We analytically consider the effect of meniscus curvature on heat transfer to laminar flow across structured surfaces. The surfaces considered are composed of ridges. Curvature of the menisci, which separates liquid in the Cassie state and gas trapped in cavities between the ridges, results from the pressure difference between the liquid and the gas. A boundary perturbation approach is used to develop expressions that account for the change in the temperature field in the limit of small curvature of a meniscus. The meniscus is considered adiabatic and a constant heat flux boundary condition is prescribed at the tips of the ridges in a semi-infinite and periodic domain. A solution for a constant temperature ridge is also presented using existing results from a mathematically equivalent hydrodynamic problem. We provide approximate expressions for the apparent thermal slip length as function of solid fraction over a range of small meniscus protrusion angles. Numerical results show good agreement with the perturbation results for protrusion angles up to 20 deg.
Micro- and nanostructured surfaces are engineered to reduce drag by using a combination of roughness and hydrophobic coatings. They are being considered for microchannel flows where surface friction forces dominate in such applications as electronics cooling  and lab-on-a-chip technologies . A meniscus separates liquid in the Cassie (unwetted) state on a micro- or nanostructured surface from gas trapped in the cavities. Meniscus curvature is often considered a secondary effect and neglected in the analysis of flow and heat transfer along these surfaces. Recent work has shown that the effect of curvature on hydrodynamic slip length is not necessarily small and depends on the geometry of the structures, solid fraction,2 and the magnitude and direction of the protrusion angle. We use a boundary perturbation approach to assess the change to apparent thermal slip length across ridge-type structures due to meniscus curvature.
A composite interface, comprised of solid–liquid and liquid–gas regions, exists along the structured surface which results in a mixed-boundary condition on the flow. The no-slip boundary condition applies at the solid–liquid interface where the primary heat transfer occurs. At the liquid–gas interface, i.e., the meniscus, the flow boundary condition is low shear and is often assumed shear-free. The thermal boundary condition is low heat flux and is often assumed adiabatic.
where T is the temperature in the liquid, is the mean temperature of the solid–liquid interface, is the mean temperature at the composite interface, bt is the apparent thermal slip length, henceforth referred to as thermal slip length, and n is the direction normal to the composite interface pointing into the liquid flow. The left-hand side of Eq. (1) is the temperature jump (or discontinuity) which results from the added thermal resistance at the composite boundary relative to that of an unstructured surface. Thermal slip length expressions are available in the literature. Enright et al.  analytically developed them for three structure geometries: parallel ridges, perpendicular ridges, and pillar arrays. They assumed a flat, adiabatic meniscus and considered isoflux and isothermal boundary conditions at the solid–liquid interface. Ng and Wang  semi-analytically developed slip length expressions for ridges which account for heat transfer across a flat meniscus. They considered an isothermal boundary condition. They found that conduction through the gas has a negligible impact on slip length when the cavity depth is on the order of the spacing of the structures and larger, i.e., the adiabatic condition at the meniscus applies. They also considered pillar and hole arrays. The case of evaporation and condensation across the meniscus has been considered by Hodes et al. , and a slip length expression is provided for isoflux ridges as a function of solid fraction and interfacial heat transfer coefficient.
The thermal slip length is found from the solution to the inner problem, near the structures. Heat transfer is governed by the Péclet number, Pea, based on the length scale of the structures, a. When it is sufficiently small, the transport is considered purely diffusive. When the domain of the inner problem can be treated as semi-infinite, the temperature field becomes uniform at a vertical distance that is on the order of the pitch of the structures. For the case of the isothermal ridge, the temperature jump is mathematically equivalent to a hydrodynamic slip velocity and produces a change in thermal energy added to the channel flow. For the isoflux case, the temperature jump is captured in the change of the mean temperature at the solid–liquid interface relative to the no-slip case. In both cases, the thermal slip length captures not only the spreading resistance but also the change to the one-dimensional resistance that results from the presence of the irregular boundary.
The effect of hydrodynamic slip on drag reduction across structured surfaces is well documented in the literature [10–13]. Heat transfer to such flows has only recently received attention [1,6,14,15].
A number of studies have shown that the shape of the meniscus and the degree of protrusion significantly affect drag. Ybert et al.  developed scaling laws for hydrodynamic slip. They derive expressions for secondary effects such as a finite shear force at the meniscus and pressure-induced curvature of the meniscus. They assumed a meniscus with a positive protrusion angle, i.e., the pressure in the gas is greater than that in the liquid. In the limit of large solid fraction, the correction to slip length is negatively proportional to the cavity fraction and the height of the meniscus. In the limit of small solid fraction, the correction to slip length varies inversely with curvature. They note that geometries with larger slip lengths are more sensitive to curvature effects. Bocquet and Barrat  discuss the analogy between thermal slip and hydrodynamic slip. We note that these scaling laws are also applicable to thermal slip in the presence of curvature.
Sbragaglia and Prosperetti  used a perturbation approach to study the effect of small curvature of the meniscus on hydrodynamic slip for parallel ridges. They identify two significant effects on the flow that result. The first is the change in the cross-sectional area of the flow due to the deformation of the meniscus, and the second is the change in the velocity field. They provided hydrodynamic slip length expressions for parallel ridges for a pressure-driven flow with a finite channel height and for a shear flow with an infinite channel height.
Several analytical studies have addressed the effect of meniscus curvature in the limit of large solid fraction for domains with semi-infinite heights. Crowdy  used a series of conformal maps to model a meniscus between parallel ridges. He derived an expression for slip length as a function of solid fraction and protrusion angle which is applicable for a periodic distribution of ridges in the limit of large solid fractions. He extended his work in Ref.  developing additional terms for his slip length expression such that it now applies for solid fractions down to . Davis and Lauga  used a two-dimensional model of shear flow past an array of bubbles trapped between transverse ridges and provided an expression to calculate the critical protrusion angle at which the slip length becomes negative. Their results also show the asymmetry between the effects of negative and positive protrusion angles. Recently, Crowdy  used a new transform scheme to model fluid flow over bubbles with partial slip at the meniscus. He provides a semi-analytical slip length expression for a meniscus with a positive protrusion angle of 90 deg. For solid fractions down to , results are within 10% of numerical data.
Teo and Khoo performed numerical studies of the effect of meniscus curvature on drag for menisci formed between parallel ridges for both Poiseuille and Couette flows  and in Poiseuille flow over transverse ridges . For transverse ridges, they report a critical protrusion angle at which the slip length becomes zero that is independent of shear-free fraction, geometry, and flow type but which decreases as channel height is reduced for a given shear-free fraction. Above θc, friction was enhanced. For parallel ridges, slip length displayed asymmetry with respect to positive and negative protrusion angles; however, there was no critical angle at which drag reduction became zero. Instead, the slip length exhibited a monotonic increase with protrusion angle even for large angle values up to . They also found that the ratio of parallel ridge slip length to transverse ridge slip length did not remain constant as protrusion angle was varied.
To the authors' knowledge, the effect of meniscus curvature on thermal slip has not been studied. This motivates the present work.
The geometry under consideration is that of a concave or convex meniscus formed between two ridges of a structured surface as shown in Fig. 1. The center-to-center spacing of the ridges is 2d, the width of the ridge is 2a, the width of the cavity is 2c, and is the cavity fraction. The solid fraction, , is the ratio of the solid–liquid area to the projected total area of the composite interface in the horizontal plane. The temperature field is symmetric about x = 0 and x = d. Typical length scales for structure width and spacing are on the order of microns [25–27]. The protrusion angle, α, is measured from the horizontal to the line tangent to the meniscus at the triple contact line per Fig. 1. A positive protrusion angle corresponds to the meniscus projecting upward into the liquid and is limited by the maximum receding contact angle, θR, where . A negative protrusion angle corresponds to the meniscus extending downward into the cavity and is limited by the maximum advancing contact angle, θA, where . Thermophysical properties are assumed constant. We address the inner problem when the Péclet number, based on the ridge width, is very small and transport is diffusive. Evaporation and condensation  and thermocapillary stress along the meniscus  are not considered here.
where is the vector normal to the meniscus.
Identification of the Perturbation Parameter.
When α = 0, corresponds to a flat meniscus. and correspond to and , respectively. For small values of α, ε is a small parameter which is used here to consider the change in the temperature field resulting from small deflections of the meniscus via a boundary perturbation approach. For example, water in the Cassie state with maximum contact angle of on a surface with solid fraction, , ε normalized to the pattern width is .
In this section, we solve for the temperature field with constant heat flux, , supplied at the solid–liquid interface. The problem is rendered dimensionless by using , and . The width of the analytical domain is unity per Fig. 2.
Boundary Condition at the Meniscus.
where is the temperature profile assuming a flat adiabatic meniscus between the structures, and is the change to the first order in that would result from the deflection of the meniscus. For small , the effect of meniscus curvature on the temperature field is captured by .
respectively. Further discussion of the perturbation expansion is provided in Appendix A.
where thermal slip length is nondimensionalized by the pitch of the structures, i.e., .
The coefficients, gm, are found from the above, and the expression is provided in Appendix B.
We compute for prescribed values of by setting n = 200 and m = 200 and increasing first n by 200 until no change is observed in the result to five digits, and then increasing m by 200 until no change is observed in the result to five digits. For example, convergence was achieved when gm was truncated at m = 3200 and n = 2600 terms. Further details of the application of the boundary perturbation method to compute the thermal slip length are discussed in Appendix C. The salient point developed therein is that when neglecting terms of , we may evaluate all quantities in the expression for thermal slip length at . The same applies to the case of an isothermal ridge discussed next.
Sbragaglia and Prosperetti  developed an expression which can be numerically integrated to calculate the hydrodynamic slip length associated with small deflections of the meniscus in a diffusive domain with infinite height. We adapt it here to provide a thermal slip length expression accounting for meniscus curvature for the case of an isothermal ridge.
Thermal slip lengths versus solid fraction is plotted for flat menisci and for a range of positive and negative protrusion angles. The cases of an isoflux ridge and isothermal ridge are shown in Figs. 3 and 4, respectively.
The presence of a flat meniscus imposes a restriction on the heat flow between the solid and the liquid domain. A thermal spreading (or constriction) resistance results, which is captured in the slip length for the flat meniscus. As solid fraction is decreased, the constriction increases and the thermal resistance of the inner liquid domain increases resulting in increased thermal slip. In the limit as , the resistance grows to resemble that of an adiabatic boundary at . In the limit as , the resistance reverts to that of a domain with a solid surface at . As the shape of the meniscus is perturbed from the flat state, it produces additional changes in the thermal resistance of the liquid domain. The shape of the irregular boundary affects the one-dimensional thermal resistance in addition to the spreading resistance in the liquid domain. The meniscus protruding downward decreases the thermal resistance resulting in lower slip lengths and enhancement to heat transfer. Conversely, as the meniscus protrudes upward, the area for heat flow becomes more constricted and the thermal resistance is increased.
Figure 5 displays the isothermal and isoflux thermal slip lengths as a function of protrusion angle for selected solid fractions. Notably, for lower solid fractions, the meniscus protrusion angle has a strong effect on slip length, and thus on heat transfer for both cases. This highlights the importance of accounting for the effect of curvature on heat transfer at low solid fractions which are desirable for the achievement of drag reduction.
where w is the width of the domain. The quantity β is equivalent to a dimensionless thermal spreading/constriction resistance .
The Laplace equation with relevant boundary conditions was solved with the finite element solver in PDE toolbox. The flow domain was discretized on average with 440,000 elements. Slip length results from this model were first compared to the analytical slip length for the flat meniscus and to comparable fluent  simulations. The values from the PDE toolbox were found to match the known flat meniscus slip length to five decimal places; therefore, it was used to evaluate the analytical values in order to determine the range of angles and solid fractions over which the analytical results are valid. The mesh at each case was adapted and refined multiple times for both the matlab and the fluent simulations to ensure mesh independence.
Tables 1 and 2 compare the slip length values obtained from the perturbation method and Refs.  and  to β obtained from the PDE toolbox. We note that, insofar as the perturbation analysis to , the mean temperatures along and are unchanged as per Eqs. (24) and (26). Thus, in Eq. (C8), the corresponding terms in the numerator, i.e., and , may be replaced by those as . Thus, the slip may also be computed from Eq. (40) for β. However, once the magnitude of is large enough that terms of become relevant, and may not be replaced by those as , and thus, we do not expect and β to agree as borne out by our results. The same argument applies for the case of an isothermal ridge. In summary, .
For the isothermal case, the perturbation results for negative angles are closer to β than those with corresponding positive angles. For the isoflux case, the perturbation results for positive angles show better agreement with the numerical values. As to be expected, the difference between and β increases with the absolute value of α commensurate with the degree of curvature of the meniscus. It is significant to note that the effect of meniscus curvature on heat transfer is not symmetrical for positive and negative protrusion angles. For a given solid fraction, positive angles produce somewhat larger changes in thermal resistance relative to their negative counterparts. As shown in Figs. 6 and 7, as increases, increases, and is an indication of the magnitude of error to be expected.
For the isoflux case, the difference between the perturbation method, and β, is less than 6.5% for protrusion angles between , and solid fractions below 0.1. For the isothermal case, for angles between , and solid fractions below 0.1, values that are typical for water-based cooling, the difference is less than 3%.
Thermal slip length expressions have been developed for flow across ridges which account for the change in thermal resistance associated with the presence of a curved adiabatic meniscus. They are applicable for a range of small protrusion angles and solid fractions which are typical of water-based cooling in a microchannel. When liquid pressure is higher than that of the gas, a negative protrusion angle exists and heat transfer is enhanced. Conversely, the presence of menisci protruding into the liquid phase because the pressure in the gas is higher than that of the liquid will reduce heat transfer to the liquid at low protrusion angles.
Two effects are important to note. First, at lower solid fractions, the meniscus protrusion angle has a significant effect on slip length, and thus on heat transfer for both cases. Second, the effect of meniscus protrusion on heat transfer is not symmetrical for positive and negative protrusion angles.
Future work will develop thermal slip lengths over a wider range of protrusion angles and solid fractions. The slip lengths presented here are applicable for use in Nusselt number expressions for channels where curvature of the meniscus is constant along the channel such as in Ref. ; however, further work must be undertaken to incorporate them into Nusselt number expressions for channels where curvature changes along the channel as in a Poiseuille flow.
The authors gratefully acknowledge Durwood Marshall and Shawn Doughty of Tufts Technology Services for their assistance and access to Tufts UIT Research Computing facilities. Funding for L.L., M.H., and G.K. was provided in part by the NSF CBET Award No. 1402783. T.K. was supported by an EPSRC doctoral scholarship. The work of S.M. was partially supported by an NSERC Discovery Grant
Appendix A: Full Perturbation Expansion
where we extend to the solid–liquid interface by taking its value to be 0 for . Note that, with this definition, the domains of and are naturally for and , since points on the meniscus are mapped to points on the -axis by this definition. Note that derivatives of with respect to must now be evaluated via the chain rule, but that the boundary condition along the meniscus can be imposed directly as a boundary condition along for and .
Note that this difference is, in fact, zero along the boundary, since for , by Eq. (21), and for as defined above. Thus, there is no change to the numerator in Eq. (34), while the denominator remains the same due to the boundary condition as . In summary, the full perturbation expansion as given by Eq. (A1) generates a distinct first-order problem from when the perturbation is applied only to the meniscus boundary condition, but this does not change the expressions presented above for the thermal slip length.
Appendix B: Coefficients gm
Appendix C: Boundary Perturbation to Compute Thermal Slip Length
Thus, when neglecting terms of , we have shown that we may evaluate all quantities in the expression for slip length at . We note this analysis applies to both isoflux and isothermal solid–liquid interfaces.
Ratio of ridge tip area to total projected area of the surface.
According to Kennard .