Non-Newtonian Couette-Poiseuille flow of a dilute gas

Mohamed Tij,Andrés Santos

doi:10.3934/krm.2011.4.361

Abstract

The steady state of a dilute gas enclosed between two infinite parallel plates in relative motion and under the action of a uniform body force parallel to the plates is considered. The Bhatnagar-Gross-Krook model kinetic equation is analytically solved for this Couette-Poiseuille flow to first order in the force and for arbitrary values of the Knudsen number associated with the shear rate. This allows us to investigate the influence of the external force on the non-Newtonian properties of the Couette flow. Moreover, the Couette-Poiseuille flow is analyzed when the shear-rate Knudsen number and the scaled force are of the same order and terms up to second order are retained. In this way, the transition from the bimodal temperature profile characteristic of the pure force-driven Poiseuille flow to the parabolic profile characteristic of the pure Couette flow through several intermediate stages in the Couette-Poiseuille flow are described. A critical comparison with the Navier-Stokes solution of the problem is carried out.

Highlights

Two paradigmatic stationary nonequilibrium flows are the plane Couette flow and the Poiseuille flow
If the Knudsen number associated with the shear rate is small enough the Navier–Stokes (NS) equations provide a satisfactory description of the Couette flow
As shearing increases, non-Newtonian effects and deviations of Fourier’s law become clearly apparent [20]. These nonlinear effects have been derived from the Boltzmann equation for Maxwell molecules [14, 28, 34, 40, 57], from the Bhatnagar–Gross–Krook (BGK) kinetic model [6, 19, 43], and from generalized hydrodynamic theories [49, 51]

Summary

Introduction

Two paradigmatic stationary nonequilibrium flows are the plane Couette flow and the Poiseuille flow. As shearing increases, non-Newtonian effects (shear thinning and viscometric properties) and deviations of Fourier’s law (generalized thermal conductivity and streamwise heat flux component) become clearly apparent [20]. These nonlinear effects have been derived from the Boltzmann equation for Maxwell molecules [14, 28, 34, 40, 57], from the Bhatnagar–Gross–Krook (BGK) kinetic model [6, 19, 43], and from generalized hydrodynamic theories [49, 51].

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Conclusion