# Analysis of extended micro-Graetz problem in a microtube

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## Abstract

In this numerical study, hydrodynamically developed but thermally developing forced convection in a microtube subjected to a step change in the wall heat flux is analysed using a finite-volume method. The slip velocity and temperature jump conditions at the wall and the axial conduction in the fluid are included in the analysis. The combined effects of the Peclet number and the Knudsen number on the local Nusselt numbers as well as on the wall and bulk temperatures are determined in the continuum and slip flow regimes (0 ≤ *Kn* ≤ 0.1). In the entrance region, large reductions are observed in the Nusselt number with decreasing Peclet number or increasing Knudsen number. The results also show that the thermal length increases with decreasing Peclet number.

## Keywords

Microtube axial conduction Knudsen number## Nomenclature

*D*diameter of the microtube (m)

*F*tangential momentum accommodation coefficient

*F*_{t}thermal accommodation coefficient

*k*thermal conductivity (W/m K)

*Kn*Knudsen number

*L*half-length of the microtube (m)

*L*^{*}dimensionless half-length of the microtube

*Nu*local Nusselt number

*Pe*Peclet number

*Pr*Prandtl number

- \( q^{\prime\prime}_{w} \)
wall heat flux (W/m

^{2})*r*radial coordinate (m)

*R*dimensionless radial coordinate

*T*temperature (K)

*u*velocity (m/s)

*x*axial direction (m)

*X*dimensionless axial coordinate

## Greek symbols

- \( \gamma \)
specific heat ratio

- \( \lambda \)
molecular mean free path (m)

*υ*kinematic viscosity (m

^{2}/s)- \( \theta \)
dimensionless temperature, Eq. (3)

- \( \theta_{b} \)
dimensionless fluid bulk temperature, Eq. (11)

- \( \theta_{s - w} \)
dimensionless temperature jump between the fluid and wall, Eq. (9)

- \( \theta_{w} \)
dimensionless wall temperature, Eq. (14)

## Subscripts

*s*fluid properties at the wall

*w*wall

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