0
Research Papers: Heat and Mass Transfer

# Effect of Meniscus Curvature on Apparent Thermal SlipOPEN ACCESS

[+] Author and Article Information
Lisa Steigerwalt Lam

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155
e-mail: lisa_lam@alum.mit.edu

Marc Hodes, Georgios Karamanis

Department of Mechanical Engineering,
Tufts University,
Medford, MA 02155

Toby Kirk

Department of Mathematics,
Imperial College,
London SW7 2AZ, UK

Scott MacLachlan

Department of Mathematics and Statistics,
Memorial University of Newfoundland,
St. John's, NL A1C 5S7, Canada

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received January 18, 2016; final manuscript received July 12, 2016; published online August 16, 2016. Assoc. Editor: Dr. Antonio Barletta.

J. Heat Transfer 138(12), 122004 (Aug 16, 2016) (9 pages) Paper No: HT-16-1019; doi: 10.1115/1.4034189 History: Received January 18, 2016; Revised July 12, 2016

## Abstract

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.

<>

## Introduction

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 [1] and lab-on-a-chip technologies [2]. 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.

Slip length expressions may be used to model the velocity and temperature discontinuities at the surface. When considering channel flow where the height of the channel is large compared to the length scale of the structures, the hydrodynamic and thermal boundary conditions on structured surfaces are treated similarly to those in the molecular slip regime of rarefied gas flows using Navier's [3] and Poisson's3 slip conditions, respectively. These provide the means to solve the outer flow and temperature problems to determine key engineering parameters, Poiseuille and Nusselt numbers, as a function of slip lengths [57]. For the case of a flat boundary, the thermal boundary condition is Display Formula

(1)$T¯sl−T¯ĉ=−bt∂T¯∂n|ĉ$

where T is the temperature in the liquid, $T¯sl$ is the mean temperature of the solid–liquid interface, $T¯ĉ$ 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. [6] 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 [8] 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. [9], 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.

## Previous Work

The effect of hydrodynamic slip on drag reduction across structured surfaces is well documented in the literature [1013]. 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. [16] 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 [17] 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 [18] 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 [19] 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. [20] developing additional terms for his slip length expression such that it now applies for solid fractions down to $ϕ=0.1$. Davis and Lauga [21] 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 [22] 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 $ϕ=0.1$, 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 [23] and in Poiseuille flow over transverse ridges [24]. For transverse ridges, they report a critical protrusion angle $θc≈62−65 deg$ 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 $90deg$. 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.

## Problem Formulation

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 $c/d=1−ϕ$ is the cavity fraction. The solid fraction, $ϕ$, 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 [2527]. 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 $α=π−θR$. A negative protrusion angle corresponds to the meniscus extending downward into the cavity and is limited by the maximum advancing contact angle, θA, where $α=θA−π/2$. 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 [9] and thermocapillary stress along the meniscus [28] are not considered here.

We consider a unit cell of width d and infinite height. The governing equation is the Laplace equation Display Formula

(2)$∇2T=0$

Symmetry boundary conditions exist at the midpoint of the meniscus and the midpoint of the ridge such that Display Formula

(3)$∂T∂x=0 for x=0,d and y>0$

Constant heat flux leaves the domain as $y→∞$Display Formula

(4)$∂T∂y=−q″∞k for 0

In Sec. 3.2, we consider a constant heat flux boundary condition at the solid–liquid interface Display Formula

(5)$∂T∂y=−q″slk for c

where $q″sl=aq″∞/d$. In Sec. 3.3, we present a solution from the literature for the case when a constant temperature boundary condition applies at the solid–liquid interface such that Display Formula

(6)$T=Tsl for c

The adiabatic condition along the meniscus is Display Formula

(7)$n̂·∇T=0$

where $n̂$ is the vector normal to the meniscus.

###### Identification of the Perturbation Parameter.

We follow the approach of Sbragaglia and Prosperetti [18] by approximating the shape of the meniscus. It is an arc of radius, R, which results from a pressure difference across the meniscus, $Δplg$, between the liquid above the composite interface and the gas in the cavities per the Laplace–Young equation Display Formula

(8)$Δplg=σlgR$

where $σlg$ is the liquid–gas surface tension. We note that, for a pressure-driven flow, the pressure gradient in the channel causes R to change in the streamwise direction, which is not addressed here. A circular arc pinned at the corners of the ridges is of the form Display Formula

(9)$y=±(R2−x2−R2−c2)$

Using the binomial approximation of the square roots results in Display Formula

(10)$y≃±{R[1−12(xR)2]−R[1−12(cR)2]}$

In the limit of a small protrusion angle, as would potentially be the case with water, the curvature of the meniscus is taken to be small and its shape is defined as Display Formula

(11)$y+εη(x)=0$

where ε is a small quantity [18]. It follows that Display Formula

(12)$η(x)≃c2−x2$

(13)$ε≃±12R$

Per the Young–Laplace equation Display Formula

(14)$ε≃±Δplg2σlg$

Notably, ε is a small parameter which varies inversely with R and represents the degree of curvature of the meniscus. Using a force balance and geometry, $Δplg$ can be expressed in terms of surface tension, solid fraction, and meniscus protrusion angle, α, as discussed in Ref. [1]; ε becomes Display Formula

(15)$ε≃− sin α2d(1−ϕ)$

When α = 0, $ε=0$ corresponds to a flat meniscus. $+ε$ and $−ε$ correspond to $−α$ and $+α$, 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 $θA=110deg$ on a surface with solid fraction, $ϕ=.01$, ε normalized to the pattern width is $−0.17$.

###### Isoflux Ridge.

In this section, we solve for the temperature field with constant heat flux, $q″sl$, supplied at the solid–liquid interface. The problem is rendered dimensionless by using $T̃=Tk/(q″sld), x̃=x/d, ỹ=y/d, c̃=c/d, η̃=η/d2$, and $ε̃=εd$. The width of the analytical domain is unity per Fig. 2.

###### Boundary Condition at the Meniscus.

Expressing the dimensionless isoflux temperature field as a Taylor series in $ε̃$, it takes the form Display Formula

(16)$T̃(x̃,ỹ)=T̃0+ε̃T̃1+o(ε̃)$

where $T̃0$ is the temperature profile assuming a flat adiabatic meniscus between the structures, and $T̃1$ is the change to the first order in $ε̃$ that would result from the deflection of the meniscus. For small $ε̃$, the effect of meniscus curvature on the temperature field is captured by $T̃1$.

Taking the normal vector as $n̂=ε̃(dη̃/dx̃)î+ĵ$, the adiabatic condition along the meniscus is Display Formula

(17)$ε̃dη̃dx̃∂T̃∂x̃|ỹ=−ε̃η̃+∂T̃∂ỹ|ỹ=−ε̃η̃=0 for 0

where $η̃=c̃2−x̃2$. Then, using Taylor expansions of $∂T̃/∂x̃$ and $∂T̃/∂ỹ$ about $ỹ=0$Display Formula

(18)$∂T̃∂x̃|ỹ=−ε̃η̃=∂T̃∂x̃|ỹ=0+o(1)$
Display Formula
(19)$∂T̃∂ỹ|ỹ=−ε̃η̃=∂T̃∂ỹ|ỹ=0−ε̃η̃∂2T̃∂ỹ2|ỹ=0+o(ε̃)$

Using Eq. (16) and the above, Eq. (17) becomes Display Formula

(20)$ε̃dη̃dx̃(∂T̃0∂x̃|ỹ=0+ε̃∂T̃1∂x̃|ỹ=0+o(1))+∂T̃0∂ỹ|ỹ=0+ε̃∂T̃1∂ỹ|ỹ=0+o(ε̃)−ε̃η̃(∂2T̃0∂ỹ2|ỹ=0+ε̃∂2T̃1∂ỹ2|ỹ=0+o(ε̃))=0$

Matching powers of $ε̃$, the meniscus boundary conditions for $T̃0$ and $T̃1$ become Display Formula

(21)$∂T̃0∂ỹ|ỹ=0=0, for 0

(22)$∂T̃1∂ỹ|ỹ=0=−dη̃dx̃∂T̃0∂x̃|ỹ=0+η̃∂2T̃0∂ỹ2|ỹ=0 for 0

respectively. Further discussion of the perturbation expansion is provided in Appendix A.

###### Zeroth-Order Problem.

The temperature profile takes the following form: Display Formula

(23)$T̃0(x̃,ỹ)=−ϕỹ+ũ(x̃,ỹ)$

The first term on the RHS is the temperature profile in the absence of a meniscus, and $ũ$ is the change to the temperature field that results from the presence of an adiabatic, flat meniscus. In the context of spreading resistance, Mikic and Rohsenow [29] obtained the temperature profile for $T̃0$ to within an inconsequential constant as Display Formula

(24)$T̃0=−ϕỹ−2π2∑n=1∞1n2e−nπỹ sin(nπ(1−ϕ))cos(nπx̃)$

Enright et al. [6] showed that the thermal slip length is analogous to a thermal spreading resistance and adapted the above to provide an expression for thermal slip length for a flat meniscus. The thermal slip length associated with $T̃0$ is Display Formula

(25)$b̃t,0=∑n=1∞ sin2(nπ(ϕ−1))ϕ2n3π3$

where thermal slip length is nondimensionalized by the pitch of the structures, i.e., $b̃t=bt/(2d)$.

###### First-Order Problem.

The correction to the temperature field in the presence of deflection of the meniscus is $ε̃T̃1$. The governing equation for $T̃1$ is the Laplace equation. It is Display Formula

(26)$∇2T̃1=0$

subjected to Eq. (22) and the following Display Formula

(27a)$∂T̃1∂ỹ=0, 1−ϕ
Display Formula
(27b)$∂T̃1∂ỹ=0,0
Symmetry boundary conditions apply at $x̃=0$ and $x̃=1$. The method of separation of variables is used to solve the system. Applying the homogeneous boundary conditions leads to Display Formula
(28)$T̃1=∑n=1∞gn cos (nπx̃)e−nπỹ$

At y = 0, the boundary condition is composed of the boundary condition at the meniscus (Eq. (22)) and the boundary condition at the ridge (Eq. (27a)). These are applied to Eq. (28) and yield a set of dual series equations Display Formula

(29)$∑n=1∞−nπgn cos (nπx̃)=f(x̃), 0
Display Formula
(30)$∑n=1∞nπgn cos (nπx̃)=0, 1−ϕ

where Display Formula

(31)$f(x̃)=∑n=1∞ sin [nπ(1−ϕ)]{4nπx̃ sin(nπx̃)+2[x̃2−(1−ϕ)2]cos(nπx̃)}$

Using orthogonality and combining Eqs. (29) and (30) results in Display Formula

(32)$∫01−mπgm cos2(mπx̃)dx̃=∑n=1∞ sin (nπ(1−ϕ))∫01−ϕ{4nπx̃ sin(nπx̃)+2[x̃2−(1−ϕ)2]cos(nπx̃)} cos(mπx̃)dx̃$

The coefficients, gm, are found from the above, and the expression is provided in Appendix B.

The total thermal slip associated with an adiabatic meniscus with small curvature in a semi-infinite domain for the isoflux ridge is Display Formula

(33)$b̃t=b̃t,0+ε̃b̃t,1+⋯$

The contribution to thermal slip length associated with the deflection of the meniscus is found from Eq. (1). Noting that $T̃¯1,ĉ=0$Display Formula

(34)$ε̃b̃t,1=ε̃T̃¯1,sl2∂T̃¯/∂ỹ|ỹ=0$

We note that the perturbation domain, $T̃1$, is adiabatically connected to the unperturbed domain such that the denominator of Eq. (34) is two times the heat flux in the unperturbed domain, namely, $−2ϕ$. It follows that Display Formula

(35)$b̃t,1=∑m=1∞gm sin(mπ(ϕ−1))2mπϕ2$

which yields, to a first order in $ε̃$Display Formula

(36)$b̃t=∑m=1∞{ sin2(mπ(ϕ−1))ϕ2m3π3+ε̃gm sin(mπ(ϕ−1))2mπϕ2}$

We compute $b̃t,1$ for prescribed values of $ϕ$ 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, $ϕ=0.01$ 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 $o(ε̃)$, we may evaluate all quantities in the expression for thermal slip length at $ỹ=0$. The same applies to the case of an isothermal ridge discussed next.

###### Isothermal Ridge.

The dimensionless thermal slip length expression for a flat meniscus and isothermal ridge is provided by Enright et al. [6] Display Formula

(37)$b̃t,0=1πln[sec((1−ϕ)(π/2))]$

Sbragaglia and Prosperetti [18] 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.

The contribution to thermal slip length associated with the deflection of the meniscus is found using Eq. (B5) from Ref. [18] Display Formula

(38)$ε̃b̃t,1π=−ε̃2∫0c̃(1−cos(πx̃))(c̃2−x̃2)(cos(πx̃)−cos(πc̃))dx̃$

It follows that the total thermal slip length accounting for meniscus curvature is Display Formula

(39)$b̃t=1πln[sec((1−ϕ)(π/2))]− sin α4(1−ϕ)∫0c̃(1−cos(πx̃))(c̃2−x̃2)(cos(πx̃)−cos(πc̃))dx̃$

## Results

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 $ϕ→0$, the resistance grows to resemble that of an adiabatic boundary at $ỹ=0$. In the limit as $ϕ→1$, the resistance reverts to that of a domain with a solid surface at $ỹ=0$. 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.

###### Model Validation.

The partial differential equation (PDE) toolbox in matlab [30] was used to compute the temperature field in the liquid domain with the meniscus, TPDE. The change in temperature in the solution domain was compared to that of a domain with no meniscus, $T1D$, having the same height, width, thermal conductivity, and heat flux through it per Display Formula

(40)$β=(T¯PDE,sl−T¯PDE,∞)−(T¯1D,sl−T¯1D,∞)∂T¯/∂y·w$

where w is the width of the domain. The quantity β is equivalent to a dimensionless thermal spreading/constriction resistance [31].

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 [32] 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. [6] and [18] to β obtained from the PDE toolbox. We note that, insofar as the perturbation analysis to $o(ε̃)$, the mean temperatures along $ỹ=0$ and $ỹ→∞$ are unchanged as per Eqs. (24) and (26). Thus, in Eq. (C8), the corresponding terms in the numerator, i.e., $T̃¯0(x̃,0)$ and $T̃1¯(x̃,0)$, may be replaced by those as $ỹ→∞$. Thus, the slip may also be computed from Eq. (40) for β. However, once the magnitude of $ϵ̃$ is large enough that terms of $o(ε̃)$ become relevant, $T̃¯0(x̃,0)$ and $T̃¯1(x̃,0)$ may not be replaced by those as $ỹ→∞$, and thus, we do not expect $b̃t$ and β to agree as borne out by our results. The same argument applies for the case of an isothermal ridge. In summary, $β=b̃t+o(ε̃)$.

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 $b̃t$ 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 $|α|$ increases, $ε̃$ increases, and $ε̃$ is an indication of the magnitude of error to be expected.

For the isoflux case, the difference between the perturbation method, $b̃t$ and β, is less than 6.5% for protrusion angles between $±20 deg$, and solid fractions below 0.1. For the isothermal case, for angles between $±20 deg$, and solid fractions below 0.1, values that are typical for water-based cooling, the difference is less than 3%.

## Conclusions

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. [7]; 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.

## Acknowledgements

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

## Appendices

###### Appendix A: Full Perturbation Expansion

The Taylor expansion of Eq. (17) (previously used in Ref. [18]) is problematic when $ỹ=0$ falls outside of the physical domain (for $ε̃<0$). A more rigorous treatment of the perturbation expansion in Eq. (16) is possible through a domain transformation, taking $Ỹ=ỹ+ε̃η̃(x̃)$ and writing Display Formula

(A1)$T̃(x̃,ỹ)=T̃0′(x̃,Ỹ)+ε̃T̃1′(x̃,Ỹ)+o(ε̃)$

where we extend $η̃(x̃)$ to the solid–liquid interface by taking its value to be 0 for $c̃≤x̃≤1$. Note that, with this definition, the domains of $T̃0′$ and $T̃1′$ are naturally for $0 and $0, since points $(x̃,−ε̃η̃(x̃))$ on the meniscus are mapped to points on the $x̃$ -axis by this definition. Note that derivatives of $T̃(x̃,Ỹ)$ with respect to $x̃$ 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 $Ỹ=0$ for $T̃0′$ and $T̃1′$.

Expanding all terms, we recover the same zeroth-order problem as discussed in Sec. 3.2.2, so $T̃0′(x̃,Ỹ)=T̃0(x̃,Ỹ)$ as defined in Eq. (24); however, there are two key changes to the first-order problem. First, the boundary condition along the meniscus simplifies to Display Formula

(A2)$∂T̃1′∂Ỹ|Ỹ=0=−dη̃dx̃∂T̃0∂x̃|Ỹ=0$

Second, the governing PDE no longer satisfies the homogeneous Laplace's equation in Eq. (26), but a nonhomogeneous form, given by Display Formula

(A3)$∇2T̃1′=−2dη̃dx̃∂2T̃0∂x̃∂ỹ−d2η̃dx̃2∂T̃0∂ỹ$

Writing $T̃1′(x̃,Ỹ)=T̃1(x̃,Ỹ)−η̃(x̃)(∂T̃0/∂ỹ)(x̃,Ỹ)$, we find that $T̃1(x̃,Ỹ)$ satisfies both Laplace's equation and the boundary condition given in Eq. (22), so that its solution is, indeed, $T̃1$ as given in Eq. (28), with coefficients as given in Eq. (32). Considering, then, the perturbation in the thermal slip length defined in Eq. (34), the only changes from that given in Eq. (35) come from the difference Display Formula

(A4)$T̃1′(x̃,Ỹ)−T̃1(x̃,Ỹ)=−η̃(x̃)∂T̃0∂ỹ(x̃,Ỹ)$

Note that this difference is, in fact, zero along the boundary, since $(∂T̃0/∂ỹ)(x̃,0)=0$ for $0, by Eq. (21), and $η̃(x̃)=0$ for $c̃≤x̃≤1$ 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 $ỹ→∞$. 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

Display Formula

(B1)$gm=∑n=1,n≠m∞[−4mnπ4sin[nπ(1−ϕ)]{π(ϕ−1)cos[(m+n)π(ϕ−1)](m+n)+ sin[(m+n)π(1−ϕ)](m+n)2+π(1−ϕ)cos[(m−n)π(ϕ−1)](m−n)+ sin[(m−n)π(ϕ−1)](m−n)2}−8n sin2[nπ(1−ϕ)]m(m2−n2)3π4{(3m2+n2)cos[mπ(1−ϕ)]+2(1−ϕ)mπ(m2−n2)sin[mπ(1−ϕ)]}−4 sin[2nπ(1−ϕ)]m(m2−n2)3π4{(1−ϕ)(m4−n4)π cos[mπ(1−ϕ)]−m(m2+3n2)}]+4 sin[mπ(1−ϕ)]4m3π4{(2m)π(1−ϕ)cos[(2m)π(ϕ−1)]+sin[(2m)π(ϕ−1)]}+ sin[(1−ϕ)mπ]6m4π4{8(1−ϕ)3m3π3−6(1−ϕ)mπ cos[2(1−ϕ)mπ]+3 sin[2(1−ϕ)mπ]}$

###### Appendix C: Boundary Perturbation to Compute Thermal Slip Length

The dimensionless form of Eq. (1) is Display Formula

(C1)$b̃t=T̃¯sl−T̃¯ĉ2∂T̃∂ñ|ĉ¯$

where $ĉ$ denotes the composite interface. Defining $s̃$ as a coordinate (normalized to $d̃$) along the composite interface from the center of the (curved) meniscus $(s̃=0)$ to the center of the solid–liquid interface, it becomes Display Formula

(C2)$b̃t=T̃¯sl∫ĉds̃−∫ĉT̃ds̃2∫ĉ∂T̃∂ñds̃$

A general expression for $η̃(x̃)$ that applies along the entire length of the composite interface is $η̃(x̃)=H(1−ϕ−x̃)(c̃2−x̃2)$, where H denotes the Heaviside step function. Utilizing the formula for arc length to integrate along $x̃$ rather than $s̃$ and noting that $∫ĉds̃=1+o(ε̃)$, Eq. (C1) becomes Display Formula

(C3)$b̃t=T̃¯sl[1+o(ε̃)]−∫01T̃(x̃,−ε̃η̃)[1+(ε̃dη̃dx̃)2]1/2dx̃2∫01∂T̃∂ñ(x̃,−ε̃η̃)[1+(ε̃dη̃dx̃)2]1/2dx̃$

Expanding the terms in brackets as per the binomial theorem and noting that, since the meniscus is adiabatic, we need only evaluate the integral in the denominator from $1−ϕ$ to 1 Display Formula

(C4)$b̃t=T̃¯sl[1+o(ε̃)]−∫01T̃(x̃,−ε̃η̃)dx̃+o(ε̃)2∫1−ϕ1∂T̃∂ñ(x̃,−ε̃η̃)dx̃+o(ε̃)$

Expanding the terms evaluated at $(x̃,−ε̃η̃)$ in the numerator such that we may evaluate them at $(x̃,0)$ and recalling that $η̃=0$ along the solid–liquid interface when evaluating the integral in the denominator, we find that Display Formula

(C5)$b̃t=T̃¯sl[1+o(ε̃)]−∫01[T̃(x̃,0)−ε̃η̃∂T̃∂ỹ(x̃,0)+o(ε̃)]dx̃+o(ε̃)2∫1−ϕ1[∂T̃∂ỹ(x̃,0)]dx̃+o(ε̃)$

Too, as per the expansion $T̃=T̃0+ε̃T̃1+o(ε̃)$, Eq. (C5) becomes Display Formula

(C6)$b̃t=[1+o(ε̃)](T̃¯0,sl+ε̃T̃¯1,sl)+o(ε̃)−∫01[T̃0(x̃,0)+ε̃T̃1(x̃,0)−ε̃η̃∂T̃0∂ỹ(x̃,0)+o(ε̃)]dx̃+o(ε̃)2∫1−ϕ1[∂T̃0∂ỹ(x̃,0)+ε̃∂T̃1∂ỹ(x̃,0)+o(ε̃)]dx̃+o(ε̃)$

We note that the term $−ε̃η̃(∂T̃0/∂ỹ)|x̃,0$ in the numerator equals zero because the partial derivative equals 0 for $0≤x̃≤1−ϕ$ and $η̃=0$ over the solid–liquid interface. Since the perturbation domain adds no net heat flux to the problem, $∫1−ϕ1(∂T̃1/∂ỹ)(x̃,0)dx̃=0$ and can be eliminated. Thus, ignoring terms of $o(ε̃)$Display Formula

(C7)$b̃t=T̃¯0,sl+ε̃T̃¯1,sl−∫01[T̃0(x̃,0)+ε̃T̃1(x̃,0)]dx̃2∫1−ϕ1[∂T̃0∂ỹ(x̃,0)]dx̃$

or using overbars to denote mean quantities Display Formula

(C8)$b̃t=T̃¯0,sl−T̃0¯(x̃,0)+ε̃[T̃¯1,sl−T̃1¯(x̃,0)]2ϕ[∂T̃¯0,sl∂ỹ]$

Thus, when neglecting terms of $o(ε̃)$, we have shown that we may evaluate all quantities in the expression for slip length at $ỹ=0$. We note this analysis applies to both isoflux and isothermal solid–liquid interfaces.

## References

Steigerwalt Lam, L. , Hodes, M. , and Enright, R. , 2015, “ Analysis of Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces,” ASME J. Heat Transfer, 137(9), p. 091003.
Zhao, Y. , Liu, Y. , Xu, Q. , Barahman, M. , and Lyons, A. M. , 2015, “ Catalytic, Self-Cleaning Surface With Stable Superhydrophobic Properties: Printed Polydimethylsiloxane (PDMS) Arrays Embedded With TiO2 Nanoparticles,” ACS Appl. Mater. Interfaces, 7(4), pp. 2632–2640. [PubMed]
Navier, C. , 1823, “ Mémoire sur les du mouvement des fluids,” Mém. l'Académie R. Sci. l'Institut Fr., 6, pp. 389–440.
Kennard, E. H. , 1938, Kinetic Theory of Gases, 1st ed., McGraw-Hill, New York.
Inman, R. M. , 1964, “ Laminar Slip Flow Heat Transfer in a Parallel-Plate Channel or a Round Tube With Uniform Wall Heating,” NASA, Washington, DC, Technical Note D-2393.
Enright, R. , Hodes, M. , Salamon, T. , and Muzychka, Y. , 2014, “ Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels,” ASME J. Heat Transfer, 136(1), p. 012402.
Steigerwalt Lam, L. , 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.
Ng, C.-O. , and Wang, C. Y. , 2014, “ Temperature Jump Coefficient for Superhydrophobic Surfaces,” ASME J. Heat Transfer, 136(6), p. 064501.
Hodes, M. , Steigerwalt Lam, L. , 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.
Davies, J. , Maynes, D. , Webb, B. W. , and Woolford, B. , 2006, “ Laminar Flow in a Microchannel With Superhydrophobic Walls Exhibiting Transverse Ribs,” Phys. Fluids, 18(8), p. 087110.
Maynes, D. , Jeffs, K. , Woolford, B. , and Webb, B. W. , 2007, “ Laminar Flow in a Microchannel With Hydrophobic Surface Patterned Microribs Oriented Parallel to the Flow Direction,” Phys. Fluids, 19(9), p. 093603.
Teo, C. J. , and Khoo, B. C. , 2008, “ Analysis of Stokes Flow in Microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves,” Microfluid. Nanofluid., 7(3), pp. 353–382.
Cheng, Y. , Teo, C. , and Khoo, B. , 2009, “ Microchannel Flows With Superhydrophobic Surfaces: Effects of Reynolds Number and Pattern Width to Channel Height Ratio,” Phys. Fluids, 21(12), p. 122004.
Maynes, D. , and Crockett, J. , 2014, “ 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.
Cheng, Y. , Xu, J. , and Sui, Y. , 2014, “ Numerical Study on Drag Reduction and Heat Transfer Enhancement in Microchannels With Superhydrophobic Surfaces for Electronic Cooling,” Appl. Therm. Eng., 88, pp. 1–11.
Ybert, C. , Barentin, C. , Cottin-Bizonne, C. , Joseph, P. , and Bocquet, L. , 2007, “ Achieving Large Slip With Superhydrophobic Surfaces: Scaling Laws for Generic Geometries,” Phys. Fluids, 19(12), p. 123601.
Bocquet, L. , and Barrat, J.-L. , 2007, “ Flow Boundary Conditions From Nano- to Micro-Scales,” Soft Matter, 3(6), pp. 685–693.
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.
Crowdy, D. , 2010, “ Slip Length for Longitudinal Shear Flow Over a Dilute Periodic Mattress of Protruding Bubbles,” Phys. Fluids, 22(12), p. 121703.
Crowdy, D. , 2016, “ Analytical Formulae for Longitudinal Slip Lengths Over Unidirectional Superhydrophobic Surfaces With Curved Menisci,” J. Fluid Mech., 791, p. 065507.
Davis, A. M. J. , and Lauga, E. , 2009, “ Geometric Transition in Friction for Flow Over a Bubble Mattress,” Phys. Fluids, 21(1), p. 011701.
Crowdy, D. , 2016, “ Effective Slip Lengths for Longitudinal Shear Flow Over Partial-Slip Circular Bubble Mattresses,” Fluid Dyn. Res., 47(6), p. 065507.
Teo, C. J. , and Khoo, B. C. , 2010, “ Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature,” Microfluid. Nanofluid., 9(2–3), pp. 499–511.
Teo, C. J. , and Khoo, B. C. , 2014, “ Effects of Interface Curvature on Poiseuille Flow Through Microchannels and Microtubes Containing Superhydrophobic Surfaces With Transverse Grooves and Ribs,” Microfluid. Nanofluid., 17(5), pp. 891–905.
Ou, J. , and Rothstein, J. , 2005, “ Direct Velocity Measurements of the Flow Past Drag-Reducing Ultrahydrophobic Surfaces,” Phys. Fluids, 17(10), p. 103606.
Woolford, B. , Prince, J. , Maynes, D. , and Webb, B. W. , 2009, “ Particle Image Velocimetry Characterization of Turbulent Channel Flow With Rib Patterned Superhydrophobic Walls,” Phys. Fluids, 21(8), p. 085106.
Schäffel, D. , Koynov, K. , Vollmer, D. , Butt, H.-J. , and Schönecker, C. , 2016, “ Local Flow Field and Slip Length of Superhydrophobic Surfaces,” Phys. Rev. Lett., 116(13), p. 134501. [PubMed]
Hodes, M. , Karamanis, G. , Steigerwalt Lam, L. , Maclachlan, S. , Kirk, T. , and Papageorgiou, D. , 2015, “ Conformal Map and Asymptotic Solutions for Apparent Slip Lengths in the Presence of Thermocapillary Stress,” 23rd National Heat and Mass Transfer Conference and 1st International ISHMT-ASTFE Heat and Mass Transfer Conference, Thiruvananthapuram, India, Dec. 17–20, Paper No. IHMTC2015-1254.
Mikic, B. B. , and Rohsenow, W. M. , 1966, “ Thermal Contact Resistance,” Department of Mechanical Engineering, MIT, Cambridge, MA, Technical Report No. DSR-74542-41.
MathWorks, 2004, “ Partial Differential Equation ToolboxTM User's Guide,” MathWorks, Inc., Natick, MA.
Yovanovich, M. , and Marotta, E. , 2003, “ Thermal Spreading and Contact Resistances,” Handbook of Heat Transfer, A. Bejan and A. D. Kraus , eds., Wiley, Hoboken, NJ, pp. 261–394.
ANSYS, 2013, “ ANSYS Fluent Theory Guide,” ANSYS, Inc., Canonsburg, PA.
View article in PDF format.

## References

Steigerwalt Lam, L. , Hodes, M. , and Enright, R. , 2015, “ Analysis of Galinstan-Based Microgap Cooling Enhancement Using Structured Surfaces,” ASME J. Heat Transfer, 137(9), p. 091003.
Zhao, Y. , Liu, Y. , Xu, Q. , Barahman, M. , and Lyons, A. M. , 2015, “ Catalytic, Self-Cleaning Surface With Stable Superhydrophobic Properties: Printed Polydimethylsiloxane (PDMS) Arrays Embedded With TiO2 Nanoparticles,” ACS Appl. Mater. Interfaces, 7(4), pp. 2632–2640. [PubMed]
Navier, C. , 1823, “ Mémoire sur les du mouvement des fluids,” Mém. l'Académie R. Sci. l'Institut Fr., 6, pp. 389–440.
Kennard, E. H. , 1938, Kinetic Theory of Gases, 1st ed., McGraw-Hill, New York.
Inman, R. M. , 1964, “ Laminar Slip Flow Heat Transfer in a Parallel-Plate Channel or a Round Tube With Uniform Wall Heating,” NASA, Washington, DC, Technical Note D-2393.
Enright, R. , Hodes, M. , Salamon, T. , and Muzychka, Y. , 2014, “ Isoflux Nusselt Number and Slip Length Formulae for Superhydrophobic Microchannels,” ASME J. Heat Transfer, 136(1), p. 012402.
Steigerwalt Lam, L. , 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.
Ng, C.-O. , and Wang, C. Y. , 2014, “ Temperature Jump Coefficient for Superhydrophobic Surfaces,” ASME J. Heat Transfer, 136(6), p. 064501.
Hodes, M. , Steigerwalt Lam, L. , 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.
Davies, J. , Maynes, D. , Webb, B. W. , and Woolford, B. , 2006, “ Laminar Flow in a Microchannel With Superhydrophobic Walls Exhibiting Transverse Ribs,” Phys. Fluids, 18(8), p. 087110.
Maynes, D. , Jeffs, K. , Woolford, B. , and Webb, B. W. , 2007, “ Laminar Flow in a Microchannel With Hydrophobic Surface Patterned Microribs Oriented Parallel to the Flow Direction,” Phys. Fluids, 19(9), p. 093603.
Teo, C. J. , and Khoo, B. C. , 2008, “ Analysis of Stokes Flow in Microchannels With Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves,” Microfluid. Nanofluid., 7(3), pp. 353–382.
Cheng, Y. , Teo, C. , and Khoo, B. , 2009, “ Microchannel Flows With Superhydrophobic Surfaces: Effects of Reynolds Number and Pattern Width to Channel Height Ratio,” Phys. Fluids, 21(12), p. 122004.
Maynes, D. , and Crockett, J. , 2014, “ 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.
Cheng, Y. , Xu, J. , and Sui, Y. , 2014, “ Numerical Study on Drag Reduction and Heat Transfer Enhancement in Microchannels With Superhydrophobic Surfaces for Electronic Cooling,” Appl. Therm. Eng., 88, pp. 1–11.
Ybert, C. , Barentin, C. , Cottin-Bizonne, C. , Joseph, P. , and Bocquet, L. , 2007, “ Achieving Large Slip With Superhydrophobic Surfaces: Scaling Laws for Generic Geometries,” Phys. Fluids, 19(12), p. 123601.
Bocquet, L. , and Barrat, J.-L. , 2007, “ Flow Boundary Conditions From Nano- to Micro-Scales,” Soft Matter, 3(6), pp. 685–693.
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.
Crowdy, D. , 2010, “ Slip Length for Longitudinal Shear Flow Over a Dilute Periodic Mattress of Protruding Bubbles,” Phys. Fluids, 22(12), p. 121703.
Crowdy, D. , 2016, “ Analytical Formulae for Longitudinal Slip Lengths Over Unidirectional Superhydrophobic Surfaces With Curved Menisci,” J. Fluid Mech., 791, p. 065507.
Davis, A. M. J. , and Lauga, E. , 2009, “ Geometric Transition in Friction for Flow Over a Bubble Mattress,” Phys. Fluids, 21(1), p. 011701.
Crowdy, D. , 2016, “ Effective Slip Lengths for Longitudinal Shear Flow Over Partial-Slip Circular Bubble Mattresses,” Fluid Dyn. Res., 47(6), p. 065507.
Teo, C. J. , and Khoo, B. C. , 2010, “ Flow Past Superhydrophobic Surfaces Containing Longitudinal Grooves: Effects of Interface Curvature,” Microfluid. Nanofluid., 9(2–3), pp. 499–511.
Teo, C. J. , and Khoo, B. C. , 2014, “ Effects of Interface Curvature on Poiseuille Flow Through Microchannels and Microtubes Containing Superhydrophobic Surfaces With Transverse Grooves and Ribs,” Microfluid. Nanofluid., 17(5), pp. 891–905.
Ou, J. , and Rothstein, J. , 2005, “ Direct Velocity Measurements of the Flow Past Drag-Reducing Ultrahydrophobic Surfaces,” Phys. Fluids, 17(10), p. 103606.
Woolford, B. , Prince, J. , Maynes, D. , and Webb, B. W. , 2009, “ Particle Image Velocimetry Characterization of Turbulent Channel Flow With Rib Patterned Superhydrophobic Walls,” Phys. Fluids, 21(8), p. 085106.
Schäffel, D. , Koynov, K. , Vollmer, D. , Butt, H.-J. , and Schönecker, C. , 2016, “ Local Flow Field and Slip Length of Superhydrophobic Surfaces,” Phys. Rev. Lett., 116(13), p. 134501. [PubMed]
Hodes, M. , Karamanis, G. , Steigerwalt Lam, L. , Maclachlan, S. , Kirk, T. , and Papageorgiou, D. , 2015, “ Conformal Map and Asymptotic Solutions for Apparent Slip Lengths in the Presence of Thermocapillary Stress,” 23rd National Heat and Mass Transfer Conference and 1st International ISHMT-ASTFE Heat and Mass Transfer Conference, Thiruvananthapuram, India, Dec. 17–20, Paper No. IHMTC2015-1254.
Mikic, B. B. , and Rohsenow, W. M. , 1966, “ Thermal Contact Resistance,” Department of Mechanical Engineering, MIT, Cambridge, MA, Technical Report No. DSR-74542-41.
MathWorks, 2004, “ Partial Differential Equation ToolboxTM User's Guide,” MathWorks, Inc., Natick, MA.
Yovanovich, M. , and Marotta, E. , 2003, “ Thermal Spreading and Contact Resistances,” Handbook of Heat Transfer, A. Bejan and A. D. Kraus , eds., Wiley, Hoboken, NJ, pp. 261–394.
ANSYS, 2013, “ ANSYS Fluent Theory Guide,” ANSYS, Inc., Canonsburg, PA.

## Figures

Fig. 1

Schematic of a curved meniscus between the ridges. The protrusion angle, α, is that between the line tangent to the meniscus at the triple contact line and the horizontal. The pitch of the structures is 2d. The width of the cavity is 2c.

Fig. 2

Schematic of isoflux domain, a half-ridge and half-cavity of a parallel ridge structured surface with meniscus between the ridges. Dashed vertical lines indicate symmetry.

Fig. 3

Dimensionless thermal slip length versus solid fraction, ϕ, for selected protrusion angles, α, for the case of an isoflux ridge

Fig. 4

Dimensionless thermal slip length versus solid fraction, ϕ, for selected protrusion angles, α, for the case of an isothermal ridge

Fig. 5

Dimensionless thermal slip length versus protrusion angle, α, for selected solid fractions, ϕ. Solid lines correspond to an isoflux ridge. Dashed lines correspond to an isothermal ridge.

Fig. 6

Dimensionless slip length versus solid fraction for the case of an isoflux ridge. Solid lines are perturbation method results, and Δ's are values for β from the numerical results from matlab.

Fig. 7

Dimensionless slip length versus solid fraction for the case of an isothermal ridge. Solid lines are perturbation method results from Ref. [18], and Δ's are values for β from the numerical results from matlab.

## Tables

Table 1 Comparison of perturbation method values, $b̃t$, with numerical values, β, for the isoflux ridge case
Table 2 Comparison of perturbation method values, $b̃t$, with numerical values, β, for the isothermal ridge case

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections