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Research Papers: Fundamental Issues and Canonical Flows

# Analytical Modeling of Gaseous Slip Flow in Parabolic Microchannels

[+] Author and Article Information

Department of Mathematics,
University of Pune,
Pune 411007, India;
Faculty of Islamic Azad University,
Kermanshah Branch,
Kermanshah 6718997551, Iran
e-mail: tahmouresif@yahoo.com; tahmouresif@iauksh.ac.ir

Samir K. Das

Professor
Department of Applied Mathematics,
Defense Institute of Advanced Technology,
Girinagar, Pune 411025, India
e-mail: samirkdas@diat.ac.in; samirkumar_d@yahoo.com

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 24, 2013; final manuscript received November 29, 2013; published online May 6, 2014. Assoc. Editor: Prashanta Dutta.

J. Fluids Eng 136(7), 071201 (May 06, 2014) (9 pages) Paper No: FE-13-1114; doi: 10.1115/1.4026663 History: Received February 24, 2013; Revised November 29, 2013

## Abstract

The paper presents an analytical solution of velocity, mass flow rate, and pressure distribution for fully developed gaseous slip flow in nonsymmetric and symmetric parabolic microchannels. The flow is considered to be steady, laminar, and incompressible with constant fluid properties. Fully developed gaseous slip flow in microchannels of parabolic cross section is solved analytically for various aspect ratios using a parabolic cylindrical coordinate system on applying the method of separation of variables. Prior to apply separation of variables, Arfken transform [Arfken, 1970, Mathematical Methods for Physicists, Academic Press, Orlando, FL, Ch. 2] was used on momentum equations and first-order slip boundary conditions at each channel wall were imposed. A simple model is proposed to predict the friction factor and Reynolds number product $fRe$ for slip flow in parabolic microchannels. Through the selection of a characteristic length scale, the square root of cross-sectional area and the effect of duct shape have been minimized. The results of a normalized Poiseuille number for symmetric parabolic microchannels ($ɛ=1$) shows good agreement with the previous results [Morini et al., 2004, “The Rarefaction Effect on the Friction Factor of Gas Flow in Micro/Nano-Channels,” Superlattices Microstruct., 35(3–6), pp. 587–599; Khan and Yovanovich, 2008, “Analytical Modeling of Fluid Flow and Heat Transfer in Microchannel/Nanochannel Heat Sinks,” J. Thermophys. Heat Transf., 22(3), pp. 352–359] for rectangular microchannels. The developed model can be used to predict mass flow rate and pressure distribution of slip flow in parabolic microchannels.

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## Figures

Fig. 1

A parabolic microchannel

Fig. 2

A symmetric parabolic microchannel

Fig. 3

Fully developed fReDhfor parabolic microchannels

Fig. 4

Fully developed fReA for parabolic microchannels

Fig. 5

Normalized Po results as a function of ε and β for parabolic microchannels

Fig. 6

Comparison of the model for parabolic microchannels

Fig. 7

Comparison of the linear model for parabolic microchannels

Fig. 8

The pressure distribution for different pressure ratios

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