Theory and simulation of micropolar fluid dynamics J Chen*, C Liang, and J D Lee Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC, USA The manuscript was received on 29 October 2010 and was accepted after revision for publication on 21 January 2011. DOI: 10.1177/1740349911400132 Abstract: This paper reviews the fundamentals of micropolar fluid dynamics (MFD), and pro- poses a numerical scheme integrating Chorin’s projection method and time-centred split method (TCSM) for solving unsteady forms of MFD equations. It has been known that Navier–Stokes equations are incapable of explaining the phenomena at micro and nano scales. On the contrary, MFD can naturally pick up the physical phenomena at micro and nano scales owingto its additional degrees of freedom for gyration. In this study, the analyti- cal and exact solutions of Couette and Hagen–Poiseuille flow are provided. Though this study is limited to the steady flow cases, the unsteady term in the MFD has been taken into account. This present work initiates the development of a general-purpose code of computa- tional micropolar fluid dynamics (CMFD). The discretization scheme in space is demon- strated with nearly second-order accuracy on multiple meshes. Keywords: micropolar fluid dynamics (MFD), microfluidics, computational micropolar fluid dynamics (CMFD), finite difference method, projection method, time-centre split method (TCSM) 1 INTRODUCTION Research activities aiming to explore fluid physics at nano and micro scales have been increasing over the past 20 years. There are existing literatures that have analysed fluid mechanics in microchannels and micromachined fluid systems (e.g. pumps and valves) using Navier–Stokes equations [1]. Fluid flow moves differently in the micro scale than that in the macro scale. There are situations in which the Navier–Stokes equations, derived from classical continuum, become incapable of explaining the micro scale fluid transport phenomena [2]. The reason is that when the channel size is comparable to the molecular size, the spinning of molecules, which have been observed in molecular dynamics (MD) simulations [3, 4], affects significantly the flow field. This effect of molecular spin is not taken into account in the Navier–Stokes equations. A novel approach, microcontinuum theory, consisting of micropolar, microstretch, and micromorphic (3M) theories, developed by Eringen [5–8] and Lee et al.[9], offers a mathematical foundation to cap- ture such motions. In 3M theories, each particle has a finite size and contains a microstructure that can rotate and deform independently, regardless of the motion of the centroid of the particle. The formula- tion of the micropolar theory has additional degrees of freedom – gyration – to determine the rotation of the microstructure. Hence, the balance law of angu- lar momentum are given for solving gyration. This equation introduces a mechanism to take into account the effect of molecular spin. The micropo- lar theory thus represents a promising alternative approach to numerically solving micro scale fluid dynamics that can be much more computationally efficient than the MD simulations. Papautsky [10] was the first one to adopt the micropolar fluid model to explain the experimental observation of volume flow rate reduction for the flow in a rectangular microchannel. In addition, Gad-El-Hak [11] explicitly states that microscale flows are essentially different from flows in the *Corresponding author: Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC, 20052, USA email: [email protected]31 Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
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Theory and simulation of micropolar fluid dynamicsJ Chen*, C Liang, and J D Lee
Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC, USA
The manuscript was received on 29 October 2010 and was accepted after revision for publication on 21 January 2011.
DOI: 10.1177/1740349911400132
Abstract: This paper reviews the fundamentals of micropolar fluid dynamics (MFD), and pro-poses a numerical scheme integrating Chorin’s projection method and time-centred splitmethod (TCSM) for solving unsteady forms of MFD equations. It has been known thatNavier–Stokes equations are incapable of explaining the phenomena at micro and nanoscales. On the contrary, MFD can naturally pick up the physical phenomena at micro andnano scales owingto its additional degrees of freedom for gyration. In this study, the analyti-cal and exact solutions of Couette and Hagen–Poiseuille flow are provided. Though this studyis limited to the steady flow cases, the unsteady term in the MFD has been taken intoaccount. This present work initiates the development of a general-purpose code of computa-tional micropolar fluid dynamics (CMFD). The discretization scheme in space is demon-strated with nearly second-order accuracy on multiple meshes.
Theory and simulation of micropolar fluid dynamics 37
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
of 20 fluid particles, 20323 max jk k, while a fluid
particle refers to either a fluid molecule or a group
of fluid molecules. For the overlapping regions, the
value of total velocity is averaged. Using the con-
cept of total velocity, it enables observation of the
gyration effect. In this example, the gyration tends
to induce the formation of vortices in the bottom
corners (see Figure 10).
9 CONCLUDING REMARKS
Microcontinuum field theories provide additional
degrees of freedom to incorporate the micro-
structure of the continuous medium. In this paper,
the micropolar theory is briefly introduced. Extra
Fig. 6 Centre velocity in cavity flow
Fig. 10 Total velocity in cavity flowFig. 7 Pressure distribution in cavity flow
Fig. 8 Gyration in cavity flow
Fig. 9 Vorticity in cavity flow
38 J Chen, C Liang, and J D Lee
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
rotating degrees of freedom not only widen the
physical background of microfluidics andthe fluid
mechanics at micro- and nanoscales, but also en-
large the capacity to address various features miss-
ing from the Navier–Stokes equations.
The second-order accurate TCSM successfully
incorporated with Chorin’s projection method to
solve the MFD. This work discusses only the steady
flow cases. Nevertheless, the unsteady terms in the
MFD are taken into account rigorously and com-
pletely in the proposed numerical scheme. The
developed discretization schemes in space are dem-
onstrated with nearly second-order accuracy on
multiple meshes.
This study initiates the development of a general
purpose numerical solver for computational MFD.
Interested readers may adopt the numerical meth-
ods developed in this paper to explore the feasibility
of micropolar fluid dynamics on multiscale fluid
mechanics problems.
� Authors 2011
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APPENDIX
Notation
e internal energy density
fk body force tensor
h energy source density
i microinertia
K/u Fourier heat-conduction coefficient
ll body moment density
mkl coupled stress tensor
p pressure
qk heat vector
tkl Cauchy stress tensor
xk,K deformation gradient
a, b, g total velocity
dkl, dKL Kronecker delta
eklm permutation symbols
h entropy density
u absolute temperature
l, m, k viscosity coefficients for stress
nk centre velocity
nk,l velocity gradient
vk total velocity
r mass density
xkK micromotion
c = e – hu Helmholtz free energy
vk gyration vector
vkl gyration tensor
Theory and simulation of micropolar fluid dynamics 39
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems