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.

## Introduction

### 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 [8–12]. 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* [22–24].^{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 $|x|\u2264d$ and 0 ≤ *y* ≤ *H* and where 2*d* is the pitch of the ridges and *H* is the distance between the parallel plates. The hydraulic diameter of the domain (*D*_{h}) is 2*H*. The width of the meniscus is 2*a*. 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 *T*_{in} (*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 $(\varphi =(d\u2212a)/d)$ and the aspect ratio of the domain (*H*/*d*).

## Analysis

### Hydrodynamic Problem.

*w*is the streamwise velocity, d

*p*/d

*z*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

where $d\u0303=d/a$ is the dimensionless (half) pitch of the ridges.

*f*Re), where

is the dimensionless mean velocity of the flow.

### Thermal Problem.

*T*and

*α*are the temperature and the thermal diffusivity of the liquid, respectively. Defining the dimensionless temperature $T\u0303$ and the dimensionless streamwise coordinate $z\u0303$ as

*T*

_{sl}is the constant temperature of the ridge and

*T*

_{ref}is a reference temperature for the problem, Eq. (16) becomes

where $T\u0303in(x\u0303,y\u0303)$ is the prescribed dimensionless temperature profile at the inlet of the domain $(z\u0303=0)$.

*λ*real and positive.

^{3}Note that $\psi (x\u0303,y\u0303)$ cannot be separated further into a product of a function of $x\u0303$ and one of $y\u0303$ since the velocity field, $w\u0303=w\u0303(x\u0303,y\u0303)$, is not separable in such a way. Equation (25) satisfies the boundary conditions

*λ*and the corresponding eigenfunction

*ψ*, with weight function $w\u0303(x\u0303,y\u0303)$. Assuming that the eigenvalues are discrete and there are infinitely many, let

*λ*and

_{i}*ψ*denote the

_{i}*i*th eigenvalue and eigenfunction, respectively, ordered such that $0<\lambda 1<\lambda 2<\cdots <\lambda i<\cdots \u2192\u221e$. The eigenfunctions are orthogonal with respect to the inner product defined by

*ψ*such that

_{i}The eigenvalue problem is solved numerically. The calculation of *ψ _{i}* and

*λ*is detailed in Sec. 2.4, and for the rest of the present analysis, they are assumed to be known.

_{i}*c*. The

_{i}*c*are determined by taking the inner product of Eq. (33) with

_{i}*ψ*at the inlet $z\u0303=0$, where $T\u0303(x\u0303,y\u0303,0)=T\u0303in(x\u0303,y\u0303)$, giving

_{i}### Nusselt Number Expressions.

*h*

_{l}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

*T*

_{b}is the bulk temperature of the liquid defined as

_{l,fd}) follows from the evaluation of Eq. (41) as $z\u0303\u2192\u221e$. Given that the

*λ*are real and 0 <

_{i}*λ*

_{1}<

*λ*

_{2}< … <

*λ*< … →

_{i}*∞*, upon dividing both the numerator and the denominator of Eq. (41) by $e\u2212\lambda 1z\u0303$ and letting $z\u0303\u2192\u221e$, only the first term of each sum remains. It follows that

Nu_{l,fd} is a function only of the first eigenfunction and it is independent of the inlet temperature profile. However, in the thermally developing region, Nu_{l} is a function of $T\u0303in$.

*y-*axis and the boundary condition given by Eq. (26) yields

_{fd}) follows in the same manner as Eq. (42) and it is given by

Nu_{fd} is a function only of the first eigenvalue *λ*_{1} and it is independent of $T\u0303in$.

*T*

_{ref}, we have $T\u0303in=1$. Then, the foregoing expression reduces to

It is emphasized that Eqs. (41), (48), (51), and (52) hold for all streamwise locations $z\u0303$; however, to achieve a given accuracy, more terms are required in the evaluation of each sum as $z\u0303$ 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 $x\u0303=0$ and $x\u0303=d\u0303$ 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 $w\u0303(x\u0303,y\u0303)$ 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 ≤ *λ _{i}* ≤ UB along with their corresponding eigenfunctions

*ψ*[27]. The upper bound (UB) was varied depending on the number of the eigenvalues sought (MATLAB

_{i}^{®}requires prescription of the aforementioned interval for the sought

*λ*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 Nu

_{i}_{fd}is less than 0.01%—typically this required 3.5 × 10

^{5}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 $\varphi =0.25$ and various values of *H*/*d* as per Fig. 3; agreement was within 0.006%. Second, *f*Re and Nu_{fd} were computed in the limit of $\varphi \u21921$, i.e., for fully developed flow between two smooth parallel plates where one is isothermal and the other is adiabatic and *f*Re* _{s}* = 96 and Nu

_{fd,}

*= 4.86 [28]. Agreement was within 0.03% of*

_{s}*f*Re and 0.009% of Nu

_{fd}. Thirdly, the boundary condition at $y\u0303=H\u0303$ was changed to an isothermal one and Nu

_{UIT}and $Nu\xafUIT$ were computed at various streamwise locations in the limit $\varphi \u21921$. 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 $z*=z/(DhPe)=z\u0303/(4H\u03032w\u0303\xaf)=1.5\xd710\u22124$. 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 $z\u0303$ to enable direct comparison of the results with those for nonstructured channels. However, $z\u0303$ is a more appropriate quantity for the case at hand given that $w\u0303\xaf$ is a function of the solid fraction and the aspect ratio in the case of channels with textured surfaces.

Finally, semi-analytical values of Nu_{fd} 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 $H/d=4,\u2009\varphi =0.3$, 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 × 10^{5} hexahedral elements. The computed Nu_{fd} 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 Nu_{fd} 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 $z\u0303$, quantities which are prohibitively expensive to compute using a general CFD code.

## Results

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, Nu_{fd}, versus the solid fraction $\varphi $ 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 Nu_{fd,}* _{s}* = 4.86. The results obey the expected asymptotic behavior as $\varphi \u21921$, with $Nufd\u2192Nufd,s$, irrespective of the aspect ratio. Additionally, as $\varphi \u21920,\u2009Nufd$ tends to zero because the available area for heat transfer vanishes. Moreover, for a given $\varphi $ (excluding the aforementioned limits) as

*H*/

*d*→ 0 and $H/d\u2192\u221e,\u2009Nufd$ tends to zero and to Nu

_{fd,}

*, respectively. This is because for*

_{s}*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, Nu_{l,fd}, versus the normalized coordinate along the ridge $(x\u0303\u22121)/(d\u0303\u22121)$ for *H*/*d* = 10 and $\varphi =0.01$, 0.1 and 0.99. The results show that Nu_{l,fd} increases with decreasing $\varphi $, 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 $\varphi \u21920$, 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, Nu_{fd,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 $\varphi <1$ 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 Nu_{UIT} and $Nu\xafUIT$ versus *z** for $\varphi =0.01$ 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 Nu_{UIT} and $Nu\xafUIT$ increase monotonically with decreasing *z**, and in the latter case, they tend to Nu_{fd}. The first ten eigenvalues and the corresponding expansion coefficients that were computed for *H*/*d* = 4 at $\varphi =0.01$ and 0.1 are provided in Table 1.

H/d = 4 | |||
---|---|---|---|

$\varphi =0.01$ | $\varphi =0.1$ | ||

λ_{i} | c_{i} | λ_{i} | c_{i} |

4.375 × 10^{−2} | 3.441 | 6.578 × 10^{−2} | 3.428 |

6.078 × 10^{−1} | 4.248 × 10^{−1} | 6.923 × 10^{−1} | 6.707 × 10^{−1} |

1.976 | 1.309 × 10^{−1} | 2.039 | 2.504 × 10^{−1} |

4.174 | 4.645 × 10^{−2} | 4.130 | 9.351 × 10^{−2} |

5.478 | 5.006 × 10^{−2} | 5.453 | 5.008 × 10^{−2} |

6.832 | 3.990 × 10^{−2} | 6.791 | 2.538 × 10^{−2} |

7.400 | 4.909 × 10^{−2} | 7.367 | 7.305 × 10^{−2} |

9.063 | 5.237 × 10^{−2} | 9.289 | 8.425 × 10^{−2} |

1.115 × 10^{1} | 7.095 × 10^{−3} | 1.075 × 10^{1} | 7.194 × 10^{−3} |

1.197 × 10^{1} | 4.493 × 10^{−2} | 1.230 × 10^{1} | 8.377 × 10^{−2} |

H/d = 4 | |||
---|---|---|---|

$\varphi =0.01$ | $\varphi =0.1$ | ||

λ_{i} | c_{i} | λ_{i} | c_{i} |

4.375 × 10^{−2} | 3.441 | 6.578 × 10^{−2} | 3.428 |

6.078 × 10^{−1} | 4.248 × 10^{−1} | 6.923 × 10^{−1} | 6.707 × 10^{−1} |

1.976 | 1.309 × 10^{−1} | 2.039 | 2.504 × 10^{−1} |

4.174 | 4.645 × 10^{−2} | 4.130 | 9.351 × 10^{−2} |

5.478 | 5.006 × 10^{−2} | 5.453 | 5.008 × 10^{−2} |

6.832 | 3.990 × 10^{−2} | 6.791 | 2.538 × 10^{−2} |

7.400 | 4.909 × 10^{−2} | 7.367 | 7.305 × 10^{−2} |

9.063 | 5.237 × 10^{−2} | 9.289 | 8.425 × 10^{−2} |

1.115 × 10^{1} | 7.095 × 10^{−3} | 1.075 × 10^{1} | 7.194 × 10^{−3} |

1.197 × 10^{1} | 4.493 × 10^{−2} | 1.230 × 10^{1} | 8.377 × 10^{−2} |

Figures 9 and 10 compare the computed values of *f*Re and Nu_{fd}, 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 *f*Re is significantly reduced if both plates are textured, especially as $\varphi \u21920$, Nu_{fd} changes by only a small fraction due to texturing. More importantly, as per Fig. 10, Nu_{fd} 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 $w\u0303/w\u0303\xaf$ for the cases at hand for *H*/*d* = 4 and $\varphi =0.3$. 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., $0\u2264x\u0303\u2264d\u0303$ and $0\u2264y\u0303\u2264H\u0303/2$, over the mean velocity of the flow

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 $\varphi $, 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 *f*Re and Nu_{fd} and the contour plot of the scaled dimensionless streamwise velocity are presented in the Appendix.

Finally, Fig. 13 compares the computed values of Nu_{fd} when both plates are textured with isothermal aligned ridges against those calculated for the same configuration but for isoflux ridges (Nu_{fd,if}) by Kirk et al. [20]. The results show that depending on the aspect ratio *H*/*d,* there is a range for $\varphi $ 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.

## Conclusions

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.

## Acknowledgment

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.

## Nomenclature

*a*=half meniscus width, m

*c*=_{i}expansion coefficients

*d*=half ridge pitch, m

*D*=domain

- $d\u0303$ =
dimensionless half ridge pitch,

*d*/*a**D*_{h}=hydraulic diameter, 2

*H*- d
*p*/d*z*= prescribed pressure gradient, Pa/m

*f*=friction factor, $2Dh(\u2212dp/dz)/(\rho w\xaf2)$

*f*Re =Poiseuille number

*H*=distance between parallel plates, m

- $H\u0303$ =
dimensionless distance between parallel plates,

*H*/*a**h*_{l}=local heat transfer coefficient, W/(m

^{2}K)*k*=thermal conductivity of liquid, W/(m K)

- LB =
lower bound of

*λ*_{i}- $n\u0302$ =
outward pointing unit normal vector on boundaries

- Nu =
Nusselt number averaged over the composite interface

- $Nu\xaf$ =
Nusselt number averaged over the composite interface and the streamwise length of the domain

- Nu
_{l}= local Nusselt number,

*h*_{l}*D*_{h}/*k*- Pe =
Péclet number, $w\xafDh/\alpha $

*R*=average velocity ratio close to the ridge, $(2\u222b0H\u0303/2\u222b0d\u0303w\u0303dx\u0303dy\u0303)/(d\u0303H\u0303w\u0303\xaf)$

- Re =
Reynolds number, $\rho w\xafDh/\mu $

- $S\u0303$ =
dimensionless coordinate along

*∂D**T*=temperature, °C

- $T\u0303$ =
dimensionless temperature, $(T\u2212Tsl)/(Tin\u2212Tsl)$

*T*_{b}=bulk temperature

- $T\u0303b$ =
dimensionless bulk temperature

*T*_{in}=inlet temperature, °C

- $T\u0303in$ =
dimensionless inlet temperature

*T*_{ref}=reference temperature, °C

*T*_{sl}=ridge temperature, °C

- UB =
upper bound of

*λ*_{i}*w*=streamwise velocity, m/s

- $w\xaf$ =
mean velocity, m/s

- $w\u0303$ =
dimensionless velocity, $2\mu w/[aH(\u2212dp/dz)]$

- $w\u0303\xaf$ =
dimensionless mean velocity

*x*=lateral coordinate, m

- $x\u0303$ =
dimensionless lateral coordinate,

*x*/*a**y*=vertical coordinate, m

- $y\u0303$ =
dimensionless vertical coordinate,

*y*/*a**z*=streamwise coordinate, m

- $z\u0303$ =
dimensionless streamwise coordinate, $2\alpha \mu z/[a3H(\u2212dp/dz)]$

*z** =dimensionless streamwise coordinate for the thermal entrance region,

*z*/(*D*_{h}Pe)*∂D*=boundary of the dimensionless domain

### 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 $y\u0303=H\u0303$. 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 Nu_{l}, Nu_{l,fd}, Nu, Nu_{fd}, $Nu\xaf$, and $Nu\xafUIT$ 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 Nu_{fd} 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 $y\u0303=H\u0303$. In terms of the eigenvalue problem, the boundary conditions given by Eqs. (26) and (27) apply rather than Eq. (28) at $y\u0303=H\u0303$.

The computed Nu_{fd} is presented in Fig. 17.

#### Both Plates Textured, Staggered Ridges

The computed Poiseuille numbers are presented in Fig. 19. Figure 20 presents the contour plot of the scaled dimensionless streamwise velocity $w\u0303/w\u0303\xaf$ 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 Nu_{l}, Nu_{l,fd}, Nu, Nu_{fd}, $Nu\xaf$, and $Nu\xafUIT$ are identical to those given by Eqs. (41), (42), (48), (49), (51), and (52). The computed Nu_{fd} is presented in Fig. 21.

The expressions for Nu_{l}, Nu_{l,fd}, Nu, Nu_{fd}, $Nu\xaf$, and $Nu\xafUIT$ are identical to those given by Eqs. (41), (42), and (A3)–(A6). The computed Nu_{fd} is presented in Fig. 22.

Parallel and transverse relative to the flow direction.

We use the term *Graetz–Nusselt problem* rather than *Graetz problem* because they refer to flow between parallel plates and through circular duct, respectively, as per the distinction made in Shah and London [21].

To show that *λ* is real and positive we multiply Eq. (25) by the complex conjugate of *ψ*, integrate over the domain, and use the divergence theorem.

If 28 eigenvalues are used instead the maximum discrepancies for the presented values of Nu_{UIT} and $Nu\xafUIT$ are less than 0.002% and 0.0003%, respectively, and if 25 eigenvalues are used instead the maximum discrepancies are less than 0.09% and 0.02%, respectively.