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

Karamanis, Georgios; Hodes, Marc; Kirk, Toby; Papageorgiou, Demetrios T.

doi:10.1115/1.4039085

We consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. 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. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz–Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.

Introduction

A sessile droplet on a structured surface characterized by small periodic length scales compared to the capillary length may be stable in the Cassie state [1,2] where solid–liquid contact is exclusively at the tips of the structures. A liquid flowing through a microchannel with structured surfaces may be as well and the necessary criteria are provided by Lam et al. [3]. Then, a mixed boundary condition of no-slip [4,5] and low-shear applies along the solid–liquid and the liquid–gas¹ interfaces (menisci), respectively. The low-shear boundary condition provides a lubrication effect and thus reduces both the hydrodynamic resistance and the caloric part of the thermal resistance. However, the reduction in the solid–liquid interfacial area reduces the available area for heat transfer and thus increases the convective part of the thermal resistance. A net reduction of the total, i.e., caloric plus convective, thermal resistance can be achieved with proper sizing of the structures [3] and it requires the knowledge of the Nusselt number (Nu). Such Nusselt numbers are especially relevant to direct liquid cooling applications [3] as per Fig. 1 that depicts a structured microchannel etched into the upper portion of a microprocessor die.

Fig. 1

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Depiction of a structured microchannel etched into the upper portion of a microprocessor die

The channel surfaces can be textured with a variety of periodic structures such as pillars, transverse ridges, or parallel ridges [6]. The latter configuration is examined here since it is more favorable from a heat transfer perspective [3,7]. The hydrodynamic effects of structured surfaces with parallel ridges in parallel plate channels have been studied for flat and curved menisci [8–15]. In terms of the 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. Their analysis applies to both symmetrically and asymmetrically heated channels with large plate spacing to structure pitch ratio. Enright et al. [7] too developed analytical expressions for apparent slip lengths for structured surfaces with parallel or transverse ridges or pillar arrays assuming flat and adiabatic menisci. Ng and Wang [16] derived semi-analytical expressions for the apparent thermal slip length for isothermal parallel ridges while accounting for conduction through the gas phase. Lam et al. [17] derived expressions for the apparent thermal slip length for isoflux and isothermal parallel ridges accounting for small meniscus curvature. Hodes et al. [18] captured the effects of evaporation and condensation along menisci on the apparent thermal slip length for isoflux ridges. Lam et al. [19] developed expressions for the Nusselt number for thermally developing Couette flow as a function of apparent slip lengths for various boundary conditions. Also, Lam et al. [19] discuss when Nu results accounting for molecular slip can be used to capture the effects of apparent slip. Maynes and Crockett [20] developed expressions for the Nusselt number and the thermal slip length for microchannels with isoflux parallel ridges assuming flat menisci and using the Navier slip approximation for the velocity profile. Kirk et al. [21] also developed expressions for the Nusselt number for this configuration, but without invoking the Navier slip approximation. Kirk et al. [21] also accounted for small meniscus curvature using a boundary perturbation method. Karamanis et al. [22] developed 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 [23–25].

The present work extends the analysis in Ref. [22] to the case of flow with finite axial conduction, i.e., to the extended Graetz–Nusselt problem [26–28]. Viscous dissipation [29,30] and (uniform) volumetric heat generation [31] are also captured. The menisci are assumed to be flat [17] and adiabatic. The configurations for the isothermal ridges are either both plates textured, as per Fig. 2, or one plate textured and the other one smooth and adiabatic, as per Fig. 3. The solution approach is similar in both configurations. It therefore suffices to present the detailed analysis for the first configuration. The second configuration is considered in Appendix A.

Fig. 2

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Schematic of the periodic domain when both plates are textured; the computational domain is indicated with dotted line

Fig. 3

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Schematic of the periodic domain when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; the computational domain is indicated with dotted line

The cross-sectional view of one period of the domain (D) considered is depicted in Fig. 2. It extends from minus to plus infinity in the streamwise direction z, and $| x | \leq d$ and $0 \leq y \leq H$ ⁠, where 2d is the pitch of the ridges and H is the distance between the ridge tips on opposing plates. The hydraulic diameter of the domain $(D_{h})$ is 2H. The width of the meniscus is 2a. The triple contact lines coincide with the corners of the ridges at $| x | = a$ at both y = 0 and y = H. Along the composite interfaces at y = 0 and y = H, a no-shear boundary condition applies for $| x | < a$ and a no-slip one is imposed for $a < | x | < d$ [4,5]. Symmetry boundary conditions apply along the boundaries at $x = | d |$ ⁠. The temperature of the ridges is $T_{r -}$ and $T_{r +}$ for $z \leq 0$ and z > 0, respectively. The flow is pressure driven, steady, laminar, hydrodynamically developed, and thermally developing with constant thermophysical properties. The temperature profile becomes uniform throughout the cross section as $z \to - \infty$ and $z \to + \infty$ ⁠, where it is $T_{r -}$ and $T_{r +}$ ⁠, respectively. Effects due to Marangoni stresses [32,33], evaporation and condensation [18], and gas diffusion in the liquid phase are neglected. The independent dimensionless geometric variables are the solid fraction of the ridge, $ϕ = (d - a) / d$ ⁠, and the aspect ratio of the domain, H/d. Finally, the analysis utilizes the symmetry of the domain with respect to the yz and zx planes through x = 0 and $y = H / 2$ ⁠, respectively, and therefore, we further restrict to $0 \leq x \leq d$ and $0 \leq y \leq H / 2$ ⁠.

Analysis

Hydrodynamic Problem.

Assuming hydrodynamically developed laminar flow with constant thermophysical properties, the streamwise-momentum equation takes the form

\nabla^{2} w = \frac{1}{μ} \frac{d p}{d z}

(1)

where w is the streamwise velocity, μ is the dynamic viscosity, and

d p / d z

is the prescribed (constant) pressure gradient. Denoting nondimensional variables with tildes, Eq. is rendered dimensionless by defining

\tilde{x} = \frac{x}{D_{h}}

(2)

\tilde{y} = \frac{y}{D_{h}}

(3)

\tilde{w} = \frac{μ}{D_{h}^{2}} {(- \frac{d p}{d z})}^{- 1} w

(4)

It becomes

\nabla^{2} \tilde{w} = - 1

(5)

subjected to

\frac{\partial \tilde{w}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{a}, \tilde{y} = 0 \end{matrix}

(6)

\tilde{w} = 0 \begin{matrix} for \tilde{a} < \tilde{x} < \tilde{d}, \tilde{y} = 0 \end{matrix}

(7)

\frac{\partial \tilde{w}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 4 \end{matrix}

(8)

\frac{\partial \tilde{w}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 4 \end{matrix}

(9)

where $\tilde{d} = d / D_{h}$ and $\tilde{a} = a / D_{h}$ are the dimensionless (half) pitch of the ridges and width of the meniscus, respectively. This hydrodynamic problem has been studied analytically [15], semi-analytically [11,14], and numerically [10] in the past. Here, we solve it numerically (see Appendix B) to facilitate the solution of the thermal energy equation.

To proceed with the formulation of the Nusselt number, we first compute the Poiseuille number

f Re

where

Re = \frac{ρ \bar{w} D_{h}}{μ}

(10)

f = \frac{2 D_{h}}{ρ {\bar{w}}^{2}} (- \frac{d p}{d z})

(11)

\bar{w} = \frac{2}{d H} \int_{0}^{H / 2} \int_{0}^{d} w d x d y

(12)

are the Reynolds number based on hydraulic diameter, the friction factor and the mean velocity of the liquid, respectively, and ρ is the density. Combining Eqs. – and –, it follows that

f Re = \frac{2}{\bar{\tilde{w}}}

(13)

where

\bar{\tilde{w}} = \frac{4}{\tilde{d}} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} d \tilde{x} d \tilde{y}

(14)

is the dimensionless mean velocity of the flow. Henceforth,

\tilde{w} (\tilde{x}, \tilde{y})

and thus

f Re

are considered to be known given that the hydrodynamic problem can be solved independently from the thermal one.

Thermal Problem.

Capturing axial conduction, viscous dissipation, and volumetric heat generation, the relevant form of the thermal energy equation is

\begin{matrix} ρ c_{p} w \frac{\partial T}{\partial z} = k (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} + \frac{\partial^{2} T}{\partial z^{2}}) + μ [{(\frac{\partial w}{\partial x})}^{2} + {(\frac{\partial w}{\partial y})}^{2}] + \dot{q} \end{matrix}

(15)

where T, k, and c_p are the temperature, thermal conductivity, and specific heat at constant pressure of the liquid, respectively, and $\dot{q}$ is the (constant) volumetric heat generation rate within the liquid.

Next, we introduce the dimensionless streamwise coordinate

\tilde{z}

and temperature

\tilde{T}

⁠, as per

\tilde{z} = \frac{z}{Pe D_{h}}

(16)

\tilde{T} = \frac{T - T_{r -}}{T_{r +} - T_{r -}}

(17)

respectively, where

Pe = RePr

(18)

is the Péclet number, i.e., the scale of the ratio of advective to axially diffusive heat transfer and

Pr = \frac{c_{p} μ}{k}

(19)

is the Prandtl number of the liquid. Combining Eqs. –, , and –, the dimensionless form of the (inhomogeneous) thermal energy equation is

\begin{matrix} \frac{f Re}{2} \tilde{w} \frac{\partial \tilde{T}}{\partial \tilde{z}} = \frac{\partial^{2} \tilde{T}}{\partial {\tilde{x}}^{2}} + \frac{\partial^{2} \tilde{T}}{\partial {\tilde{y}}^{2}} + \frac{1}{{Pe}^{2}} \frac{\partial^{2} \tilde{T}}{\partial {\tilde{z}}^{2}} \\ + \frac{Br {(f Re)}^{2}}{4} [{(\frac{\partial \tilde{w}}{\partial \tilde{x}})}^{2} + {(\frac{\partial \tilde{w}}{\partial \tilde{y}})}^{2}] + \tilde{\dot{q}} \end{matrix}

(20)

where

Br = \frac{μ {\bar{w}}^{2}}{k (T_{r +} - T_{r -})}

(21)

\tilde{\dot{q}} = \frac{\dot{q} D_{h}^{2}}{k (T_{r +} - T_{r -})}

(22)

are the Brinkman number and the dimensionless heat generation rate, respectively. It is subjected to the following boundary conditions:

\frac{\partial \tilde{T}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{a}, \tilde{y} = 0 \end{matrix}

(23)

\tilde{T} = 0 \begin{matrix} for \tilde{a} < \tilde{x} < \tilde{d}, \tilde{y} = 0, \tilde{z} \leq 0 \end{matrix}

(24)

\tilde{T} = 1 \begin{matrix} for \tilde{a} < \tilde{x} < \tilde{d}, \tilde{y} = 0, \tilde{z} > 0 \end{matrix}

(25)

\frac{\partial \tilde{T}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 4 \end{matrix}

(26)

\frac{\partial \tilde{T}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 4 \end{matrix}

(27)

\tilde{T} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, 0 < \tilde{y} < 1 / 4, \tilde{z} \to - \infty \end{matrix}

(28)

\tilde{T} = 1 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, 0 < \tilde{y} < 1 / 4, \tilde{z} \to + \infty \end{matrix}

(29)

The dimensionless temperature field is decomposed as

\tilde{T} = {\tilde{T}}_{h} + {\tilde{T}}_{p}

(30)

where ${\tilde{T}}_{h} (\tilde{x}, \tilde{y}, \tilde{z}, H / d, ϕ, Pe)$ and ${\tilde{T}}_{p} (\tilde{x}, \tilde{y}, H / d, ϕ, Br, \tilde{\dot{q}})$ are the homogeneous and particular solutions, respectively. Thus, ${\tilde{T}}_{h}$ satisfies the homogeneous form of Eq. (20), with viscous dissipation and heat generation absent, and ${\tilde{T}}_{p}$ satisfies Eq. (20) but with homogeneous boundary conditions.

Homogeneous Solution.

Here, we consider the homogeneous form of Eq. , i.e.,

Br

and

\tilde{\dot{q}}

are set to zero. We seek solutions of the form

{\tilde{T}}_{h} (\tilde{x}, \tilde{y}, \tilde{z}) = {\begin{matrix} ψ_{-} (\tilde{x}, \tilde{y}) g_{-} (\tilde{z}) & for \tilde{z} \leq 0 \\ 1 - ψ_{+} (\tilde{x}, \tilde{y}) g_{+} (\tilde{z}) & for \tilde{z} > 0 \end{matrix}

which separate the dependence of

\tilde{T}

on the streamwise coordinate

\tilde{z}

from that on the transverse coordinates

\tilde{x}

and

\tilde{y}

⁠. It follows that

g_{\pm} (\tilde{z}) = exp (- λ_{\pm} \tilde{z})

(32)

and

ψ_{\pm} (\tilde{x}, \tilde{y})

satisfies

\nabla^{2} ψ_{\pm} = - λ_{\pm} (\frac{f Re}{2} \tilde{w} + \frac{λ_{\pm}}{{Pe}^{2}}) ψ_{\pm}

(33)

subject to the boundary conditions

\frac{\partial ψ_{\pm}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{a}, \tilde{y} = 0 \end{matrix}

(34)

ψ_{\pm} = 0 \begin{matrix} for \tilde{a} < \tilde{x} < \tilde{d}, \tilde{y} = 0 \end{matrix}

(35)

\frac{\partial ψ_{\pm}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 4 \end{matrix}

(36)

\frac{\partial ψ_{\pm}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 4 \end{matrix}

(37)

Therefore, $(ψ_{+}, λ_{+})$ and $(ψ_{-}, λ_{-})$ are solutions of the same nonlinear eigenvalue problem, given by Eqs. (33)–(37). The eigenvalues are real. We assume that they are discrete and there are infinitely many and let λ_i and ψ_i denote the ith eigenvalue and eigenfunction, respectively, ordered such that $- \infty \leftarrow < \dots < λ_{- 2} < λ_{- 1} < 0 < λ_{1} < λ_{2} < \dots \to + \infty$ ⁠. Then, the eigensolutions for $\tilde{z} > 0$ and $\tilde{z} \leq 0$ correspond to those with $λ_{i} > 0$ and $λ_{i} < 0$ ⁠, respectively, so that there is exponential decay in the upstream (⁠ $\tilde{z} \to - \infty$ ⁠) and downstream (⁠ $\tilde{z} \to + \infty$ ⁠) directions. The set of ψ_i and λ_i is determined numerically (see Appendix B) and henceforth assumed to be known.

We proceed by expressing the general homogeneous solution

{\tilde{T}}_{h} (\tilde{x}, \tilde{y}, \tilde{z})

as a linear combination of the appropriate eigenfunctions in each region, i.e.,

{\tilde{T}}_{h} (\tilde{x}, \tilde{y}, \tilde{z}) = {\begin{matrix} \sum_{i = - \infty}^{- 1} c_{i} ψ_{i} (\tilde{x}, \tilde{y}) exp (- λ_{i} \tilde{z}) & for \tilde{z} \leq 0 \\ 1 - \sum_{i = 1}^{+ \infty} c_{i} ψ_{i} (\tilde{x}, \tilde{y}) exp (- λ_{i} \tilde{z}) & for \tilde{z} > 0 \end{matrix}

(38)

The expansion coefficients c_i follow from the requirement that both temperature and heat flux are continuous at

\tilde{z} = 0

⁠, for

0 < \tilde{x} < \tilde{d}

and

0 < \tilde{y} < 1 / 4

⁠, i.e.,

\lim_{\tilde{z} \to 0^{-}} {\tilde{T}}_{h} = \lim_{\tilde{z} \to 0^{+}} {\tilde{T}}_{h}

(39)

\lim_{\tilde{z} \to 0^{-}} \frac{\partial {\tilde{T}}_{h}}{\partial \tilde{z}} = \lim_{\tilde{z} \to 0^{+}} \frac{\partial {\tilde{T}}_{h}}{\partial \tilde{z}}

(40)

Substituting Eq. in Eqs. and , the latter become, respectively,

\sum_{i = - \infty, i \neq 0}^{+ \infty} c_{i} ψ_{i} = 1

(41)

\sum_{i = - \infty, i \neq 0}^{+ \infty} c_{i} λ_{i} ψ_{i} = 0

(42)

Given that the eigenvalue problem is nonlinear, we lack a natural orthogonality condition for the eigenfunctions ψ_i. Therefore, we modify the analysis by Deavours [27] to derive an orthogonality condition, to enable us to determine the expansion coefficients from Eqs. and . First, we multiply both sides of Eq. for the ith eigenvalue by

λ_{j} ψ_{j}

to give

λ_{j} ψ_{j} \nabla^{2} ψ_{i} = - λ_{j} ψ_{j} λ_{i} (\frac{f Re}{2} \tilde{w} + \frac{λ_{i}}{{Pe}^{2}}) ψ_{i}

(43)

Interchanging i and j in Eq. and subtracting the result from Eq. , it follows that

λ_{j} ψ_{j} \nabla^{2} ψ_{i} - λ_{i} ψ_{i} \nabla^{2} ψ_{j} = \frac{λ_{i} λ_{j}}{{Pe}^{2}} (λ_{j} - λ_{i}) ψ_{i} ψ_{j}

(44)

Using the identity

ψ_{j} \nabla^{2} ψ_{i} = \nabla \cdot (ψ_{j} \nabla ψ_{i}) - \nabla ψ_{j} \cdot \nabla ψ_{i}

(45)

Equation can be rewritten as

\begin{matrix} λ_{j} \nabla \cdot (ψ_{j} \nabla ψ_{i}) - λ_{i} \nabla \cdot (ψ_{i} \nabla ψ_{j}) = (λ_{j} - λ_{i}) (\frac{λ_{i} λ_{j}}{{Pe}^{2}} ψ_{i} ψ_{j} + \nabla ψ_{i} \cdot \nabla ψ_{j}) \end{matrix}

(46)

Integrating Eq. over the domain yields

\begin{matrix} λ_{j} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \nabla \cdot (ψ_{j} \nabla ψ_{i}) d \tilde{x} d \tilde{y} - λ_{i} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \nabla \cdot (ψ_{i} \nabla ψ_{j}) d \tilde{x} d \tilde{y} \\ = (λ_{j} - λ_{i}) \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{λ_{i} λ_{j}}{{Pe}^{2}} ψ_{i} ψ_{j} + \nabla ψ_{i} \cdot \nabla ψ_{j}) d \tilde{x} d \tilde{y} \end{matrix}

(47)

However, employing the Divergence Theorem and utilizing Eqs. –, we can show that

\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \nabla \cdot (ψ_{i} \nabla ψ_{j}) d \tilde{x} d \tilde{y} = 0

(48)

Hence, from Eq. and given that

λ_{j} \neq λ_{i}

⁠, Eq. yields

\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{λ_{i} λ_{j}}{{Pe}^{2}} ψ_{i} ψ_{j} + \nabla ψ_{i} \cdot \nabla ψ_{j}) d \tilde{x} d \tilde{y} = 0, i \neq j

(49)

or, in vector notation

\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} {[\begin{matrix} λ_{i} ψ_{i} \\ \partial ψ_{i} / \partial \tilde{x} \\ \partial ψ_{i} / \partial \tilde{y} \end{matrix}]}^{T} [\begin{matrix} \frac{1}{{Pe}^{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} λ_{j} ψ_{j} \\ \partial ψ_{j} / \partial \tilde{x} \\ \partial ψ_{j} / \partial \tilde{y} \end{matrix}] d \tilde{x} d \tilde{y} = 0

(50)

Thus, the required orthogonality condition is that the vectors

{[λ_{i} ψ_{i}, \partial ψ_{i} / \partial \tilde{x}, \partial ψ_{i} / \partial \tilde{y}]}^{T}

and

{[λ_{j} ψ_{j}, \partial ψ_{j} / \partial \tilde{x}, \partial ψ_{j} / \partial \tilde{y}]}^{T}

are orthogonal with respect to the matrix

B = [\begin{matrix} \frac{1}{{Pe}^{2}} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

(51)

With this orthogonality relation, we can now proceed to compute the expansion coefficients c_i. We multiply each vector

{[λ_{j} ψ_{j}, \partial ψ_{j} / \partial \tilde{x}, \partial ψ_{j} / \partial \tilde{y}]}^{T}

by the corresponding expansion coefficient c_j and sum the resulting expressions over all indices j. Then, it follows from Eq. that

\sum_{j = - \infty, j \neq 0}^{+ \infty} c_{j} [\begin{matrix} λ_{j} ψ_{j} \\ \partial ψ_{j} / \partial \tilde{x} \\ \partial ψ_{j} / \partial \tilde{y} \end{matrix}] = \sum_{j = - \infty, j \neq 0}^{+ \infty} c_{j} [\begin{matrix} 0 \\ \partial ψ_{j} / \partial \tilde{x} \\ \partial ψ_{j} / \partial \tilde{y} \end{matrix}]

(52)

Next, taking the dot product of both sides of Eq. with the vector

B {[λ_{i} ψ_{i}, \partial ψ_{i} / \partial \tilde{x}, \partial ψ_{i} / \partial \tilde{y}]}^{T}

and integrating the resulting expression over the domain, it follows that

\begin{matrix} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \sum_{j = - \infty, j \neq 0}^{+ \infty} c_{j} (\frac{λ_{j} λ_{i}}{{Pe}^{2}} ψ_{j} ψ_{i} + \nabla ψ_{j} \cdot \nabla ψ_{i}) d \tilde{x} d \tilde{y} \\ = \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \sum_{j = - \infty, j \neq 0}^{+ \infty} c_{j} (\nabla ψ_{j} \cdot \nabla ψ_{i}) d \tilde{x} d \tilde{y} \end{matrix}

(53)

Switching the order of integration and summation on the left-hand side of Eq. and employing Eq. yields

\begin{matrix} c_{i} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{λ_{i}^{2}}{{Pe}^{2}} ψ_{i}^{2} + \nabla ψ_{i} \cdot \nabla ψ_{i}) d \tilde{x} d \tilde{y} \\ = \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \sum_{j = - \infty, j \neq 0}^{+ \infty} c_{j} (\nabla ψ_{j} \cdot \nabla ψ_{i}) d \tilde{x} d \tilde{y} \end{matrix}

(54)

Then, using Eq. (which is valid for i = j as well as

i \neq j

⁠), Eq. becomes

\begin{matrix} c_{i} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} [\frac{λ_{i}^{2}}{{Pe}^{2}} ψ_{i}^{2} + \nabla \cdot (ψ_{i} \nabla ψ_{i}) - ψ_{i} \nabla^{2} ψ_{i}] d \tilde{x} d \tilde{y} \\ = \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \sum_{j = - \infty, j \neq 0}^{+ \infty} c_{j} [\nabla \cdot (ψ_{j} \nabla ψ_{i}) - ψ_{j} \nabla^{2} ψ_{i}] d \tilde{x} d \tilde{y} \end{matrix}

(55)

Switching the order of integration and summation on the right-hand side of Eq. and employing Eqs. and yields

\begin{matrix} c_{i} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{2 λ_{i}}{{Pe}^{2}} + \frac{f Re}{2} \tilde{w}) ψ_{i}^{2} d \tilde{x} d \tilde{y} \\ = \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{λ_{i}}{{Pe}^{2}} + \frac{f Re}{2} \tilde{w}) ψ_{i} (\sum_{j = - \infty, j \neq 0}^{+ \infty} c_{j} ψ_{j}) d \tilde{x} d \tilde{y} \end{matrix}

(56)

Then, using condition Eq. , it follows that

\begin{matrix} c_{i} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{2 λ_{i}}{{Pe}^{2}} + \frac{f Re}{2} \tilde{w}) ψ_{i}^{2} d \tilde{x} d \tilde{y} = \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{λ_{i}}{{Pe}^{2}} + \frac{f Re}{2} \tilde{w}) ψ_{i} d \tilde{x} d \tilde{y} \end{matrix}

(57)

Thus, rearranging Eq. gives the expansion coefficients as per

c_{i} = \frac{\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} [\tilde{w} + 2 λ_{i} / (f {RePe}^{2})] ψ_{i} d \tilde{x} d \tilde{y}}{\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} [\tilde{w} + 4 λ_{i} / (f {RePe}^{2})] ψ_{i}^{2} d \tilde{x} d \tilde{y}}

(58)

Finally, our attention shifts to compute the integral over the ridge of

{\partial ψ_{i} / \partial \tilde{y} |}_{\tilde{y} = 0}

that is used later in the formulation of the Nusselt number. Integrating Eq. over the domain yields

\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \nabla^{2} ψ_{i} d \tilde{x} d \tilde{y} = - λ_{i} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{f Re}{2} \tilde{w} + \frac{λ_{i}}{{Pe}^{2}}) ψ_{i} d \tilde{x} d \tilde{y}

(59)

Applying the Divergence Theorem on the left-hand side and utilizing Eqs. –, Eq. becomes

\int_{\tilde{a}}^{\tilde{d}} {\frac{\partial ψ_{i}}{\partial \tilde{y}} |}_{\tilde{y} = 0} d \tilde{x} = λ_{i} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\frac{f Re}{2} \tilde{w} + \frac{λ_{i}}{{Pe}^{2}}) ψ_{i} d \tilde{x} d \tilde{y}

(60)

Particular Solution.

We choose the particular solution to be the solution constant in

\tilde{z}

of Eq. (the inhomogeneous equation) with homogeneous boundary conditions. Viscous dissipation and volumetric heat generation are considered separately and the solutions are superimposed. Therefore, we express the particular solution as

{\tilde{T}}_{p} = \frac{Br {(f Re)}^{2}}{4} {\tilde{T}}_{p, Br} + \tilde{\dot{q}} {\tilde{T}}_{p, \tilde{\dot{q}}}

(61)

The quantity

{\tilde{T}}_{p, Br}

satisfies

\nabla^{2} {\tilde{T}}_{p, Br} = - {| \nabla \tilde{w} |}^{2}

(62)

with boundary conditions

\frac{\partial {\tilde{T}}_{p, i}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{a}, \tilde{y} = 0 \end{matrix}

(63)

{\tilde{T}}_{p, i} = 0 \begin{matrix} for \tilde{a} < \tilde{x} < \tilde{d}, \tilde{y} = 0 \end{matrix}

(64)

\frac{\partial {\tilde{T}}_{p, i}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 4 \end{matrix}

(65)

\frac{\partial {\tilde{T}}_{p, i}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 4 \end{matrix}

(66)

where

i = Br

⁠. Recall that once the velocity field is computed from Eqs. –, the right-hand side of Eq. is known. An important result for the formulation of the Nusselt number follows by combining Eqs. , , and to show that

\nabla^{2} {\tilde{T}}_{p, Br} = - [\nabla \cdot (\tilde{w} \nabla \tilde{w}) + \tilde{w}]

(67)

Then, integrating both sides of Eq. over the domain, applying the Divergence Theorem utilizing the boundary conditions in Eqs. – and –, and using Eq. , it follows that

\int_{\tilde{a}}^{\tilde{d}} {\frac{\partial {\tilde{T}}_{p, Br}}{\partial \tilde{y}} |}_{\begin{matrix} \tilde{y} = 0 \end{matrix}} d \tilde{x} = \frac{\tilde{d}}{2 f Re}

(68)

It follows, from Eqs. and , that

{\tilde{T}}_{p, \tilde{\dot{q}}}

satisfies

\nabla^{2} {\tilde{T}}_{p, \tilde{\dot{q}}} = - 1

(69)

with boundary conditions given by Eqs. – where

i = \tilde{\dot{q}}

⁠. Comparing Eq. with Eq. , it follows that the problem for

{\tilde{T}}_{p, \tilde{\dot{q}}}

is identical to the hydrodynamic problem. However, this is not the case for the second configuration considered in Appendix A, where one plate is smooth. Integrating both sides of Eq. over the domain and applying the Divergence Theorem utilizing the boundary conditions in Eqs. – yields

\int_{\tilde{a}}^{\tilde{d}} {\frac{\partial {\tilde{T}}_{p, \tilde{\dot{q}}}}{\partial \tilde{y}} |}_{\begin{matrix} \tilde{y} = 0 \end{matrix}} d \tilde{x} = \frac{\tilde{d}}{4}

(70)

We note that ${\tilde{T}}_{p, Br}$ and ${\tilde{T}}_{p, \tilde{\dot{q}}}$ are only functions of the transverse coordinates, the aspect ratio, and the solid fraction of the domain. They are determined numerically (see Appendix C), and for the rest of the analysis, they are assumed to be known.

Local Nusselt Number.

The local Nusselt number is defined as

{Nu}_{l \pm} = \frac{h_{l \pm} D_{h}}{k}

(71)

where

h_{l \pm}

is the local heat transfer coefficient for

\tilde{z} > 0

and

\tilde{z} \leq 0

⁠, respectively. An energy balance at a point along the ridges yields

- k {\frac{\partial T_{-}}{\partial y} |}_{y = 0} = h_{l -} (T_{r -} - T_{b -}) for z \leq 0

(72)

- k {\frac{\partial T_{+}}{\partial y} |}_{y = 0} = h_{l +} (T_{r +} - T_{b +}) for z > 0

(73)

where

T_{b \pm}

is the bulk temperature of the liquid defined as

T_{b \pm} = \frac{2}{d H \bar{w}} \int_{0}^{H / 2} \int_{0}^{d} w T_{\pm} d x d y

(74)

Substituting Eqs. – into Eq. yields

{Nu}_{l -} = \frac{1}{{\tilde{T}}_{b -}} {\frac{\partial {\tilde{T}}_{-}}{\partial \tilde{y}} |}_{\tilde{y} = 0} for \tilde{z} \leq 0

(75)

{Nu}_{l +} = \frac{1}{({\tilde{T}}_{b +} - 1)} {\frac{\partial {\tilde{T}}_{+}}{\partial \tilde{y}} |}_{\tilde{y} = 0} for \tilde{z} > 0

(76)

where

\begin{matrix} {\tilde{T}}_{b -} = \frac{2 f Re}{\tilde{d}} \sum_{i = - \infty}^{- 1} c_{i} exp (- λ_{i} \tilde{z}) \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} ψ_{i} d \tilde{x} d \tilde{y} + {\tilde{T}}_{p, b} \end{matrix}

(77)

\begin{matrix} {\tilde{T}}_{b +} = 1 - \frac{2 f Re}{\tilde{d}} \sum_{i = 1}^{+ \infty} c_{i} exp (- λ_{i} \tilde{z}) \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} ψ_{i} d \tilde{x} d \tilde{y} + {\tilde{T}}_{p, b} \end{matrix}

(78)

are the dimensionless bulk temperatures of the liquid for

\tilde{z} \leq 0

and

\tilde{z} > 0

⁠, respectively, and

\begin{matrix} {\tilde{T}}_{p, b} = \frac{2 f Re}{\tilde{d}} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} [\frac{Br {(f Re)}^{2}}{4} {\tilde{T}}_{p, Br} + \tilde{\dot{q}} {\tilde{T}}_{p, \tilde{\dot{q}}}] d \tilde{x} d \tilde{y} \end{matrix}

(79)

is the contribution to these quantities from the particular solution.

Thus, from Eqs. , , , and –, it follows that the local Nusselt number is given by

{Nu}_{l \pm} = \frac{\tilde{d} (F_{l, 1 \pm} \mp F_{l, 2})}{2 f Re (F_{3 \pm} \mp F_{4})}

(80)

where²

F_{l, 1 \pm} = \sum_{i = \pm 1}^{\pm \infty} c_{i} exp (- λ_{i} \tilde{z}) {\frac{\partial ψ_{i}}{\partial \tilde{y}} |}_{\tilde{y} = 0}

(81)

F_{l, 2} = \frac{Br {(f Re)}^{2}}{4} {\frac{\partial {\tilde{T}}_{p, Br}}{\partial \tilde{y}} |}_{\tilde{y} = 0} + \tilde{\dot{q}} {\frac{\partial {\tilde{T}}_{p, \tilde{\dot{q}}}}{\partial \tilde{y}} |}_{\tilde{y} = 0}

(82)

F_{3 \pm} = \sum_{i = \pm 1}^{\pm \infty} c_{i} exp (- λ_{i} \tilde{z}) \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} ψ_{i} d \tilde{x} d \tilde{y}

(83)

F_{4} = \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} [\frac{Br {(f Re)}^{2}}{4} {\tilde{T}}_{p, Br} + \tilde{\dot{q}} {\tilde{T}}_{p, \tilde{\dot{q}}}] d \tilde{x} d \tilde{y}

(84)

We note that $F_{l, 1 \pm}$ and $F_{l, 2}$ are functions of $\tilde{x}$ ⁠, but that $F_{3 \pm}$ and F₄ are not; therefore, only the former have subscript l.

The Nusselt number averaged over the composite interface is

{Nu}_{\pm} = \frac{1}{d} \int_{0}^{d} {Nu}_{l \pm} d x

(85)

Combining Eqs. – and utilizing Eqs. , , and yields

{Nu}_{\pm} = \frac{F_{1 \pm} \mp F_{2}}{4 (F_{3 \pm} \mp F_{4})}

(86)

where

\begin{matrix} F_{1 \pm} = \sum_{i = \pm 1}^{\pm \infty} λ_{i} c_{i} exp (- λ_{i} \tilde{z}) \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} (\tilde{w} + \frac{2 λ_{i}}{f {RePe}^{2}}) ψ_{i} d \tilde{x} d \tilde{y} \end{matrix}

(87)

F_{2} = \frac{\tilde{d}}{4} (Br + \frac{2 \tilde{\dot{q}}}{f Re})

(88)

Fully Developed Nusselt Number.

In this section, our attention shifts to the asymptotic values that the Nusselt number attains in the streamwise direction as a function of the Péclet and Brinkman numbers and the dimensionless volumetric heat generation rate. Two regions can be identified where the Nusselt number does not depend on

\tilde{z}

⁠. First, where aside from geometrical effects, those of

Pe

are dominant and, second, when those of

Br

and

\tilde{\dot{q}}

are dominant. First, notice that

F_{l, 1 \pm}, F_{1 \pm}

and

F_{3 \pm}

decay exponentially with increasing

| \tilde{z} |

⁠. Comparing the two leading terms of

F_{l \pm}

⁠, it follows that when

| \tilde{z} | ≫ | {\tilde{z}}_{Pe \pm} |

⁠, where

{\tilde{z}}_{Pe \pm} = \frac{\ln | λ_{\pm 2} | - \ln | λ_{\pm 1} |}{(λ_{\pm 2} - λ_{\pm 1})}

(89)

F_{l, 1 \pm}, F_{1 \pm}

⁠, and

F_{3 \pm}

can be approximated with their leading term. Next, comparing the leading terms of

F_{1 \pm}

and

F_{3 \pm}

with F₂ and F₄, respectively, it follows that when

| \tilde{z} | ≪ \min (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)

where

{\tilde{z}}_{Br \pm} = \frac{1}{λ_{\pm 1}} \ln (\frac{1}{Br})

(90)

{\tilde{z}}_{\tilde{\dot{q}} \pm} = \frac{1}{λ_{\pm 1}} \ln (\frac{1}{\tilde{\dot{q}}})

(91)

respectively,

F_{1 \pm} ≫ F_{2}

and

F_{3 \pm} ≫ F_{4}

⁠.

Regions Where Pe Effects are Dominant.

From Eqs. – and –, it follows that when

| {\tilde{z}}_{Pe \pm} | ≪ | \tilde{z} | ≪ \min (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)

⁠, the fully developed local Nusselt number

({Nu}_{l, fd, Pe \pm})

and the fully developed Nusselt number averaged over the composite interface

({Nu}_{fd, Pe \pm})

are given by

{Nu}_{l, fd, Pe \pm} = \frac{\tilde{d} {\frac{\partial ψ_{\pm 1}}{\partial \tilde{y}} |}_{\tilde{y} = 0}}{2 f Re \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} ψ_{\pm 1} d \tilde{x} d \tilde{y}}

(92)

and

{Nu}_{fd, Pe \pm} = \frac{λ_{\pm 1}}{4} (1 + \frac{2 λ_{\pm 1}}{f {RePe}^{2}} \frac{\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} ψ_{\pm 1} d \tilde{x} d \tilde{y}}{\int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} ψ_{\pm 1} d \tilde{x} d \tilde{y}})

(93)

respectively.

Regions Where Br and q˙̃ Effects are Dominant.

When

| \tilde{z} | ≫ \max (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)

such that the effects of

Br

and/or

\tilde{\dot{q}}

are dominant, the corresponding fully developed local Nusselt number

({Nu}_{l, fd, Br, \tilde{\dot{q}} \pm})

and fully developed Nusselt number averaged over the composite interface

({Nu}_{fd, Br, \tilde{\dot{q}} \pm})

are found to be

{Nu}_{l, fd, Br, \tilde{\dot{q}} \pm} = \frac{\tilde{d} F_{l, 2}}{2 f Re F_{4}}

(94)

and

{Nu}_{fd, Br, \tilde{\dot{q}} \pm} = \frac{F_{2}}{4 F_{4}}

(95)

respectively.

Equations and indicate that when

| \tilde{z} | ≫ \max (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)

⁠, the fully developed Nusselt number is independent of the Péclet number. Moreover, when

\tilde{\dot{q}} ≫ Br

⁠, Eq. becomes

{Nu}_{fd, \tilde{\dot{q}} \pm} = \frac{\tilde{d}}{8 f Re \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} {\tilde{T}}_{p, \tilde{\dot{q}}} d \tilde{x} d \tilde{y}}

(96)

and if

\tilde{\dot{q}} = 0

⁠, it becomes

{Nu}_{fd, Br \pm} = \frac{\tilde{d}}{4 {(f Re)}^{2} \int_{0}^{1 / 4} \int_{0}^{\tilde{d}} \tilde{w} {\tilde{T}}_{p, Br} d \tilde{x} d \tilde{y}}

(97)

Equations (96) and (97) state that when volumetric heat generation is either much larger than viscous dissipation or absent, and $| \tilde{z} | ≫ | {\tilde{z}}_{Br \pm} |$ ⁠, the fully developed Nusselt number is only function of the geometry of the domain.

Results

This section contains three subsections. The first two consider separately the effects of axial conduction, and of viscous dissipation and volumetric heat generation, respectively, on the fully developed (local and averaged over the composite interface) Nusselt number. The third one considers the combined effects of axial conduction and viscous dissipation on the developing Nusselt number averaged over the composite interface. The results are for the first configuration of the ridges and those for the second configuration are presented in Appendix A. When a variable $(Pe, Br, \tilde{\dot{q}})$ appears as a subscript of $Nu$ ⁠, it signifies that the corresponding physical effects are dominant in that scenario or at that streamwise location. Two subscript variables signify that both are equally important. Note, however, that a subscript variable may not appear in the corresponding $Nu$ expression, see for example Eqs. (96) and (97).

Effects of Axial Conduction on Fully Developed Nusselt Number

Effects of ϕ and H/d for Pe=1.

Figures 4 and 5 plot the fully developed Nusselt number averaged over the composite interface versus the solid fraction $ϕ$ for aspect ratios of $H / d = 1, 2, 4, 6, 10$ ⁠, and 100, and $Pe = 1$ ⁠. They apply when $\min ({\tilde{z}}_{Br -}, {\tilde{z}}_{\tilde{\dot{q}} -}) ≪ \tilde{z} ≪ {\tilde{z}}_{Pe -}$ and ${\tilde{z}}_{Pe +} ≪ \tilde{z} ≪ \min ({\tilde{z}}_{Br +}, {\tilde{z}}_{\tilde{\dot{q}} +})$ ⁠, i.e., they provide ${Nu}_{fd, Pe -}$ and ${Nu}_{fd, Pe +}$ ⁠, respectively. Recall that in this part (when it exists) of the fully developed region, the effects of $Br$ and $\tilde{\dot{q}}$ on the fully developed Nusselt number are negligible. The dashed lines correspond to smooth plates with Nusselt numbers ${Nu}_{fd, Pe -, s} = 8.26$ and ${Nu}_{fd, Pe +, s} = 8.01$ [28], respectively. This difference between ${Nu}_{fd, Pe -}$ and ${Nu}_{fd, Pe +}$ was also observed by Agrawal [26] for the case of smooth parallel plates. Physically, this is expected since advection prevents symmetry arguments to be used pertaining to the upstream and downstream portions of the domain. Moreover, the difference between the computed ${Nu}_{fd, Pe +, s} = 8.01$ and the corresponding value of 7.54 when $Pe \to \infty$ is a manifestation of the effects of axial conduction which provides an additional path to heat transfer as discussed in Sec. 3.1.2.

Fig. 4

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${Nu}_{fd, Pe -}$ versus $ϕ$ for $Pe = 1$ and selected H/d when both plates are textured with isothermal ridges

Fig. 5

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${Nu}_{fd, Pe +}$ versus $ϕ$ for $Pe = 1$ and selected H/d when both plates are textured with isothermal ridges

The results obey the same trends with respect to H/d and $ϕ$ as observed in Ref. [22]. In the limit as $ϕ \to 1$ ⁠, ${Nu}_{fd, Pe \pm} \to {Nu}_{fd, Pe \pm, s}$ ⁠, irrespective of the aspect ratio, as they should. Additionally, as $ϕ \to 0$ ⁠, both ${Nu}_{fd, Pe -}$ and ${Nu}_{fd, Pe +}$ tend to zero because the available area for heat transfer vanishes. Furthermore, and excluding the aforementioned limits, for fixed $ϕ$ as $H / d \to 0$ and $H / d \to \infty$ ⁠, both ${Nu}_{fd, Pe -}$ and ${Nu}_{fd, Pe +}$ tend to zero and to their corresponding counterparts for smooth plates, respectively. This is because as $H / d \to 0$ and $H / d \to \infty$ ⁠, the difference between the temperature of the ridge and the mean temperature of the composite interface becomes significant and negligible, respectively, compared to the difference between the temperature of the ridge and the bulk temperature of the liquid.

Figure 6 plots the fully developed local Nusselt number $({Nu}_{l, fd, Pe +})$ versus the normalized coordinate along the ridge $(\tilde{x} - \tilde{a}) / (\tilde{d} - \tilde{a})$ for $Pe = 1, H / d = 10$ ⁠, and $ϕ = 0.01, 0.1$ and 0.99. The maximum and minimum values of ${Nu}_{l, fd, Pe +}$ in each case are observed at the triple contact line $(\tilde{x} = \tilde{a})$ and at the center of the ridge $(\tilde{x} = \tilde{d})$ ⁠, respectively. Moreover, ${Nu}_{l, fd, Pe +}$ increases with decreasing $ϕ$ indicating a local enhancement of heat transfer due to the higher velocities of the liquid close to the ridge as $ϕ \to 0$ ⁠. Both trends are consistent with the previous studies [22,34]. In summary, the overall effect of the decrease in the available heat transfer area and the local enhancement of heat transfer for $ϕ < 1$ is an increase in the convective portion of the total thermal resistance that is completely captured in Figs. 4 and 5.

Fig. 6

Nul,fd,Pe+ versus the normalized coordinate along the ridge (x̃−ã)/(d̃−ã) for Pe=1, H/d=10 and selected values of ϕ when both plates are textured with isothermal ridges

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${Nu}_{l, fd, Pe +}$ versus the normalized coordinate along the ridge $(\tilde{x} - \tilde{a}) / (\tilde{d} - \tilde{a})$ for $Pe = 1, H / d = 10$ and selected values of $ϕ$ when both plates are textured with isothermal ridges

Effects of ϕ and Pe for H/d=1 and 10.

Figures 7 and 8 plot the computed ${Nu}_{fd, Pe -}$ and ${Nu}_{fd, Pe +}$ ⁠, respectively, versus the solid fraction for $Pe = 0.01, 1$ ⁠, and 10 for $H / d = 1$ ⁠. The latter also includes the $Pe \to \infty$ limit [22] for comparison. Figures 9 and 10 apply when $H / d = 10$ ⁠. The results show that as $Pe \to 0$ ⁠, ${Nu}_{fd, Pe -, s}$ approaches ${Nu}_{fd, Pe +, s}$ and they become approximately equal to 8.12 [28]. This is expected as in this limit the primary mode of heat transfer is conduction and thus the problem becomes antisymmetric with respect to $\tilde{z} = 0$ ⁠, where $\tilde{T} = 0.5$ ⁠.

Fig. 7

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${Nu}_{fd, Pe -}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $H / d = 1$ when both plates are textured with isothermal ridges

Fig. 8

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, and H/d=1 when both plates are textured with isothermal ridges

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${Nu}_{fd, Pe +}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $Pe \to \infty$ ⁠, and $H / d = 1$ when both plates are textured with isothermal ridges

Fig. 9

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${Nu}_{fd, Pe -}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $H / d = 10$ when both plates are textured with isothermal ridges

Fig. 10

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, and H/d=10 when both plates are textured with isothermal ridges

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${Nu}_{fd, Pe +}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $Pe \to \infty$ ⁠, and $H / d = 10$ when both plates are textured with isothermal ridges

Comparing Figs. 7 and 9 with Figs. 8 and 10, respectively, shows that as the Péclet number increases, ${Nu}_{fd, Pe -}$ and ${Nu}_{fd, Pe +}$ respond differently. ${Nu}_{fd, Pe -}$ tends to infinity as $Pe$ increases because the temperature field for $\tilde{z} \leq 0$ becomes essentially isothermal and thus ${\tilde{T}}_{-} \to 0$ and ${\partial {\tilde{T}}_{-} / \partial \tilde{y} |}_{\tilde{y} = 0} \to 0$ ⁠. Note that despite the fact that ${Nu}_{fd, Pe -}$ tends to infinity in this case, there is no heat transfer from the ridge to the liquid given that ${\partial {\tilde{T}}_{-} / \partial \tilde{y} |}_{\tilde{y} = 0} \to 0$ ⁠. This behavior is consistent with the trends observed by Agrawal [26] for the case of smooth isothermal plates. Contrary, ${Nu}_{fd, Pe +}$ decreases as $Pe$ increases because the axial conduction enhancement to heat transfer is reduced, and in the limit $Pe \to \infty$ ⁠, ${Nu}_{fd, Pe +, s}$ tends to finite values. These trends are reversed, however, when only one plate is textured with isothermal ridges and the other one is smooth and adiabatic. The slower velocity field in this case causes ${\partial {\tilde{T}}_{-} / \partial \tilde{y} |}_{\tilde{y} = 0}$ to tend to zero faster than ${\tilde{T}}_{-}$ does [22] and therefore ${Nu}_{fd, Pe -} \to 0$ ⁠, as $Pe \to \infty$ as per the corresponding results in Appendix A. Also, the adiabatic boundary condition along the smooth plate leads to convection dominated heat transfer and thus ${Nu}_{fd, Pe +}$ increases as $Pe$ increases with ${Nu}_{fd, Pe +}$ tending to finite values as $Pe \to \infty$ ⁠. For both ridge configurations, the change of ${Nu}_{fd, Pe -}$ and ${Nu}_{fd, Pe +}$ for an increase of the Péclet number is small for $Pe < 1$ as in this region the heat transfer is predominantly diffusive, but the change becomes large when $Pe > 1$ and advection becomes important.

Finally, comparing Figs. 7 and 8 with Figs. 9 and 10, respectively, it follows that the effects of Péclet number become important as the solid fraction increases, and for ${Nu}_{fd, Pe +}$ ⁠, the range of values of $ϕ$ where change is observed increases with H/d. Moreover, the effects are more pronounced on ${Nu}_{fd, Pe -}$ which, as explained earlier, has a stronger dependence on $Pe$ than ${Nu}_{fd, Pe +}$ ⁠.

Effects of Viscous Dissipation and Volumetric Heat Generation on Fully Developed Nusselt Number.

The computed fully developed Nusselt numbers averaged over the composite interface when $| \tilde{z} | ≫ \max (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)$ ⁠, ${Nu}_{fd, Br \pm}$ and ${Nu}_{fd, \tilde{\dot{q}} \pm}$ ⁠, are presented in Figs. 11 and 12, respectively. The results present the same trends with respect to $ϕ$ and H/d as those described for ${Nu}_{fd, Pe \pm}$ ⁠, i.e., irrespective of H/d as $ϕ \to 0$ ⁠, both ${Nu}_{fd, Br \pm}$ and ${Nu}_{fd, \tilde{\dot{q}} \pm}$ tend to zero and as $ϕ \to 1, {Nu}_{fd, Br \pm} \to {Nu}_{fd, Br \pm, s} = 17.5$ [29], and ${Nu}_{fd, \tilde{\dot{q}} \pm} \to {Nu}_{fd, \tilde{\dot{q}} \pm, s} = 10$ [31], respectively.

Fig. 11

Nufd,Br± versus ϕ for selected H/d when both plates are textured with isothermal ridges

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${Nu}_{fd, Br \pm}$ versus $ϕ$ for selected H/d when both plates are textured with isothermal ridges

Fig. 12

Nufd,q˙̃± versus ϕ for selected H/d when both plates are textured with isothermal ridges

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${Nu}_{fd, \tilde{\dot{q}} \pm}$ versus $ϕ$ for selected H/d when both plates are textured with isothermal ridges

Combined Effects of Axial Conduction and Viscous Dissipation on Developing Nusselt Number.

Here, we present results for the combined effects of the Péclet and Brinkman numbers on the Nusselt number averaged over the composite interface for $\tilde{z} > 0 ({Nu}_{+})$ ⁠. Figure 13 presents ${Nu}_{+}$ versus the dimensionless streamwise coordinate for $ϕ = 0.01, H / d = 10$ ⁠, $Pe = 1, \tilde{\dot{q}} = 0$ and for three different values of the Brinkman number, namely $Br = 2.71 \times 10^{- 5}, 2.71 \times 10^{- 8}$ and as $Br \to 0$ ⁠. The second value of $Br$ is relevant to flow of liquid metals through textured microchannels [3]. In this figure, we can identify the two asymptotic values of ${Nu}_{+}$ ⁠. First, as $\tilde{z}$ increases and becomes larger than ${\tilde{z}}_{Pe +}$ ⁠, ${Nu}_{+}$ approaches ${Nu}_{fd, Pe +} = 4.33$ ⁠. Then, as $\tilde{z}$ continues to increase, the effects of the step change of the ridge temperature decay significantly and become of the same order as the viscous dissipation effects. Thus, in the region where $\tilde{z} \approx {\tilde{z}}_{Br +}, {Nu}_{+}$ starts to increase until $\tilde{z} ≫ {\tilde{z}}_{Br +}$ where ${Nu}_{+} \to {Nu}_{fd, Br \pm} = 8.92$ ⁠. Also, as $Br$ increases, the location of the transition moves further upstream but its limiting value remains 8.92. The same trends were reported in Ref. [30] for the case of smooth plates. Figure 14 presents ${Nu}_{+}$ versus $\tilde{z}$ for the same domain geometry and $Br$ as in Fig. 13, but for $Pe = 10$ ⁠. Comparing Figs. 13 and 14, we see that as $Pe$ increases, the transitions of ${Nu}_{+}$ to ${Nu}_{fd, Pe +}$ and then to ${Nu}_{fd, Br \pm}$ occur further upstream given that the flow becomes thermally developed faster with increasing $Pe$ ⁠.

Fig. 13

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=1, and q˙̃=0 when both plates are textured with isothermal ridges; z̃Pe+=0.13, z̃Br+,1=2.81, and z̃Br+,2=4.65

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${Nu}_{+}$ versus $\tilde{z}$ for $ϕ = 0.01, H / d = 10, Pe = 1$ ⁠, and $\tilde{\dot{q}} = 0$ when both plates are textured with isothermal ridges; ${\tilde{z}}_{Pe +} = 0.13, {\tilde{z}}_{Br +, 1} = 2.81$ ⁠, and ${\tilde{z}}_{Br +, 2} = 4.65$

Fig. 14

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=10, and q˙̃=0 when both plates are textured with isothermal ridges; z̃Pe+=0.02, z̃Br+,1=0.71, and z̃Br+,2=1.17

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${Nu}_{+}$ versus $\tilde{z}$ for $ϕ = 0.01, H / d = 10, Pe = 10$ ⁠, and $\tilde{\dot{q}} = 0$ when both plates are textured with isothermal ridges; ${\tilde{z}}_{Pe +} = 0.02, {\tilde{z}}_{Br +, 1} = 0.71$ ⁠, and ${\tilde{z}}_{Br +, 2} = 1.17$

Conclusions

We considered the extended Graetz–Nusselt problem, i.e., hydrodynamically developed and thermally developing flow with finite axial conduction, for the case of textured plates (or plate) with isothermal parallel ridges. We developed semi-analytical expressions for the Nusselt number in an infinite domain, before and after a jump in ridge temperature. Effects of viscous dissipation and volumetric heat generation were included. Two different configurations for the ridges were analyzed: (1) both plates textured and (2) one plate textured and the other one smooth and adiabatic. The menisci between the ridges were considered to be flat and adiabatic. The solid–liquid interfaces and the menisci were subjected to no-slip and no-shear boundary conditions, respectively. Using separation of variables, we expressed the homogeneous part of the solution 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 nonlinear eigenvalue problem from which the eigenvalues and eigenfunctions follow numerically. The particular solution accounting for viscous dissipation and volumetric heat generation is also determined numerically.

The derived expressions for the local Nusselt number and the Nusselt number averaged over the composite interface indicate that the Nusselt number is a function of the transverse (along the ridge) and streamwise coordinates, the aspect ratio of the domain, the solid fraction of the ridges, the Péclet and Brinkman numbers, and the dimensionless volumetric heat generation rate. Expressions were also derived for the fully developed local Nusselt number and for the fully developed Nusselt number averaged over the composite interface. Two asymptotic limits were identified for the fully developed Nusselt number and expressions were derived to estimate the streamwise locations where they occur. The first limit is relevant to the effects of axial conduction, and the corresponding fully developed Nusselt number is a function of the geometry and the Péclet number. The second limit is relevant to viscous dissipation and volumetric heat generation effects, and the corresponding fully developed Nusselt number is a function of the geometry, the Brinkman number, and the dimensionless volumetric heat generation rate. If volumetric heat generation is absent, the aforementioned Nusselt number is a function of the geometry only.

The results indicate that the Nusselt number averaged over the composite interface decreases as the aspect ratio and/or the solid fraction decreases. Moreover, the fully developed Nusselt number averaged over the composite interface in the region after the temperature change tends to a finite value as the Péclet number tends to infinity for both geometries studied. On the contrary, in the region before the temperature change, the fully developed Nusselt number averaged over the composite interface tends to infinity when both plates are textured with isothermal ridges, and to zero when one plate is smooth and adiabatic, as the Péclet number tends to infinity.

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 computational fluid dynamics solver. More importantly, the analysis provides semi-analytical expressions to evaluate the local Nusselt number and the Nusselt number averaged over the composite interface at any location, which are prohibitively expensive to compute using a general computational fluid dynamics code.

Acknowledgment

The work of TK was supported by an EPSRC-UK doctoral scholarship. The computations in this paper were run on the Tufts High-performance Computing Research Cluster at Tufts University.

Funding Data

National Science Foundation (Grant No. 1402783).
Engineering and Physical Sciences Research Council (UK) (Grant Nos. EP/K041134 and EP/L020564).

Nomenclature

Roman Symbols

$a$ =: half meniscus width, m
$\tilde{a}$ =: dimensionless half meniscus width; $a / D_{h}$
$B$ =: orthogonality matrix
$Br$ =: Brinkman number; $(μ {\bar{w}}^{2}) / [k (T_{r +} - T_{r -})]$
$c_{i}$ =: expansion coefficients
$c_{p}$ =: specific heat at constant pressure, J/(kg K)
$d$ =: half ridge pitch, m
$\tilde{d}$ =: dimensionless half ridge pitch; $d / D_{h}$
$d p / d z$ =: prescribed pressure gradient, Pa/m
$D$ =: domain
$D_{h}$ =: hydraulic diameter; $2 H$
$f$ =: friction factor; $2 D_{h} (- d p / d z) / (ρ {\bar{w}}^{2})$
$f Re$ =: Poiseuille number
$h$ =: heat transfer coefficient, W/(m² K)
$h_{l \pm}$ =: local heat transfer coefficient, W/(m² K)
$H$ =: distance between parallel plates, m
$k$ =: thermal conductivity, W/(mK)
$Nu$ =: Nusselt number; $h D_{h} / k$
${Nu}_{fd, Br \pm}$ =: fully developed Nusselt number averaged over the composite interface when the effects of $Br$ are dominant, i.e., when $| \tilde{z} | ≫ | {\tilde{z}}_{Br \pm} |$ and $Br ≫ \tilde{\dot{q}}$
${Nu}_{fd, Br, \tilde{\dot{q}} \pm}$ =: fully developed Nusselt number averaged over the composite interface when the effects of $Br$ and $\tilde{\dot{q}}$ are dominant, i.e., when $| \tilde{z} | ≫ \max (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)$
${Nu}_{fd, Pe \pm}$ =: fully developed Nusselt number averaged over the composite interface when the effects of $Br$ and $\tilde{\dot{q}}$ are negligible, $i . e .$ ⁠, when $| {\tilde{z}}_{Pe \pm} | ≪ | \tilde{z} | ≪ \min (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)$
${Nu}_{fd, \tilde{\dot{q}} \pm}$ =: fully developed Nusselt number averaged over the composite interface when the effects of $\tilde{\dot{q}}$ are dominant, i.e., when $| \tilde{z} | ≫ | {\tilde{z}}_{\tilde{\dot{q}} \pm} |$ and $\tilde{\dot{q}} ≫ Br$
${Nu}_{l \pm}$ =: local Nusselt number; $h_{l \pm} D_{h} / k$
${Nu}_{l, fd, Br, \tilde{\dot{q}} \pm}$ =: fully developed local Nusselt number when the effects of $Br$ and $\tilde{\dot{q}}$ are dominant, i.e., when $| \tilde{z} | ≫ \max (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)$
${Nu}_{l, fd, Pe \pm}$ =: fully developed local Nusselt number when the effects of $Br$ and $\tilde{\dot{q}}$ are negligible, i.e., when $| {\tilde{z}}_{Pe \pm} | ≪ | \tilde{z} | ≪ \min (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)$
${Nu}_{\pm}$ =: Nusselt number averaged over the composite interface; $\int_{0}^{d} {Nu}_{l \pm} d x / d$
$Pe$ =: Péclet number; $RePr$
$Pr$ =: Prandtl number; $(c_{p} μ) / k$
$\dot{q}$ =: volumetric heat generation rate, W/m³
$\tilde{\dot{q}}$ =: dimensionless heat generation rate; $\dot{q} D_{h}^{2} / [k (T_{r +} - T_{r -})]$
$Re$ =: Reynolds number; $ρ \bar{w} D_{h} / μ$
$T$ =: temperature, ° C
$\tilde{T}$ =: dimensionless temperature; $(T - T_{r -}) / (T_{r +} - T_{r -})$
$T_{b}$ =: bulk temperature; $2 {(d H \bar{w})}^{- 1} \int_{0}^{H / 2} \int_{0}^{d} w T d x d y$
${\tilde{T}}_{b}$ =: dimensionless bulk temperature
${\tilde{T}}_{h}$ =: homogeneous solution
${\tilde{T}}_{p}$ =: particular solution
${\tilde{T}}_{p, b}$ =: dimensionless bulk temperature of particular solution
$T_{r}$ =: ridge temperature
$w$ =: streamwise velocity, m/s
$\bar{w}$ =: mean streamwise velocity, m/s
$\tilde{w}$ =: dimensionless streamwise velocity; $(μ w) {(- d p / d z)}^{- 1} / D_{h}^{2}$
$\bar{\tilde{w}}$ =: dimensionless mean streamwise velocity; $(μ \bar{w}) {(- d p / d z)}^{- 1} / D_{h}^{2}$
$x$ =: lateral coordinate, m
$\tilde{x}$ =: dimensionless lateral coordinate; $x / D_{h}$
$y$ =: vertical coordinate, m
$\tilde{y}$ =: dimensionless vertical coordinate; $y / D_{h}$
$z$ =: streamwise coordinate, m
$\tilde{z}$ =: dimensionless streamwise coordinate; $z / (Pe D_{h})$
${\tilde{z}}_{Pe \pm}$ =: estimate of the dimensionless streamwise location where $Nu$ becomes fully developed and the effects of $Br$ and $\tilde{\dot{q}}$ are negligible compared to those of $Pe$ ⁠; $(\ln | λ_{\pm 2} | - \ln | λ_{\pm 1} |) / (λ_{\pm 2} - λ_{\pm 1})$
${\tilde{z}}_{Br \pm}$ =: estimate of the dimensionless streamwise location where the effects of $Br$ and $Pe$ become of the same order; $\ln ({Br}^{- 1}) / λ_{\pm 1}$
${\tilde{z}}_{\tilde{\dot{q}} \pm}$ =: estimate of the dimensionless streamwise location where the effects of $\tilde{\dot{q}}$ and $Pe$ become of the same order; $\ln ({\tilde{\dot{q}}}^{- 1}) / λ_{\pm 1}$

Greek Symbols

$λ_{i}$ =: ith eigenvalue
$μ$ =: dynamic viscosity, Pa⋅s
$ρ$ =: density, kg/m³
$ϕ$ =: solid fraction; $(d - a) / d$
$ψ_{i}$ =: ith eigenfunction

Subscripts

est =: estimate
$fd$ =: fully developed conditions
$i$ =: index
$i$ =: index
$j$ =: index
$m$ =: index
s =: smooth plates
$-$ =: referring to streamwise location $z \leq 0$
$+$ =: referring to streamwise location $z > 0$

1

In general, the cavities beneath the menisci are filled with inert gas and vapor on account of the vapor pressure of the liquid phase, and for brevity, we refer to this mixture as “gas.”

2

The notation convention is that the summations in $F_{-}$ and $F_{+}$ are from $- \infty$ to –1 and from 1 to $\infty$ ⁠, respectively.

3

Note $λ_{i, m, j - 1, est} \neq λ_{i, m, j - 1}$ ⁠.

Appendix A

This section provides the necessary information for the extension of the present analysis to the configuration when one plate is textured with isothermal parallel ridges and the other one is smooth and adiabatic, as per Fig. 3. The domain in the present case is symmetric with respect to the yz plane through x = 0, and therefore, we further restrict to $0 \leq x \leq d$ and $0 \leq y \leq H$ ⁠.

The relevant boundary conditions for the hydrodynamic problem are composed of Eqs. , and

\tilde{w} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 2 \end{matrix}

(A1)

\frac{\partial \tilde{w}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 2 \end{matrix}

(A2)

This hydrodynamic problem has been solved analytically by Philip [8]. However, here, as in the original case, we solve it numerically (see Appendix B) to facilitate the solution of the thermal energy equation.

The relevant boundary conditions for the thermal problem are composed of Eqs. – and

\frac{\partial \tilde{T}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 2 \end{matrix}

(A3)

\frac{\partial \tilde{T}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 2 \end{matrix}

(A4)

\tilde{T} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, 0 < \tilde{y} < 1 / 2, \tilde{z} \to - \infty \end{matrix}

(A5)

\tilde{T} = 1 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, 0 < \tilde{y} < 1 / 2, \tilde{z} \to + \infty \end{matrix}

(A6)

and the two continuity conditions are identical with those provided by Eqs. (39) and (40) but they apply for $0 < \tilde{x} < \tilde{d}$ and $0 < \tilde{y} < 1 / 2$ ⁠.

The boundary conditions for the eigenvalue problem are composed of Eqs. , and

\frac{\partial ψ_{\pm}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 2 \end{matrix}

(A7)

\frac{\partial ψ_{\pm}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 2 \end{matrix}

(A8)

and for the particular solution are composed of Eqs. , and

\frac{\partial {\tilde{T}}_{p, i}}{\partial \tilde{y}} = 0 \begin{matrix} for 0 < \tilde{x} < \tilde{d}, \tilde{y} = 1 / 2 \end{matrix}

(A9)

\frac{\partial {\tilde{T}}_{p, i}}{\partial \tilde{x}} = 0 \begin{matrix} for \tilde{x} = 0, \tilde{x} = \tilde{d}, 0 < \tilde{y} < 1 / 2 \end{matrix}

(A10)

where $i = Br$ or $\tilde{\dot{q}}$ ⁠.

Then, following the same procedure as in Secs. 2.2, 2.3, and 2.4, it can be shown that the corresponding expression for the expansion coefficients is

c_{i} = \frac{\int_{0}^{1 / 2} \int_{0}^{\tilde{d}} [\tilde{w} + 2 λ_{i} / (f {RePe}^{2})] ψ_{i} d \tilde{x} d \tilde{y}}{\int_{0}^{1 / 2} \int_{0}^{\tilde{d}} [\tilde{w} + 4 λ_{i} / (f {RePe}^{2})] ψ_{i}^{2} d \tilde{x} d \tilde{y}}

(A11)

The local Nusselt number is given by the expression

{Nu}_{l \pm} = \frac{\tilde{d} (F_{l, 1 \pm} \mp F_{l, 2})}{f Re (F_{3 \pm} \mp F_{4})}

(A12)

where

F_{l, 1 \pm} = \sum_{i = \pm 1}^{\pm \infty} c_{i} exp (- λ_{i} \tilde{z}) {\frac{\partial ψ_{i}}{\partial \tilde{y}} |}_{\tilde{y} = 0}

(A13)

F_{l, 2} = \frac{Br {(f Re)}^{2}}{4} {\frac{\partial {\tilde{T}}_{p, Br}}{\partial \tilde{y}} |}_{\tilde{y} = 0} + \tilde{\dot{q}} {\frac{\partial {\tilde{T}}_{p, \tilde{\dot{q}}}}{\partial \tilde{y}} |}_{\tilde{y} = 0}

(A14)

F_{3 \pm} = \sum_{i = \pm 1}^{\pm \infty} c_{i} exp (- λ_{i} \tilde{z}) \int_{0}^{1 / 2} \int_{0}^{\tilde{d}} \tilde{w} ψ_{i} d \tilde{x} d \tilde{y}

(A15)

F_{4} = \int_{0}^{1 / 2} \int_{0}^{\tilde{d}} [\frac{Br {(f Re)}^{2}}{4} {\tilde{T}}_{p, Br} + \tilde{\dot{q}} {\tilde{T}}_{p, \tilde{\dot{q}}}] \tilde{w} d \tilde{x} d \tilde{y}

(A16)

The Nusselt number averaged over the composite interface is

{Nu}_{\pm} = \frac{F_{1 \pm} \mp F_{2}}{2 (F_{3 \pm} \mp F_{4})}

(A17)

where

\begin{matrix} F_{1 \pm} = \sum_{i = \pm 1}^{\pm \infty} λ_{i} c_{i} exp (- λ_{i} \tilde{z}) \int_{0}^{1 / 2} \int_{0}^{\tilde{d}} [\tilde{w} + \frac{2 λ_{i}}{f {RePe}^{2}}] ψ_{i} d \tilde{x} d \tilde{y} \end{matrix}

(A18)

F_{2} = \frac{\tilde{d}}{2} (Br + \frac{2 \tilde{\dot{q}}}{f Re})

(A19)

The fully developed local Nusselt number and the fully developed Nusselt number averaged over the composite interface when

| {\tilde{z}}_{Pe \pm} | ≪ | \tilde{z} | ≪ \min (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)

are given by the expressions

{Nu}_{l, fd, Pe \pm} = \frac{\tilde{d} {\frac{\partial ψ_{\pm 1}}{\partial \tilde{y}} |}_{\tilde{y} = 0}}{f Re \int_{0}^{1 / 2} \int_{0}^{\tilde{d}} \tilde{w} ψ_{\pm 1} d \tilde{x} d \tilde{y}}

(A20)

and

{Nu}_{fd, Pe \pm} = \frac{λ_{\pm 1}}{2} (1 + \frac{2 λ_{\pm 1}}{f {RePe}^{2}} \frac{\int_{0}^{1 / 2} \int_{0}^{\tilde{d}} ψ_{\pm 1} d \tilde{x} d \tilde{y}}{\int_{0}^{1 / 2} \int_{0}^{\tilde{d}} \tilde{w} ψ_{\pm 1} d \tilde{x} d \tilde{y}})

(A21)

respectively. When

| \tilde{z} | ≫ \max (| {\tilde{z}}_{Br \pm} |, | {\tilde{z}}_{\tilde{\dot{q}} \pm} |)

⁠, the expression for the corresponding fully developed local Nusselt number and the fully developed Nusselt number averaged over the composite interface becomes

{Nu}_{l, fd, Br, \tilde{\dot{q}} \pm} = \frac{\tilde{d} F_{l, 2}}{f Re F_{4}}

(A22)

and

{Nu}_{fd, Br, \tilde{\dot{q}} \pm} = \frac{F_{2}}{2 F_{4}}

(A23)

respectively. Moreover, when

\tilde{\dot{q}} ≫ Br

or

\tilde{\dot{q}} = 0

⁠, Eq. yields

{Nu}_{fd, \tilde{\dot{q}} \pm} = \frac{\tilde{d}}{2 f Re \int_{0}^{1 / 2} \int_{0}^{\tilde{d}} \tilde{w} {\tilde{T}}_{p, \tilde{\dot{q}}} d \tilde{x} d \tilde{y}}

(A24)

and

{Nu}_{fd, Br \pm} = \frac{\tilde{d}}{{(f Re)}^{2} \int_{0}^{1 / 2} \int_{0}^{\tilde{d}} \tilde{w} {\tilde{T}}_{p, Br} d \tilde{x} d \tilde{y}}

(A25)

respectively.

Finally, we present the corresponding computed ${Nu}_{fd, Pe \pm}$ ⁠, ${Nu}_{fd, Br \pm}, {Nu}_{fd, \tilde{\dot{q}} \pm}$ ⁠, and ${Nu}_{\pm}$ in Figs. 15–24 for the same prescribed parameters as those in Figs. 4, 5, 7–14, respectively. Overall, the results exhibit the same trends as those described in Sec. 3 with the exception that ${Nu}_{fd, Pe -}$ tends to zero as $Pe \to \infty$ ⁠.

Fig. 15

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${Nu}_{fd, Pe -}$ versus $ϕ$ for $Pe = 1$ and selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 16

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${Nu}_{fd, Pe +}$ versus $ϕ$ for $Pe = 1$ selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 17

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${Nu}_{fd, Pe -}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $H / d = 1$ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 18

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, H/d=1 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

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${Nu}_{fd, Pe +}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $Pe \to \infty, H / d = 1$ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 19

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${Nu}_{fd, Pe -}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $H / d = 10$ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 20

Nufd,Pe+ versus ϕ for Pe=0.01,1,10 and Pe→∞, and H/d=10 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

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${Nu}_{fd, Pe +}$ versus $ϕ$ for $Pe = 0.01, 1, 10$ and $Pe \to \infty$ ⁠, and $H / d = 10$ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 21

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

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${Nu}_{fd, Br \pm}$ versus $ϕ$ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 22

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

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${Nu}_{fd, \tilde{\dot{q}} \pm}$ versus $ϕ$ for selected H/d when one plate is textured with isothermal ridges and the other one is smooth and adiabatic

Fig. 23

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=1, and q˙̃=0 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; z̃Pe+=0.24, z̃Br+,1=5.03, and z̃Br+,2=8.34

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${Nu}_{+}$ versus $\tilde{z}$ for $ϕ = 0.01, H / d = 10, Pe = 1$ ⁠, and $\tilde{\dot{q}} = 0$ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; ${\tilde{z}}_{Pe +} = 0.24$ ⁠, ${\tilde{z}}_{Br +, 1} = 5.03$ ⁠, and ${\tilde{z}}_{Br +, 2} = 8.34$

Fig. 24

Nu+ versus z̃ for ϕ=0.01, H/d=10, Pe=10, and q˙̃=0 when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; z̃Pe+=0.05, z̃Br+,1=1.65, and z̃Br+,2=2.74

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${Nu}_{+}$ versus $\tilde{z}$ for $ϕ = 0.01, H / d = 10, Pe = 10$ ⁠, and $\tilde{\dot{q}} = 0$ when one plate is textured with isothermal ridges and the other one is smooth and adiabatic; ${\tilde{z}}_{Pe +} = 0.05$ ⁠, ${\tilde{z}}_{Br +, 1} = 1.65$ ⁠, and ${\tilde{z}}_{Br +, 2} = 2.74$

Appendix B

The nonlinear eigenvalue problem presented in Sec. 2.2.1 was solved numerically using the Finite Element Method. The algorithm was coded in matlab and results were obtained for multiple values of the aspect ratio of the domain (H/d), solid fraction of the ridges $(ϕ)$ ⁠, and Péclet number of the flow.

An overview of the algorithm is presented here and the detailed steps follow. For each pair of H/d and

ϕ

values, the algorithm initially solves the hydrodynamic problem to compute

\tilde{w}

and the corresponding

f Re

⁠. Then, for the prescribed value of

Pe

⁠, the ith eigenvalue of interest

(λ_{i})

is computed by carrying out iterations on inner and outer loops. Each iteration of the outer loop (indexed by m) refines the spatial discretization of the domain (mesh) to ensure mesh independence of the final results. Each iteration of the inner loop (indexed by j) computes a refined estimate of λ_i for the current mesh. This is accomplished by linearizing and solving Eq. in the form

\nabla^{2} ψ_{i, m, j} = - λ_{i, m, j} (\frac{f {Re}_{m}}{2} {\tilde{w}}_{m} + \frac{λ_{i, m, j - 1, est}}{{Pe}^{2}}) ψ_{i, m, j}

(B1)

where

{\tilde{w}}_{m}

and

f {Re}_{m}

are the previously computed dimensionless velocity and Poiseuille number, respectively, from the current mesh, and

λ_{i, m, j - 1, est}

is an estimate of λ_i from the previous

(j - 1)

inner iteration.³ This approach allows the use of linear eigenvalue problem theory to solve the problem at hand at the expense that the process is iterative and only the computed eigenvalue of interest λ_i is valid, i.e., the solution process needs to be repeated if, for example,

λ_{i + 1}

is also of interest. Moreover, in order to use the same code to calculate both the positive and negative eigenvalues, Eq. is written in the form

\nabla^{2} ψ_{i, m, j} = - λ_{i, m, j}^{*} (\frac{i}{| i |} \frac{f {Re}_{m}}{2} {\tilde{w}}_{m} + \frac{λ_{i, m, j - 1, est}^{*}}{{Pe}^{2}}) ψ_{i, m, j}

(B2)

where

λ_{i, m, j}^{*} = \frac{i}{| i |} λ_{i, m, j}

(B3)

The detailed steps of the solution process are as follows. In the first iteration of the outer loop $(m = 1)$ ⁠, the domain is discretized with an initial number of finite elements. Next, Eq. (5) is solved subject to the boundary conditions given by Eqs. (6)–(9) to determine ${\tilde{w}}_{m} (\tilde{x}, \tilde{y})$ and, subsequently, $f {Re}_{m}$ required in Eq. (B2). Then, Eq. (B2) subject to the boundary conditions given by Eqs. (34)–(37) is solved iteratively within the inner loop to compute the eigenvalue of interest λ_i and the corresponding eigenfunction ψ_i for the current spatial discretization. At each iteration j of the inner loop, the code first uses the Arnoldi algorithm [35] to compute $λ_{i, m, j}^{*}$ ⁠. Then, if $λ_{i, m, j}^{*} > λ_{i, m, j - 1, est}^{*}$ ⁠, the new estimate is $λ_{i, m, j, est}^{*} = λ_{i, m, j - 1, est}^{*} + β$ ⁠, where $β > 0$ ⁠, and if $λ_{i, m, j}^{*} < λ_{i, m, j - 1, est}^{*}$ ⁠, the new estimate is $λ_{i, m, j, est}^{*} = λ_{i, m, j}^{*}$ and $β = β / 10$ ⁠. The inner loop stops when $| λ_{i, m, j}^{*} - λ_{i, m, j - 1, est}^{*} | / λ_{i, m, j - 1, est}^{*} \leq 0.01 %$ and the corresponding value of j is recorded as $j - final$ ⁠. Next, the mesh is refined by adaptively placing elements in regions of sharp gradients and the algorithm proceeds from step two. The outer loop stops when $| λ_{i, m, j - final}^{*} - λ_{i, m - 1, j - final}^{*} | / λ_{i, m - 1, j - final}^{*} \leq 0.01 %$ ⁠.

The code was validated by computing λ_i and ${Nu}_{fd, Pe \pm}$ in the limit $ϕ \to 1$ for the first ridge configuration, i.e., for smooth isothermal parallel plates, for different values of the Péclet number. The results are compared with those available in the literature in Tables 1 and 2. It is noted that due to a different nondimensionlization scheme, the results in Agrawal [26] and Deavours [27] correspond to $Pe / 4$ and to $Pe / 2$ ⁠, respectively, and the eigenvalues in both cases are multiplied by a factor of 1/16. The discrepancies are less than 0.1%, except for the case of ${Nu}_{fd, Pe -}$ for $Pe = 2$ ⁠, where it is 1.17%. This is attributed to the lack of more significant digits in the Nusselt number provided in Ref. [26], since the discrepancy in the corresponding eigenvalue is 0.02%.

Table 1

Comparison of computed λ_i for $ϕ \to 1$ against the literature

$Pe = 2$
	Code	[27]	%
λ₁	10.2236	10.2235	0.00
$\| λ_{- 1} \|$	15.4358	15.4358	0.00

$Pe = 4$
	Code	[26]	%
λ₁	16.7665	16.7664	0.00
$\| λ_{- 1} \|$	37.5717	37.5776	0.02
λ₂	67.5856	67.5943	0.01
λ₃	117.8576	117.8707	0.01
λ₄	168.1265	168.0928	0.02
λ₅	218.4054	218.4320	0.01

Table 2

Comparison of computed ${Nu}_{fd, Pe \pm}$ for $ϕ \to 1$ against the literature

$Pe = 0.01$
	Code	[28]	%
${Nu}_{fd, Pe +}$	8.1160	8.1155	0.01

$Pe = 2$
	Code	[26]	%
${Nu}_{fd, Pe +}$	7.7909	7.7960	0.07
${Nu}_{fd, Pe -}$	8.9030	8.8	1.17

Appendix C

The two-dimensional particular problems for ${\tilde{T}}_{p, Br}$ and ${\tilde{T}}_{p, \tilde{\dot{q}}}$ presented in Sec. 2.2.2 were numerically solved in an iterative manner using the Finite Element Method. The corresponding algorithms were coded in matlab and results were obtained for multiple values of the aspect ratio of the domain $(H / d)$ and the solid fraction $(ϕ)$ ⁠. Recall that ${\tilde{T}}_{p, \tilde{\dot{q}}}$ is identical to $\tilde{w}$ for the first ridge configuration and thus the corresponding problem does not need to be solved.

The steps of the algorithm for ${\tilde{T}}_{p, Br}$ are as follows. First, the domain is discretized with an initial number of finite elements. Next, Eq. (5) is solved subject to the boundary conditions in Eqs. (6)–(9) to compute $\tilde{w} (\tilde{x}, \tilde{y})$ and then $\nabla \tilde{w} (\tilde{x}, \tilde{y})$ that is required in Eq. (62). Then, Eq. (62) is solved subject to the boundary conditions in Eqs (63)–(66) to compute ${\tilde{T}}_{p, Br} (\tilde{x}, \tilde{y})$ and consequently ${Nu}_{fd, Br \pm}$ ⁠. Next, the mesh is refined by adaptively increasing the element density in regions of sharp gradients of ${\tilde{T}}_{p, Br}$ ⁠, and the algorithm proceeds from step two. The process is repeated until the change in the computed value of ${Nu}_{fd, Br \pm}$ is less than 0.01%. The code was validated by computing ${Nu}_{fd, Br \pm}$ at the limit $ϕ \to 1$ for the first ridge configuration, i.e., for smooth isothermal parallel plates, and the discrepancy with the corresponding result in the literature ${Nu}_{fd, Br \pm, s} = 17.5$ [29] was found to be less than 0.05%.

The steps of the algorithm for ${\tilde{T}}_{p, \tilde{\dot{q}}}$ are as follows. First, the domain is discretized with an initial number of finite elements. Next, Eq. (69) is solved subject to the boundary conditions given by Eqs. (63)–(66) where $i = \tilde{\dot{q}}$ ⁠, to compute ${\tilde{T}}_{p, \tilde{\dot{q}}} (\tilde{x}, \tilde{y})$ and consequently ${Nu}_{fd, \tilde{\dot{q}} \pm}$ ⁠. Then, the mesh is adaptively refined and the algorithm proceeds from step two. The process is repeated until the change in the computed value of ${Nu}_{fd, \tilde{\dot{q}} \pm}$ is less than 0.01%.

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

Introduction

Analysis

Hydrodynamic Problem.

Thermal Problem.

Homogeneous Solution.

Particular Solution.

Local Nusselt Number.

Fully Developed Nusselt Number.

Regions Where Pe Effects are Dominant.

Regions Where Br and q˙̃ Effects are Dominant.

Results

Effects of Axial Conduction on Fully Developed Nusselt Number

Effects of ϕ and H/d for Pe=1.

Effects of ϕ and Pe for H/d=1 and 10.

Effects of Viscous Dissipation and Volumetric Heat Generation on Fully Developed Nusselt Number.

Combined Effects of Axial Conduction and Viscous Dissipation on Developing Nusselt Number.

Conclusions

Acknowledgment

Funding Data

Nomenclature

Roman Symbols

Greek Symbols

Subscripts

Appendix A

Appendix B

Appendix C

References

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

Introduction

Analysis

Hydrodynamic Problem.

Thermal Problem.

Homogeneous Solution.

Particular Solution.

Local Nusselt Number.

Fully Developed Nusselt Number.

Regions Where Pe Effects are Dominant.

Regions Where Br and q˙̃ Effects are Dominant.

Results

Effects of Axial Conduction on Fully Developed Nusselt Number

Effects of ϕ and H/d for Pe=1.

Effects of ϕ and Pe for H/d=1 and 10.

Effects of Viscous Dissipation and Volumetric Heat Generation on Fully Developed Nusselt Number.

Combined Effects of Axial Conduction and Viscous Dissipation on Developing Nusselt Number.

Conclusions

Acknowledgment

Funding Data

Nomenclature

Roman Symbols

Greek Symbols

Subscripts

Appendix A

Appendix B

Appendix C

References

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