas in the cavities. Maynes et al. [18] and Maynes and Crockett [19] developed expressions for the Nusselt number and the thermal slip length for microchannels with isoflux transverse and parallel ridges, respectively, assuming flat menisci and using the Navier slip approximation for the velocity profile. Kirk et al. [20] also developed expressions for the Nusselt number for isoflux parallel ridges using the fully resolved velocity field in the thermal energy equation. Furthermore, Kirk et al. [20] accounted for small meniscus curvature using a boundary perturbation method.

The present work develops semi-analytical expressions for the Nusselt number for the case of isothermal parallel ridges for hydrodynamically developed and thermally developing flow with negligible axial conduction, i.e., for the Graetz–Nusselt problem [2224].2 It is emphasized that we do not assume diffusive heat transfer near the composite interface.

Assumptions.

We consider three different configurations for the parallel ridges: (1) one plate textured and the other one smooth, as per Fig. 2; (2) both plates textured and the ridges aligned in the transverse direction (see Fig. 14); and (3) both plates textured and the ridges staggered in the transverse direction by half a pitch (see Fig. 18). The solution approach is similar in all three configurations. Therefore, it suffices to present here the detailed analyses for the first one and the relevant parts of the analysis for the other two configurations in the Appendix.

The domain (D) for the first configuration is depicted in Fig. 2, where |x|d and 0 ≤ y ≤ H and where 2d is the pitch of the ridges and H is the distance between the parallel plates. The hydraulic diameter of the domain (Dh) is 2H. The width of the meniscus is 2a. The curvature of the meniscus is neglected [14,20] and the triple contact lines coincide with the corners of the ridges at x=|a| and y = 0. The cavities may be filled with inert gas and/or vapor. Along the composite interface (y = 0), a no-shear boundary condition is applied for |x|<a and a no-slip one is imposed for a<|x|<d. A no-slip boundary condition is also imposed on the smooth upper plate. Symmetry boundary conditions apply at the x=|d| boundaries. The flow is pressure driven, steady, laminar, hydrodynamically developed, and thermally developing with constant thermophysical properties and negligible axial conduction and viscous dissipation. The ridges on the lower plate are isothermal, whereas the upper plate and the meniscus are considered adiabatic. The temperature profile in the liquid starts developing at z = 0 from an arbitrary (unless otherwise stated) two-dimensional distribution Tin (x, y). Effects due to Marangoni stresses [25,26], evaporation and condensation [15], and gas diffusion in the liquid phase are neglected. The independent dimensionless variables are the solid fraction of the ridge (ϕ=(da)/d) and the aspect ratio of the domain (H/d).

Hydrodynamic Problem.

The relevant form of the streamwise momentum equation is Display Formula

(1)2wx2+2wy2=1μdpdz

where w is the streamwise velocity, dp/dz is the prescribed pressure gradient, and μ is the dynamic viscosity. Denoting nondimensional variables with tildes, Eq. (1) and the boundary conditions imposed on it are rendered dimensionless by defining Display Formula

(2)x̃=xa
Display Formula
(3)ỹ=ya
Display Formula
(4)H̃=Ha
Display Formula
(5)w̃=2μwaH(dp/dz)

Then, Eq. (1) becomes Display Formula

(6)2w̃x̃2+2w̃ỹ2=2H̃

subjected to Display Formula

(7)w̃ỹ=0for |x̃|<1, ỹ=0
Display Formula
(8)w̃=0for 1<|x̃|<d̃, ỹ=0
Display Formula
(9)w̃=0for |x̃|<d̃, ỹ=H̃
Display Formula
(10)w̃x̃=0for |x̃|=d̃, 0<ỹ<H̃

where d̃=d/a is the dimensionless (half) pitch of the ridges.

This hydrodynamic problem has been solved analytically [8] and semi-analytically [9,11]. However, no analytical solutions have been found for the problems in the Appendix. Therefore, the velocity field is numerically determined here for all cases, which also facilitates the numerical solution of the thermal energy equation in Sec. 2.4. The numerical results were validated against the analytical solution [8] using the computed Poiseuille number (fRe), where Display Formula

(11)Re=ρw¯Dhμ
Display Formula
(12)f=dpdz2Dhρw¯2
Display Formula
(13)w¯=12dH0Hddwdxdy

are the Reynolds number based on the hydraulic diameter, the friction factor and the mean velocity of the flow, respectively, and ρ is the liquid density. From Eqs. (5) and (11)(13), it follows that the Poiseuille number is given by Display Formula

(14)fRe=16H̃w̃¯

where Display Formula

(15)w̃¯=12d̃H̃0H̃d̃d̃w̃dx̃dỹ

is the dimensionless mean velocity of the flow.

Thermal Problem.

The relevant form of the dimensional thermal energy equation is Display Formula

(16)wTz=α(2Tx2+2Ty2)

where T and α are the temperature and the thermal diffusivity of the liquid, respectively. Defining the dimensionless temperature T̃ and the dimensionless streamwise coordinate z̃ as Display Formula

(17)T̃=TTslTrefTsl
Display Formula
(18)z̃=2αμza3H(dp/dz)

where Tsl is the constant temperature of the ridge and Tref is a reference temperature for the problem, Eq. (16) becomes Display Formula

(19)w̃T̃z̃=2T̃x̃2+2T̃ỹ2

It is subject to the following boundary conditions: Display Formula

(20)T̃ỹ=0for |x̃|<1, ỹ=0
Display Formula
(21)T̃=0for 1<|x̃|<d̃, ỹ=0
Display Formula
(22)T̃ỹ=0for |x̃|<d̃, ỹ=H̃
Display Formula
(23)T̃x̃=0for |x̃|=d̃, 0<ỹ<H̃
Display Formula
(24)T̃=T̃inat z̃=0

where T̃in(x̃,ỹ) is the prescribed dimensionless temperature profile at the inlet of the domain (z̃=0).

Seeking separable solutions of the form T̃=ψ(x̃,ỹ)g(z̃), which separate the streamwise coordinate z̃ from the transverse coordinates x̃ and ỹ, it can be shown that g(z̃)=exp(λz̃) and ψ(x̃,ỹ) satisfies Display Formula

(25)2ψ=λw̃ψ

with λ real and positive.3 Note that ψ(x̃,ỹ) cannot be separated further into a product of a function of x̃ and one of ỹ since the velocity field, w̃=w̃(x̃,ỹ), is not separable in such a way. Equation (25) satisfies the boundary conditions Display Formula

(26)ψỹ=0for |x̃|<1, ỹ=0
Display Formula
(27)ψ=0for 1<|x̃|<d̃, ỹ=0
Display Formula
(28)ψỹ=0for |x̃|<d̃, ỹ=H̃
Display Formula
(29)ψx̃=0for |x̃|=d̃, 0<ỹ<H̃

and so constitutes a two-dimensional Sturm–Liouville eigenvalue problem for λ and the corresponding eigenfunction ψ, with weight function w̃(x̃,ỹ). Assuming that the eigenvalues are discrete and there are infinitely many, let λi and ψi denote the ith eigenvalue and eigenfunction, respectively, ordered such that 0<λ1<λ2<<λi<. The eigenfunctions are orthogonal with respect to the inner product defined by Display Formula

(30)F1,F2=0H̃d̃d̃w̃F1F2dx̃dỹ

that is Display Formula

(31)ψi,ψj=0 for ij

Moreover, the eigenfunctions are unique up to a multiplication by a constant. Thus, for the rest of the present analysis, we normalize ψi such that Display Formula

(32)ψi,ψi=1

The eigenvalue problem is solved numerically. The calculation of ψi and λi is detailed in Sec. 2.4, and for the rest of the present analysis, they are assumed to be known.

We proceed by expressing the solution T̃(x̃,ỹ,z̃) as a linear combination of the eigenfunctions Display Formula

(33)T̃(x̃,y,̃z̃)=i=1ciψi(x̃,ỹ)exp(λiz̃)

using the expansion coefficients ci. The ci are determined by taking the inner product of Eq. (33) with ψi at the inlet z̃=0, where T̃(x̃,ỹ,0)=T̃in(x̃,ỹ), giving Display Formula

(34)ci=ψi,T̃in

Substituting Eq. (34) into Eq. (33), the dimensionless temperature profile takes the form Display Formula

(35)T̃(x̃,ỹ,z̃)=i=1ψi,T̃inψi(x̃,ỹ)exp(λiz̃)

Nusselt Number Expressions.

The local Nusselt number is defined as Display Formula

(36)Nul=hlDhk

where hl is the local heat transfer coefficient and k is the thermal conductivity of the liquid. An energy balance at a point along the ridges yields Display Formula

(37)kTy|y=0=hl(TslTb)

where Tb is the bulk temperature of the liquid defined as Display Formula

(38)Tb=12dHw¯0HddwTdxdy

Combining Eqs. (36)(38), the local Nusselt number can be written in terms of dimensionless quantities as Display Formula

(39)Nul=2H̃T̃bT̃ỹ|ỹ=0

where T̃b is the dimensionless bulk temperature of the liquid defined as Display Formula

(40)T̃b=12d̃H̃w̃¯T̃,1

Next, combining Eqs. (14), (33), (39), and (40) yields Display Formula

(41)Nul=64H̃3d̃fRei=1ψi,T̃inψiỹ|ỹ=0exp(λiz̃)i=1ψi,T̃inψi,1exp(λiz̃)
The local Nusselt number for the limiting case of fully developed flow (Nul,fd) follows from the evaluation of Eq. (41) as z̃. Given that the λi are real and 0 < λ1 < λ2 < … < λi < … → , upon dividing both the numerator and the denominator of Eq. (41) by eλ1z̃ and letting z̃, only the first term of each sum remains. It follows that Display Formula
(42)Nul,fd=64H̃3d̃fReψ1ỹ|ỹ=0ψ1,1

Nul,fd is a function only of the first eigenfunction and it is independent of the inlet temperature profile. However, in the thermally developing region, Nul is a function of T̃in.

The Nusselt number averaged over the composite interface is Display Formula

(43)Nu=12dddNuldx

Substituting Eq. (41) into Eq. (43) and utilizing the symmetry of the eigenvalue problem with respect to the y-axis and the boundary condition given by Eq. (26) yields Display Formula

(44)Nu=64H̃3fRei=1ψi,T̃in1d̃ψiỹ|ỹ=0dx̃ exp(λiz̃)i=1ψi,T̃inψi,1exp(λiz̃)

Next, we express the integral in the numerator of Eq. (44) as a function of the inner product ψi,1. This is because it is more accurate to numerically evaluate ψi,1 than 1d̃ψi/ỹ|ỹ=0dx̃ as the latter requires numerical differentiation in order to evaluate the derivative at the boundary. The steps are as follows: First, we rearrange Eq. (25) and integrate it over the cross section of the domain to obtain Display Formula

(45)0H̃d̃d̃1λi2ψidx̃dỹ=ψi,1

Then, applying the divergence theorem to the left-hand side of Eq. (45), we find that Display Formula

(46)1λiDψi·n̂dS̃=ψi,1

where n̂ is the outward pointing unit normal vector on the boundary ∂D and S̃ is a dimensionless coordinate along ∂D. Then, imposing boundary conditions (26)(29), we obtain Display Formula

(47)1d̃ψiỹ|ỹ=0dx̃=λi2ψi,1

Inserting this result into Eq. (44) yields Display Formula

(48)Nu=32H̃3fRei=1ψi,T̃inψi,1λiexp(λiz̃)i=1ψi,T̃inψi,1exp(λiz̃)

The Nusselt number averaged over the composite interface for the limiting case of fully developed flow (Nufd) follows in the same manner as Eq. (42) and it is given by Display Formula

(49)Nufd=32H̃3λ1fRe

Nufd is a function only of the first eigenvalue λ1 and it is independent of T̃in.

The Nusselt number averaged over the composite interface and the streamwise length of the domain is defined as Display Formula

(50)Nu¯=1z̃0z̃Nudz̃

Substituting Eq. (48) into Eq. (50), it follows that Display Formula

(51)Nu¯=32H̃3fRe1z̃ln(i=1ψi,T̃inψi,1i=1ψi,T̃inψi,1exp(λiz̃))

In the case of a uniform inlet temperature (UIT) at Tref, we have T̃in=1. Then, the foregoing expression reduces to Display Formula

(52)Nu¯UIT=32H̃3fRe1z̃ln(1T̃b)

It is emphasized that Eqs. (41), (48), (51), and (52) hold for all streamwise locations z̃; however, to achieve a given accuracy, more terms are required in the evaluation of each sum as z̃ is decreased. Moreover, expressions for the Nusselt number averaged only over the width of the ridge rather than the composite interface follow by dividing Eqs. (48), (49), (51), and (52) by the solid fraction.

Solution of the Eigenvalue Problem.

The two-dimensional eigenvalue problem defined by Eqs. (25)(29) was numerically solved for multiple values of the aspect ratio and the solid fraction of the domain using a finite element method. The solution process is iterative and it was coded in MATLAB® employing the partial differential equation (PDE) toolbox [27]. The algorithm exploits the symmetry of the hydrodynamic and the eigenvalue problems with respect to the y-axis in order to increase computational efficiency; therefore, the boundary conditions given by Eqs. (10) and (29) were both modified to apply at the x̃=0 and x̃=d̃ boundaries.

The steps of the algorithm are as follows: First, the half domain is discretized with an initial number of finite elements. Next, Eq. (6) is solved subject to the new form of the boundary conditions given by Eqs. (7)(9) to determine the two-dimensional velocity profile w̃(x̃,ỹ) required in Eq. (25). Then, Eq. (25) subject to the new form of the boundary conditions (26)(29) is solved to determine all the eigenvalues in the interval 0 ≤ λi ≤ UB along with their corresponding eigenfunctions ψi [27]. The upper bound (UB) was varied depending on the number of the eigenvalues sought (MATLAB® requires prescription of the aforementioned interval for the sought λi because it solves the discretized eigenvalue problem by applying the Arnoldi algorithm to a shifted and inverted version of the original pencil [27]). Next, mesh refinement is implemented and the algorithm proceeds from step two until the change in the computed value of Nufd is less than 0.01%—typically this required 3.5 × 105 elements that were adaptively placed in regions of sharp gradients. Finally, the computed eigenfunctions are normalized to satisfy Eq. (32).

The computations were validated in four ways. First, computed Poiseuille numbers were compared against those that follow from an analytical solution for the velocity profile by Philip [8] at a solid fraction ϕ=0.25 and various values of H/d as per Fig. 3; agreement was within 0.006%. Second, fRe and Nufd were computed in the limit of ϕ1, i.e., for fully developed flow between two smooth parallel plates where one is isothermal and the other is adiabatic and fRes = 96 and Nufd,s = 4.86 [28]. Agreement was within 0.03% of fRe and 0.009% of Nufd. Thirdly, the boundary condition at ỹ=H̃ was changed to an isothermal one and NuUIT and Nu¯UIT were computed at various streamwise locations in the limit ϕ1. The results were compared with those provided by Shah [29]. We find that if only ten terms are used in the series given by Eqs. (48) and (52), the difference between our results and those provided in Ref. [29] was found to be less than 1.4% and 0.3%, respectively, even down to thermal entrance lengths z*=z/(DhPe)=z̃/(4H̃2w̃¯)=1.5×104. It is noted that we choose to present the results for the Nusselt number in the thermal entrance region as function of z* instead of z̃ to enable direct comparison of the results with those for nonstructured channels. However, z̃ is a more appropriate quantity for the case at hand given that w̃¯ is a function of the solid fraction and the aspect ratio in the case of channels with textured surfaces.

Finally, semi-analytical values of Nufd were compared with those obtained using FLUENT [30], which is a general three-dimensional CFD solver. This was done for the present case and for those in the Appendix when only one plate has isothermal ridges. Conditions of hydrodynamically and thermally developed flow were imposed by using translational periodic boundary conditions between the inlet and outlet—for details, see Refs. [26], [30], and [31]. The governing equations were discretized using a second-order upwind scheme and were solved using the pressure-based coupled algorithm provided by FLUENT. The aspect ratio, solid fraction, Reynolds number, and Péclet number were taken to be H/d=4,ϕ=0.3, Re = 2342.89, and Pe = 100.87, respectively. This value of the Péclet number was chosen to enable comparisons with the present analysis which assumes Pe ≫ 1 given that FLUENT accounts for axial conduction. This study considers steady flows only and so the solutions obtained are laminar even at Reynolds numbers as high as 2342.89. Adaptive mesh refinement was employed, with the final computational mesh containing as many as 9 × 105 hexahedral elements. The computed Nufd for the three geometries mentioned above are 4.124, 3.836, and 3.836 correct to three decimals, and the discrepancy with the predicted values from the analysis are 0.12%, 0.19%, and 0.19%. (The aligned and staggered values are almost identical since H/d is large enough and makes the alignment unimportant—see the Appendix for more details.)

It is important to note that the present analysis produces results for Nufd in less than 3 min on a desktop computer, whereas FLUENT requires several hours to converge. Furthermore, it provides the means to evaluate the Nusselt number averaged over the composite interface and, additionally, the streamwise length of the domain at any z̃, quantities which are prohibitively expensive to compute using a general CFD code.

In this section, we present the results for the case at hand and some representative ones for the cases in the Appendix for comparison. The additional results are presented in the Appendix.

Figure 4 plots the fully developed Nusselt number averaged over the composite interface, Nufd, versus the solid fraction ϕ for aspect ratios of H/d = 1, 1.5, 2, 4, 6, 10, and 100, when the lower plate is textured with isothermal ridges and the upper one is smooth and adiabatic. The dashed curve corresponds to smooth plates with Nusselt number Nufd,s = 4.86. The results obey the expected asymptotic behavior as ϕ1, with NufdNufd,s, irrespective of the aspect ratio. Additionally, as ϕ0,Nufd tends to zero because the available area for heat transfer vanishes. Moreover, for a given ϕ (excluding the aforementioned limits) as H/d → 0 and H/d,Nufd tends to zero and to Nufd,s, respectively. This is because for H/d → 0 heat is mainly advected by the part of the flow above the shear-free meniscus as opposed to the relatively stagnant liquid above the ridges degrading the heat transfer. In the other limit, as H/d the difference between the temperature of the ridge and the mean temperature of the composite interface becomes significantly smaller than the difference between the temperature of the ridge and the bulk temperature of the flow.

Figure 5 plots the fully developed local Nusselt number, Nul,fd, versus the normalized coordinate along the ridge (x̃1)/(d̃1) for H/d = 10 and ϕ=0.01, 0.1 and 0.99. The results show that Nul,fd increases with decreasing ϕ, indicating a local enhancement of heat transfer. The same trend has been observed in previous studies [18] and it is due to the fact that as ϕ0, the velocity of the liquid close to the ridge increases. Figure 6 plots the fully developed Nusselt number averaged over the width of the ridge, Nufd,ridge, versus the solid fraction. In summary, the overall effect of the decrease in the available heat transfer area and the local enhancement of heat transfer for ϕ<1 is an increase in the convective portion of the total thermal resistance that is completely captured in Fig. 4.

For the case of uniform inlet temperature (UIT), Figs. 7 and 8 plot NuUIT and Nu¯UIT versus z* for ϕ=0.01 and 0.1, respectively. The results were computed using the first 29 eigenvalues.4 The results exhibit the correct asymptotic behavior as z* → 0 and z* → ; in the former case, both NuUIT and Nu¯UIT increase monotonically with decreasing z*, and in the latter case, they tend to Nufd. The first ten eigenvalues and the corresponding expansion coefficients that were computed for H/d = 4 at ϕ=0.01 and 0.1 are provided in Table 1.

Figures 9 and 10 compare the computed values of fRe and Nufd, respectively, for the case when one plate is textured and the other one is smooth (solid curves), to the case in the Appendix when both plates are textured and the ridges are aligned in the transverse direction (dashed curves). In both cases, one plate has isothermal ridges and the other one is adiabatic. Although fRe is significantly reduced if both plates are textured, especially as ϕ0, Nufd changes by only a small fraction due to texturing. More importantly, as per Fig. 10, Nufd decreases if both plates are textured and heat is exchanged through the domain only through the isothermal ridges of one plate. This can be explained by comparing Figs. 11 and 12 that present the contour plots of the scaled dimensionless streamwise velocity w̃/w̃¯ for the cases at hand for H/d = 4 and ϕ=0.3. Indeed, when both plates are textured and the ridges are aligned, as per Fig. 12, the flow exhibits higher velocities closer to the center of the domain, but lower velocities closer to the ridge. Thus, the convective thermal transport is degraded. When one plate is smooth, however (see Fig. 11), the velocity in the vicinity of the ridge is higher and so enhances heat transfer. This can be quantified by considering the ratio (R) of the average velocity of the flow in an area close to the ridge, i.e., 0x̃d̃ and 0ỹH̃/2, over the mean velocity of the flow

Display Formula

(53)R=20H̃/20d̃w̃dx̃dỹd̃H̃w̃¯

When both plates are textured and the ridges are aligned, R is equal to 1 due to symmetry, but, when one plate is smooth, R becomes 1.127 for the prescribed values of H/d and ϕ, which indicates higher velocities close to the ridge. The same observations can be made for the case when both plates are textured, but the ridges are staggered in the transverse direction. The corresponding plots for fRe and Nufd and the contour plot of the scaled dimensionless streamwise velocity are presented in the Appendix.

Finally, Fig. 13 compares the computed values of Nufd when both plates are textured with isothermal aligned ridges against those calculated for the same configuration but for isoflux ridges (Nufd,if) by Kirk et al. [20]. The results show that depending on the aspect ratio H/d, there is a range for ϕ where the fully developed Nusselt number averaged over the composite interface for isothermal ridges is slightly higher than for isoflux ridges despite the fact that the fully developed Nusselt number for smooth isothermal plates is smaller than that for smooth isoflux plates.

We developed semi-analytical expressions for the Nusselt number for the case of hydrodynamically developed and thermally developing flow between parallel plates that are textured with ridges oriented parallel to the flow. The ridges of one plate are isothermal and the other plate can be smooth and adiabatic, or textured with adiabatic or isothermal ridges. When both plates are textured, the ridges can be aligned or staggered by half a pitch in the transverse direction. The menisci between the ridges were considered to be flat and adiabatic. The solid–liquid interface and the menisci were subjected to no-slip and no-shear boundary conditions, respectively. Using separation of variables, we expressed the three-dimensional temperature field as an infinite sum of the product of an exponentially decaying function of the streamwise coordinate and a second eigenfunction depending on the transverse coordinates. The latter eigenfunctions satisfy a two-dimensional Sturm–Liouville problem from which the eigenvalues and eigenfunctions follow numerically.

The derived expressions for the local Nusselt number, the Nusselt number averaged over the composite interface, and the Nusselt number averaged over the composite interface and the streamwise length of the domain indicate that the Nusselt number is a function of the dimensionless streamwise coordinate, the aspect ratio of the domain, the solid fraction, and the inlet temperature profile. Expressions were also derived for the fully developed local Nusselt number and for the fully developed Nusselt number averaged over the composite interface in terms of the first eigenfunction and of the first eigenvalue, respectively.

The results indicate that the Nusselt number averaged over the composite interface decreases as the aspect ratio and/or the solid fraction decrease. Moreover, it was observed that when one plate is adiabatic, the configuration where the adiabatic plate is smooth provides a higher Nusselt number than when it is textured. Finally, using the present analysis, the fully developed local Nusselt number and the fully developed Nusselt number averaged over the composite interface can be computed in a small fraction of the time that is required by a general CFD solver. More importantly, the analysis provides semi-analytical expressions to evaluate the Nusselt number averaged over the composite interface and, additionally, the streamwise length of the domain at any location, quantities which are prohibitively expensive to compute using a general CFD code.

The work of GK and MH was supported by the National Science Foundation under Grant No. 1402783. The work of DTP was supported in part by the Engineering and Physical Sciences Research Council (UK) Grant Nos. EP/K041134 and EP/L020564. The work of TK was supported by an EPSRC-UK doctoral scholarship.

  • a =

    half meniscus width, m

  • ci =

    expansion coefficients

  • d =

    half ridge pitch, m

  • D =

    domain

  • d̃ =

    dimensionless half ridge pitch, d/a

  • Dh =

    hydraulic diameter, 2H

  • dp/dz =

    prescribed pressure gradient, Pa/m

  • f =

    friction factor, 2Dh(dp/dz)/(ρw¯2)

  • fRe =

    Poiseuille number

  • H =

    distance between parallel plates, m

  • H̃ =

    dimensionless distance between parallel plates, H/a

  • hl =

    local heat transfer coefficient, W/(m2 K)

  • k =

    thermal conductivity of liquid, W/(m K)

  • LB =

    lower bound of λi

  • n̂ =

    outward pointing unit normal vector on boundaries

  • Nu =

    Nusselt number averaged over the composite interface

  • Nu¯ =

    Nusselt number averaged over the composite interface and the streamwise length of the domain

  • Nul =

    local Nusselt number, hlDh/k

  • Pe =

    Péclet number, w¯Dh/α

  • R =

    average velocity ratio close to the ridge, (20H̃/20d̃w̃dx̃dỹ)/(d̃H̃w̃¯)

  • Re =

    Reynolds number, ρw¯Dh/μ

  • S̃ =

    dimensionless coordinate along ∂D

  • T =

    temperature, °C

  • T̃ =

    dimensionless temperature, (TTsl)/(TinTsl)

  • Tb =

    bulk temperature

  • T̃b =

    dimensionless bulk temperature

  • Tin =

    inlet temperature, °C

  • T̃in =

    dimensionless inlet temperature

  • Tref =

    reference temperature, °C

  • Tsl =

    ridge temperature, °C

  • UB =

    upper bound of λi

  • w =

    streamwise velocity, m/s

  • w¯ =

    mean velocity, m/s

  • w̃ =

    dimensionless velocity, 2μw/[aH(dp/dz)]

  • w̃¯ =

    dimensionless mean velocity

  • x =

    lateral coordinate, m

  • x̃ =

    dimensionless lateral coordinate, x/a

  • y =

    vertical coordinate, m

  • ỹ =

    dimensionless vertical coordinate, y/a

  • z =

    streamwise coordinate, m

  • z̃ =

    dimensionless streamwise coordinate, 2αμz/[a3H(dp/dz)]

  • z* =

    dimensionless streamwise coordinate for the thermal entrance region, z/(Dh Pe)

  • ∂D =

    boundary of the dimensionless domain

 Greek Symbols
  • α =

    thermal diffusivity, m2/s

  • λi =

    ith eigenvalue

  • μ =

    dynamic viscosity, Pa·s

  • ρ =

    density, kg/m3

  • ϕ =

    solid fraction, (d − a)/d

  • ψi =

    ith eigenfunction

 Subscripts
  • al =

    textured plates with aligned ridges

  • fd =

    fully developed flow

  • if =

    isoflux ridges

  • ridge =

    indicates quantity based on the width of the ridge

  • s =

    smooth plates

  • t–s =

    one textured and one smooth plate

  • UIT =

    uniform inlet temperature

Appendix

Sections A.1 and A.2 provide the necessary information for the extension of the present analysis to the configurations when both plates are textured with parallel ridges and the ridges are either aligned or staggered, respectively. Each subsection covers the cases when the ridges of one plate are isothermal and those of the other one are either adiabatic or isothermal.

Both Plates Textured, Aligned Ridges

When both plates are textured and the ridges are aligned as indicated in Fig. 14, the boundary conditions for the hydrodynamic problem given by Eqs. (7) and (8) apply rather than Eq. (9) at ỹ=H̃. The computed Poiseuille numbers are presented in Fig. 15.

If only the lower plate has isothermal ridges and the upper one has adiabatic ridges, the boundary conditions for the thermal problem and for the eigenvalue problem are identical to those in Sec. 2.2. The expressions for Nul, Nul,fd, Nu, Nufd, Nu¯, and Nu¯UIT are identical to those given by Eqs. (41), (42), (48), (49), (51), and (52) and the reader is referred to those expressions for their detailed form. The computed Nufd is presented in Fig. 16.

If the ridges of both plates are isothermal, the thermal boundary conditions given by Eqs. (20) and (21) apply rather than Eq. (22) at ỹ=H̃. In terms of the eigenvalue problem, the boundary conditions given by Eqs. (26) and (27) apply rather than Eq. (28) at ỹ=H̃.

In this case, the expressions for Nul and Nul,fd are identical to those given by Eqs. (41) and (42). However, the Nusselt number averaged over the composite interfaces is Display Formula

(A1)Nu=12dddNul|ỹ=0dx+12dddNul|ỹ=H̃dx

Moreover, following the same steps as in Sec. 2.3, it follows that Display Formula

(A2)1d̃ψiỹ|ỹ=0dx̃=λi4ψi,1

Thus, the expressions for the averaged values of the Nusselt number take the form Display Formula

(A3)Nu=16H̃3fRei=1ψi,T̃inψi,1λiexp(λiz̃)i=1ψi,T̃inψi,1exp(λiz̃)
Display Formula
(A4)Nufd=16H̃3fReλ1
Display Formula
(A5)Nu¯=16H̃3z̃fReln(i=1ψi,T̃inψi,1i=1ψi,T̃inψi,1exp(λiz̃))   
Display Formula
(A6)Nu¯UIT=16H̃3z̃fReln(1T̃b)

The computed Nufd is presented in Fig. 17.

Both Plates Textured, Staggered Ridges

When both plates are textured and the ridges are staggered in the transverse direction by half a pitch as shown in Fig. 18, the relevant boundary conditions for the hydrodynamic problem become Display Formula

(A7)w̃ỹ=0for |x̃|<1, ỹ=0
Display Formula
(A8)w̃=0for 1<|x̃|<d̃, ỹ=0
Display Formula
(A9)w̃ỹ=0for d̃1<|x̃|<d̃, ỹ=H̃
Display Formula
(A10)w̃=0for |x̃|<d̃1, ỹ=H̃
Display Formula
(A11)w̃x̃=0for |x̃|=d̃, 0<ỹ<H̃

The computed Poiseuille numbers are presented in Fig. 19. Figure 20 presents the contour plot of the scaled dimensionless streamwise velocity w̃/w̃¯ for this case.

If only the lower plate has isothermal ridges and the upper one has adiabatic ridges, the boundary conditions for the thermal problem and for the eigenvalue problem are identical to those in Sec. 2.2. The expressions for Nul, Nul,fd, Nu, Nufd, Nu¯, and Nu¯UIT are identical to those given by Eqs. (41), (42), (48), (49), (51), and (52). The computed Nufd is presented in Fig. 21.

When the ridges on both plates are isothermal, the boundary conditions for the thermal problem become Display Formula

(A12)T̃ỹ=0for |x̃|<1, ỹ=0
Display Formula
(A13)T̃=0for 1<|x̃|<d̃, ỹ=0
Display Formula
(A14)T̃ỹ=0for d̃1<|x̃|<d̃, ỹ=H̃
Display Formula
(A15)T̃=0for |x̃|<d̃1, ỹ=H̃
Display Formula
(A16)T̃x̃=0for |x̃|=d̃, 0<ỹ<H̃
Display Formula
(A17)T̃=T̃infor z̃=0

while those of the eigenvalue problem read Display Formula

(A18)ψiỹ=0for |x̃|<1, ỹ=0
Display Formula
(A19)ψi=0for 1<|x̃|<d̃, ỹ=0
Display Formula
(A20)ψiỹ=0for d̃1<|x̃|<d̃, ỹ=H̃
Display Formula
(A21)ψi=0for |x̃|<d̃1, ỹ=H̃
Display Formula
(A22)ψix̃=0for |x̃|=d̃, 0<ỹ<H̃

The expressions for Nul, Nul,fd, Nu, Nufd, Nu¯, and Nu¯UIT are identical to those given by Eqs. (41), (42), and (A3)(A6). The computed Nufd is presented in Fig. 22.

Lam, L. S. , Hodes, M. , and Enright, R. , 2015, “ Analysis of Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces,” ASME J. Heat Transfer, 137(9), p. 091003. [CrossRef]
Lobaton, E. J. , and Salamon, T. R. , 2007, “ Computation of Constant Mean Curvature Surfaces: Application to the Gas–Liquid Interface of a Pressurized Fluid on a Superhydrophobic Surface,” J. Colloid Interface Sci., 314(1), pp. 184–198. [CrossRef] [PubMed]
Quéré, D. , 2005, “ Non-Sticking Drops,” Rep. Prog. Phys., 68(11), p. 2495. [CrossRef]
Cassie, A. B. D. , and Baxter, S. , 1944, “ Wettability of Porous Surfaces,” Trans. Faraday Soc., 40, pp. 546–551. [CrossRef]
Huang, D. M. , Sendner, C. , Horinek, D. , Netz, R. R. , and Bocquet, L. , 2008, “ Water Slippage Versus Contact Angle: A Quasiuniversal Relationship,” Phys. Rev. Lett., 101(22), p. 226101. [CrossRef] [PubMed]
Cottin-Bizonne, C. , Steinberger, A. , Cross, B. , Raccurt, O. , and Charlaix, E. , 2008, “ Nanohydrodynamics: The Intrinsic Flow Boundary Condition on Smooth Surfaces,” Langmuir, 24(4), pp. 1165–1172. [CrossRef] [PubMed]
Enright, R. , Hodes, M. , Salamon, T. , and Muzychka, Y. , 2013, “ Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels,” ASME J. Heat Transfer, 136(1), p. 012402. [CrossRef]
Philip, J. R. , 1972, “ Flows Satisfying Mixed No-Slip and No-Shear Conditions,” Z. Angew. Math. Phys., 23(3), pp. 353–372. [CrossRef]
Sbragaglia, M. , and Prosperetti, A. , 2007, “ A Note on the Effective Slip Properties for Microchannel Flows With Ultrahydrophobic Surfaces,” Phys. Fluids, 19(4), p. 043603. [CrossRef]
Rothstein, J. P. , 2010, “ Slip on Superhydrophobic Surfaces,” Annu. Rev. Fluid Mech., 42(1), pp. 89–109. [CrossRef]
Teo, C. J. , and Khoo, B. C. , 2009, “ Analysis of Stokes Flow in Microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves,” Microfluid. Nanofluid., 7(3), pp. 353–382. [CrossRef]
Teo, C. J. , and Khoo, B. C. , 2010, “ Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature,” Microfluid. Nanofluid., 9(2), pp. 499–511. [CrossRef]
Ng, C.-O. , and Wang, C. Y. , 2014, “ Temperature Jump Coefficient for Superhydrophobic Surfaces,” ASME J. Heat Transfer, 136(6), p. 064501. [CrossRef]
Lam, L. S. , Hodes, M. , Karamanis, G. , Kirk, T. , and MacLachlan, S. , 2016, “ Effect of Meniscus Curvature on Apparent Thermal Slip,” ASME J. Heat Transfer, 138(12), p. 122004. [CrossRef]
Hodes, M. , Lam, L. S. , Cowley, A. , Enright, R. , and MacLachlan, S. , 2015, “ Effect of Evaporation and Condensation at Menisci on Apparent Thermal Slip,” ASME J. Heat Transfer, 137(7), p. 071502. [CrossRef]
Lam, L. S. , Melnick, C. , Hodes, M. , Ziskind, G. , and Enright, R. , 2014, “ Nusselt Numbers for Thermally Developing Couette Flow With Hydrodynamic and Thermal Slip,” ASME J. Heat Transfer, 136(5), p. 051703. [CrossRef]
Maynes, D. , Webb, B. W. , and Davies, J. , 2008, “ Thermal Transport in a Microchannel Exhibiting Ultrahydrophobic Microribs Maintained at Constant Temperature,” ASME J. Heat Transfer, 130(2), p. 022402. [CrossRef]
Maynes, D. , Webb, B. W. , Crockett, J. , and Solovjov, V. , 2012, “ Analysis of Laminar Slip-Flow Thermal Transport in Microchannels With Transverse Rib and Cavity Structured Superhydrophobic Walls at Constant Heat Flux,” ASME J. Heat Transfer, 135(2), p. 021701. [CrossRef]
Maynes, D. , and Crockett, J. , 2013, “ Apparent Temperature Jump and Thermal Transport in Channels With Streamwise Rib and Cavity Featured Superhydrophobic Walls at Constant Heat Flux,” ASME J. Heat Transfer, 136(1), p. 011701. [CrossRef]
Kirk, T. L. , Hodes, M. , and Papageorgiou, D. T. , 2017, “ Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature,” J. Fluid Mech., 811(1), pp. 315–349. [CrossRef]
Shah, R. , and London, A. , 1978, “ Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data,” Advances in Heat Transfer: Supplement, Academic Press, New York.
Nusselt, W. , 1923, “ Der wärmeaustausch am berieselungskühler,” VDI-Z, 67(9), pp. 206–216.
Brown, G. M. , 1960, “ Heat or Mass Transfer in a Fluid in Laminar Flow in a Circular or Flat Conduit,” AIChE J., 6(2), pp. 179–183. [CrossRef]
Mikhaĭlov, M. D. , and Öz iş ik, M. N. , 1984, Unified Analysis and Solutions of Heat and Mass Diffusion, Wiley, New York.
Panton, R. L. , 2006, Incompressible Flow, Wiley, New York.
Hodes, M. , Kirk, T. L. , Karamanis, G. , and MacLachlan, S. , 2017, “ Effect of Thermocapillary Stress on Slip Length for a Channel Textured With Parallel Ridges,” J. Fluid Mech., 814(3), pp. 301–324. [CrossRef]
MathWorks, 2016, “ Partial Differential Equation Toolbox User's Guide,” MathWorks Inc., Natick, MA.
McCuen, P. , Kays, W. , and Reynolds, W. , 1962, Heat Transfer With Laminar and Turbulent Flow Between Parallel Planes With Constant and Variable Wall Temperature and Heat Flux, Stanford University, Stanford, CA.
Shah, R. , 1975, “ Thermal Entry Length Solutions for the Circular Tube and Parallel Plates,” Third National Heat Mass Transfer Conference, Indian Institute of Technology, Bombay, Mumbai, India, Dec. 11–13, Vol. 1, pp. 11–75.
ANSYS, 2013, “ ANSYS Fluent Theory Guide,” ANSYS Inc., Canonsburg, PA.
Patankar, S. V. , Liu, C. H. , and Sparrow, E. M. , 1977, “ Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area,” ASME J. Heat Transfer, 99(2), pp. 180–186. [CrossRef]
Copyright © 2017 by ASME
View article in PDF format.

References

Lam, L. S. , Hodes, M. , and Enright, R. , 2015, “ Analysis of Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces,” ASME J. Heat Transfer, 137(9), p. 091003. [CrossRef]
Lobaton, E. J. , and Salamon, T. R. , 2007, “ Computation of Constant Mean Curvature Surfaces: Application to the Gas–Liquid Interface of a Pressurized Fluid on a Superhydrophobic Surface,” J. Colloid Interface Sci., 314(1), pp. 184–198. [CrossRef] [PubMed]
Quéré, D. , 2005, “ Non-Sticking Drops,” Rep. Prog. Phys., 68(11), p. 2495. [CrossRef]
Cassie, A. B. D. , and Baxter, S. , 1944, “ Wettability of Porous Surfaces,” Trans. Faraday Soc., 40, pp. 546–551. [CrossRef]
Huang, D. M. , Sendner, C. , Horinek, D. , Netz, R. R. , and Bocquet, L. , 2008, “ Water Slippage Versus Contact Angle: A Quasiuniversal Relationship,” Phys. Rev. Lett., 101(22), p. 226101. [CrossRef] [PubMed]
Cottin-Bizonne, C. , Steinberger, A. , Cross, B. , Raccurt, O. , and Charlaix, E. , 2008, “ Nanohydrodynamics: The Intrinsic Flow Boundary Condition on Smooth Surfaces,” Langmuir, 24(4), pp. 1165–1172. [CrossRef] [PubMed]
Enright, R. , Hodes, M. , Salamon, T. , and Muzychka, Y. , 2013, “ Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels,” ASME J. Heat Transfer, 136(1), p. 012402. [CrossRef]
Philip, J. R. , 1972, “ Flows Satisfying Mixed No-Slip and No-Shear Conditions,” Z. Angew. Math. Phys., 23(3), pp. 353–372. [CrossRef]
Sbragaglia, M. , and Prosperetti, A. , 2007, “ A Note on the Effective Slip Properties for Microchannel Flows With Ultrahydrophobic Surfaces,” Phys. Fluids, 19(4), p. 043603. [CrossRef]
Rothstein, J. P. , 2010, “ Slip on Superhydrophobic Surfaces,” Annu. Rev. Fluid Mech., 42(1), pp. 89–109. [CrossRef]
Teo, C. J. , and Khoo, B. C. , 2009, “ Analysis of Stokes Flow in Microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves,” Microfluid. Nanofluid., 7(3), pp. 353–382. [CrossRef]
Teo, C. J. , and Khoo, B. C. , 2010, “ Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature,” Microfluid. Nanofluid., 9(2), pp. 499–511. [CrossRef]
Ng, C.-O. , and Wang, C. Y. , 2014, “ Temperature Jump Coefficient for Superhydrophobic Surfaces,” ASME J. Heat Transfer, 136(6), p. 064501. [CrossRef]
Lam, L. S. , Hodes, M. , Karamanis, G. , Kirk, T. , and MacLachlan, S. , 2016, “ Effect of Meniscus Curvature on Apparent Thermal Slip,” ASME J. Heat Transfer, 138(12), p. 122004. [CrossRef]
Hodes, M. , Lam, L. S. , Cowley, A. , Enright, R. , and MacLachlan, S. , 2015, “ Effect of Evaporation and Condensation at Menisci on Apparent Thermal Slip,” ASME J. Heat Transfer, 137(7), p. 071502. [CrossRef]
Lam, L. S. , Melnick, C. , Hodes, M. , Ziskind, G. , and Enright, R. , 2014, “ Nusselt Numbers for Thermally Developing Couette Flow With Hydrodynamic and Thermal Slip,” ASME J. Heat Transfer, 136(5), p. 051703. [CrossRef]
Maynes, D. , Webb, B. W. , and Davies, J. , 2008, “ Thermal Transport in a Microchannel Exhibiting Ultrahydrophobic Microribs Maintained at Constant Temperature,” ASME J. Heat Transfer, 130(2), p. 022402. [CrossRef]
Maynes, D. , Webb, B. W. , Crockett, J. , and Solovjov, V. , 2012, “ Analysis of Laminar Slip-Flow Thermal Transport in Microchannels With Transverse Rib and Cavity Structured Superhydrophobic Walls at Constant Heat Flux,” ASME J. Heat Transfer, 135(2), p. 021701. [CrossRef]
Maynes, D. , and Crockett, J. , 2013, “ Apparent Temperature Jump and Thermal Transport in Channels With Streamwise Rib and Cavity Featured Superhydrophobic Walls at Constant Heat Flux,” ASME J. Heat Transfer, 136(1), p. 011701. [CrossRef]
Kirk, T. L. , Hodes, M. , and Papageorgiou, D. T. , 2017, “ Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature,” J. Fluid Mech., 811(1), pp. 315–349. [CrossRef]
Shah, R. , and London, A. , 1978, “ Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data,” Advances in Heat Transfer: Supplement, Academic Press, New York.
Nusselt, W. , 1923, “ Der wärmeaustausch am berieselungskühler,” VDI-Z, 67(9), pp. 206–216.
Brown, G. M. , 1960, “ Heat or Mass Transfer in a Fluid in Laminar Flow in a Circular or Flat Conduit,” AIChE J., 6(2), pp. 179–183. [CrossRef]
Mikhaĭlov, M. D. , and Öz iş ik, M. N. , 1984, Unified Analysis and Solutions of Heat and Mass Diffusion, Wiley, New York.
Panton, R. L. , 2006, Incompressible Flow, Wiley, New York.
Hodes, M. , Kirk, T. L. , Karamanis, G. , and MacLachlan, S. , 2017, “ Effect of Thermocapillary Stress on Slip Length for a Channel Textured With Parallel Ridges,” J. Fluid Mech., 814(3), pp. 301–324. [CrossRef]
MathWorks, 2016, “ Partial Differential Equation Toolbox User's Guide,” MathWorks Inc., Natick, MA.
McCuen, P. , Kays, W. , and Reynolds, W. , 1962, Heat Transfer With Laminar and Turbulent Flow Between Parallel Planes With Constant and Variable Wall Temperature and Heat Flux, Stanford University, Stanford, CA.
Shah, R. , 1975, “ Thermal Entry Length Solutions for the Circular Tube and Parallel Plates,” Third National Heat Mass Transfer Conference, Indian Institute of Technology, Bombay, Mumbai, India, Dec. 11–13, Vol. 1, pp. 11–75.
ANSYS, 2013, “ ANSYS Fluent Theory Guide,” ANSYS Inc., Canonsburg, PA.
Patankar, S. V. , Liu, C. H. , and Sparrow, E. M. , 1977, “ Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-Periodic Variations of Cross-Sectional Area,” ASME J. Heat Transfer, 99(2), pp. 180–186. [CrossRef]

Figures

Grahic Jump Location
Fig. 2

Schematic of the domain when one plate is textured and the other one is smooth

Grahic Jump Location
Fig. 1

Liquid in the Cassie state and the composite interface

Grahic Jump Location
Fig. 5

Nul,fd versus the normalized coordinate (x̃−1)/(d̃−1) along the ridge for H/d = 10 and selected values of ϕ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 6

Nufd,ridge versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 3

fRe versus ϕ for selected values of H/d when one plate is textured and the other one is smooth

Grahic Jump Location
Fig. 4

Nufd versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Grahic Jump Location
Fig. 7

NuUIT and Nu¯UIT versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for ϕ=0.01

Grahic Jump Location
Fig. 8

NuUIT and Nu¯UIT versus z* for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic for ϕ=0.1

Grahic Jump Location
Fig. 9

fRe versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is adiabatic and either smooth (t–s) or textured with aligned ridges (al)

Grahic Jump Location
Fig. 10

Nufd versus ϕ for selected H/d when one plate is textured with isothermal ridges and the other one is adiabatic and either smooth (t–s) or textured with aligned ridges (al)

Grahic Jump Location
Fig. 11

Contour plot of w̃/w̃¯ when one plate is textured with isothermal ridges and the other one is adiabatic and smooth (t–s) for H/d = 4 and ϕ=0.3

Grahic Jump Location
Fig. 12

Contour plot of w̃/w̃¯ when one plate is textured with isothermal and the other one with adiabatic ridges and the ridges are aligned (al) for H/d = 4 and ϕ=0.3

Grahic Jump Location
Fig. 13

Nufd and Nufd,if versus ϕ for selected H/d when both plates are textured with aligned isothermal and isoflux ridges, respectively

Grahic Jump Location
Fig. 14

Schematic of the domain when both plates are textured and the ridges are aligned

Grahic Jump Location
Fig. 15

fRe versus ϕ for selected H/d when both plates are textured and the ridges are aligned

Grahic Jump Location
Fig. 16

Nufd versus ϕ for selected H/d when the ridges of one plate are isothermal and the ridges of the other one are aligned and adiabatic

Grahic Jump Location
Fig. 17

Nufd versus ϕ for selected H/d when the ridges of both plates are isothermal and aligned

Grahic Jump Location
Fig. 18

Schematic of the domain when both plates are textured and the ridges are staggered by half a pitch

Grahic Jump Location
Fig. 19

fRe versus ϕ for selected H/d when both plates are textured and the ridges are staggered by half a pitch

Grahic Jump Location
Fig. 20

Contour plot of w̃/w̃¯ when both plates are textured and the ridges are staggered by half a pitch for H/d = 4 and ϕ=0.3

Grahic Jump Location
Fig. 21

Nufd versus ϕ for selected H/d when the ridges of one plate are isothermal and the ridges of the other one are staggered by half a pitch and adiabatic

Grahic Jump Location
Fig. 22

Nufd versus ϕ for selected H/d when the ridges of both plates are isothermal and staggered by half a pitch

Tables

Table Grahic Jump Location
Table 1 First ten eigenvalues and corresponding expansion coefficients for H/d = 4 and ϕ=0.01 or 0.1 when one plate is textured with isothermal ridges, the other one is smooth and adiabatic and T̃in=1

Errata

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