Asymptotic Nusselt numbers for internal flow in the Cassie state

TL;DR

This paper analytically derives asymptotic expressions for Nusselt numbers and slip lengths in laminar microchannel flows with textured surfaces, improving accuracy over existing models and accounting for inertial and thermal effects.

Contribution

It provides new closed-form asymptotic formulas for Nusselt numbers in microchannels with ridges, valid for any solid fraction and including inertial and thermal effects.

Findings

New Nusselt number expressions accurate to O(ε^2) for parallel ridges.

Error analysis shows improved accuracy over existing literature.

Thermal spreading resistance expressed with polylogarithm functions.

Abstract

We consider laminar, fully-developed, Poiseuille flows of liquid in the Cassie state through diabatic, parallel-plate microchannels symmetrically textured with isoflux ridges. Through the use of matched asymptotic expansions we analytically develop expressions for (apparent hydrodynamic) slip lengths and variously-defined Nusselt numbers. Our small parameter ($\epsilon$) is the pitch of the ridges divided by the height of the microchannel. When the ridges are oriented parallel to the flow, we quantify the error in the Nusselt number expressions in the literature and provide a new closed-form result. The latter is accurate to $O\left(\epsilon^2\right)$ and valid for any solid (ridge) fraction, whereas those in the current literature are accurate to $O\left(\epsilon^1\right)$ and breakdown in the important limit when solid fraction approaches zero. When the ridges are oriented transverse to the (periodically fully-developed) flow, the error associated with neglecting inertial effects in the slip length is shown to be $O\left(\epsilon^3\mathrm{Re}\right)$, where $\mathrm{Re}$ is the channel-scale Reynolds number based on its hydraulic diameter. The corresponding Nusselt number expressions are new and their accuracy is shown to be dependent on Reynolds number, Peclet number and Prandtl number in addition to $\epsilon$. Manipulating the solution to the inner temperature problem encountered in the vicinity of the ridges shows that classic results for thermal spreading resistance are better expressed in terms of polylogarithm functions.