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Extended Graetz-Nusselt Problem for Liquid Flow in Cassie State Over Isothermal Parallel Ridges

[+] Author and Article Information
Georgios Karamanis

Department of Mechanical Engineering, Tufts University, Medford, MA 02155
georgios.karamanis@tufts.edu

Marc Hodes

Department of Mechanical Engineering, Tufts University, Medford, MA 02155
marc.hodes@tufts.edu

Toby Kirk

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

Demetrios Papageorgiou

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

1Corresponding author.

ASME doi:10.1115/1.4039085 History: Received July 05, 2017; Revised January 18, 2018

Abstract

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 non-linear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully-eveloped 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.

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