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Research Papers: Forced Convection

Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges for Arbitrary Meniscus Curvature

[+] Author and Article Information
Simon Game

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: s.game14@imperial.ac.uk

Marc Hodes

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

Toby Kirk

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

Demetrios T. Papageorgiou

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

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 1, 2017; final manuscript received November 13, 2017; published online April 19, 2018. Assoc. Editor: Sara Rainieri.

J. Heat Transfer 140(8), 081701 (Apr 19, 2018) (13 pages) Paper No: HT-17-1443; doi: 10.1115/1.4038831 History: Received August 01, 2017; Revised November 13, 2017

We numerically compute Nusselt numbers for laminar, hydrodynamically, and thermally fully developed Poiseuille flow of liquid in the Cassie state through a parallel plate-geometry microchannel symmetrically textured by a periodic array of isoflux ridges oriented parallel to the flow. Our computations are performed using an efficient, multiple domain, Chebyshev collocation (spectral) method. The Nusselt numbers are a function of the solid fraction of the ridges, channel height to ridge pitch ratio, and protrusion angle of menisci. Significantly, our results span the entire range of these geometrical parameters. We quantify the accuracy of two asymptotic results for Nusselt numbers corresponding to small meniscus curvature, by direct comparison against the present results. The first comparison is with the exact solution of the dual series equations resulting from a small boundary perturbation (Kirk et al., 2017, “Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges Accounting for Meniscus Curvature,” J. Fluid Mech., 811, pp. 315–349). The second comparison is with the asymptotic limit of this solution for large channel height to ridge pitch ratio.

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References

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Figures

Grahic Jump Location
Fig. 2

Schematic of the flow configuration, showing the liquid–solid contact and groove dimensions. Domain A is the cross section used for the analysis.

Grahic Jump Location
Fig. 1

Menisci along tee-shaped structures when pressure difference imposed across them is zero (top line), required for the triple contact line to advance along the vertical side of the top of a tee (middle curve) and 90 deg relative to the horizontal surface on the bottom side of the top of a tee (lower curve)

Grahic Jump Location
Fig. 4

Diagram indicating how the domain is decomposed into two distinct subdomains

Grahic Jump Location
Fig. 3

Domains, governing equations, and boundary conditions for the (dimensionless) w and T problems

Grahic Jump Location
Fig. 5

The problems solved by the singular part of the velocity (a) and the temperature (b)

Grahic Jump Location
Fig. 6

The average Nusselt number for the indicated values of h. In all cases, the lower, middle, and upper lines correspond to ϕ=0.01,0.05 and 0.1, respectively. The dashed lines correspond to the approximate values as calculated in Kirk et al. [1], the solid curves correspond to values calculated in the current work.

Grahic Jump Location
Fig. 9

The average Nusselt number as ϕ and h vary, given at θ=±87.1 deg

Grahic Jump Location
Fig. 7

The average Nusselt number for the indicated values of h. In each case, the lower, middle, and upper lines correspond to ϕ=0.9,0.95 and 0.99, respectively. The dashed lines correspond to the approximate values as calculated in Kirk et al. [1], the solid curves correspond to values calculated in the current work.

Grahic Jump Location
Fig. 8

The average Nusselt number for the indicated values of h. In each case, the lower, middle, and upper curves correspond to ϕ=0.01,  0.05, and 0.1 respectively. The dashed lines correspond to the approximate values as calculated using the h→∞ approximation in Kirk et al. [1], the solid curves correspond to values calculated in the current work.

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