Asymptotic Nusselt numbers for internal flow in the Cassie state

We consider laminar, fully developed, Poiseuille flows of liquid in the Cassie state through diabatic, parallel-plate microchannels symmetrically textured with isoflux ridges. Via matched asymptotic expansions, we develop expressions for (apparent hydrodynamic) slip lengths and Nusselt numbers. Our small parameter ( is an element of) 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. It is accurate to O (is an element of(2)) and valid for any solid (ridge) fraction, whereas previous ones are accurate to O (is an element of(1)) and breakdown in the important limit when the 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 (is an element of(3) Re), where Re is the channel-scale Reynolds number based on its hydraulic diameter. The corresponding Nusselt number expressions' accuracies are shown to depend on the Reynolds number, Peclet number and Prandtl number in addition to is an element of. Manipulating the solution to the inner temperature problem encountered in the vicinity of the ridges shows that classic results for the thermal spreading resistance are better expressed in terms of polylogarithm functions.

Automated summary

This article investigates laminar flows of liquid in parallel-plate microchannels with isoflux ridges in the Cassie state. Expressions for slip lengths and Nusselt numbers are developed through asymptotic expansions. The paper provides a new closed-form result for Nusselt numbers when the ridges are parallel to the flow and quantifies the error in previous expressions. When the ridges are transverse to the flow, the error associated with neglecting inertial effects in the slip length is shown to depend on the Reynolds number. The accuracy of the Nusselt number expressions is also dependent on additional fluid parameters. The article also discusses the solution to the inner temperature problem encountered near the ridges.