theory and simulation of micropolar fluid dynamics - college of
TRANSCRIPT
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.
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
macroscale. The Navier–Stokes description is inca-
pable of explaining the observed effects. The calcu-
lated hydrodynamic quantities for a fluid as
a classical continuous medium (from Navier–Stokes
equations) differ significantly from those obtained
experimentally, and the difference increases with
the decrease of the channel diameter in the flow
through narrow channels.
There are many recent developments of micro-
polar theory that have focused on numerical analy-
sis of Hagen–Poiseuille flow and its applications on
nano- and microfluidics, including Papautsky [10],
Ye [12], and Hansen [13]. However, all of the stud-
ies considered only a steady state solution and did
not solve for pressure. Their methods are therefore
unable to solve unsteady flow problems.
In this work, a numerical scheme for solving the
unsteady form of micropolar fluid dynamics (MFD)
is developed. A detailed explanation for the physical
meaning of all coefficients is provided. Analytical
and exact solutions for flat-plate Hagen–Poiseuille
flow and flat-plate Couette flow are discussed
against numerical solutions. As a numerical exam-
ple, lid-driven cavity flow is simulated by solving
the micropolar equations. Nomenclature can be
found in the Appendix section.
2 MICROPOLAR FLUID THEORY
In microcontinuum field theories, the material points
of the fluid are considered to be small deformable
particles. The macromotion and micromotion of the
material particles are expressed by [5–9]
xk5xk X ; tð Þ; k51;2;3 (1)
jk5xkK X ; tð Þ�K ; K 51;2;3 (2)
Since the material particles are considered to be
geometrical points with mass and inertia, xkK X ; tð Þhere represents the three deformable directors
attached to the material particles.
For a material body called a micropolar contin-
uum, the micromotion is further reduced to a rota-
tion. In other words, its directors are orthonormal
and rigid, that is
xkK xlK ¼ dkl; xkK xkL ¼ dKL (3)
For fluid flow, deformation-rate tensors are cru-
cial to the characterization of the viscous resistance.
Deformation-rate tensors may be deduced by
simply calculating the material time-rates of the
spatial deformation tensors. For micropolar fluid,
two objective deformation-rate tensors are [5–8]
akl ¼ vl;k1elkmvm; bkl ¼ vk;l (4)
vm is the gyration vector, which is the addi-
tional rotating degree of freedom for a particle.
Because the mean free path of fluid is larger than
solid, each fluid molecule has more space to move
around. When a group of fluid molecules or
a single fluid molecule spins, the effect of the
gyration vector appears and cannot be observed in
classical continuum theory. Therefore, the gy
ration vector is a good candidate for determining
the physics at the micro scale while adopting the
continuum assumption.
The balance laws of the micropolar continuum
can be expressed as [5, 6]:
Conservation of mass
_r1rvl;l ¼ 0 (5)
Balance of momentum
tkl;k1rðfl � _vlÞ ¼ 0 (6)
Balance of angular momentum
mkl;k1elmntmn1rðll � i _vlÞ ¼ 0 (7)
Conservation of energy
r _e � tklðvl;k1elkrvrÞ �mklvl;k 1 qk;k � rh ¼ 0 (8)
Clausius–Duhem inequality
� rð _c1h _uÞ1tklakl1mblblk �qk
uu;k � 0 (9)
The linear constitutive equations for Cauchy
stress, moment stress, and heat flux are derived to
be [5, 6]
tkl5� pdkl1l tr amnð Þdkl1 m1kð Þakl1malk
[� pdkl1Dtkl
mkl5a
ueklmu;m1a tr bmnð Þdkl1bbkl1gblk
qk5K
uu;k1aeklmvm;l (10)
Substitute the constitutive equations into all the
balance laws, and the governing field equations of
MFD can be rewritten as [5–8]:
Conservation of mass
_r1rvl;l ¼ 0 (11)
32 J Chen, C Liang, and J D Lee
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
Balance of momentum
�rp1ðl1mÞrr � v1ðm1kÞr2v
1kr3v1rf ¼ r _v(12)
Balance of angular momentum
ða1bÞrr �v1gr2v1kðr3v � 2vÞ1rl ¼ ri _v (13)
Conservation of energy
r _e � Dt : aT�m : b1r � q � rh ¼ 0 (14)
The microinertia is defined as
i [ 2jkjkh i
5 2
Rr0jkjkdv0R
r0dv0
[ l2
(15)
and l represents a hidden length scale, which can be
at the level of molecular scale, Kolmogorov micro
scale, or Taylor micro scale. These small-scale activi-
ties can possibly be measured experimentally using
Largragian velocities of tracer particles [14,15].
3 CONNECTION WITH NAVIER–STOKES
EQUATIONS
Vorticity is considered as the circulation per unit
area at a point in a fluid flow field. It is a common
practice in general vector analysis to describe
a vector function of a position having zero curl as
irrotational in view of the connection between
r 3 v and the local rotation of the fluid [16]. It has
another physical interpretation: vorticity measures
the solid-body-like rotation of a material point P’
adjacent to the primary material point P [17].
In micropolar fluid dynamics, gyration has a simi-
lar concept. One can interpret the motion in MFD
using the earth motion as an example. In the
motion of the earth, it not only revolves around the
sun, which results in seasons, but also spins on its
own axis, which makes days. A micropolar contin-
uum is considered as a continuous collection of
finite-size particles. The translation of finite-size
fluid particles can be imagined as the earth revolu-
tion with, the gyration being similar to the spin of
the earth.
The material time rates of spatial deformation
tensors can be obtained as
D
Dtðxk;K Þ ¼ vk;lxl;K
D
DtðxkK Þ ¼ vklxlK (16)
If the micromotion equals the macromotion, that
is, xl,K = xlK, this leads to
vk;l ¼ vkl (17)
In micropolar theory, the gyration tensor is anti-
symmetric, that is
vkl ¼ �eklmvm (18)
This leads to
vm ¼1
2elkmvk;l (19)
The physical picture of equation (19) is similar to
the motion of the moon; it always faces the Earth
with the same side while revolving around the Earth.
Substitute equation (19) into equation (12) and
one can obtain
�rp1ðl1m�Þrr � v1m�r2v1rf ¼ r _v (20)
where m� ¼ m11=2k. It is identical to Navier–Stokes
equations derived from Newtonian fluid. At this
point, the MFD formulation has been clearly shown
as more general than Navier–Stokes equations.
4 NUMERICAL SCHEME
The time-centre split method (TCSM) was first devel-
oped by Fu and Hodges in 2009 for unsteady advec-
tion problems [18]. Here the TCSM is further
extended for incompressible MFD. The incompress-
ible fluid implies r � v ¼ 0 and hence the pressure p
becomes the Lagrange multiplier. The condition
r � v ¼ 0 must be enforced and indeed it is used to
calculate the Lagrange multiplier. The Chorin’s pro-
jection method is incorporated with TCSM to update
the pressure gradient term for solving the Poisson
equation. Also, it is noted that the effect of thermo-
mechanical coupling is not considered.
The projection method was originally introduced
to solve time-dependent incompressible Navier–
Stokes fluid-flow problems by Chorin [19]. In
Chorin’s original version of the projection method,
the intermediate velocity v* is explicitly computed
Theory and simulation of micropolar fluid dynamics 33
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
using the momentum equations, ignoring the pres-
sure gradient term
v� � vn
ðt� � tnÞ ¼ �vn � rvn1mr2vn(21)
where vn is the velocity at the nth time step. In the
next step, the velocity is updated with
vn11 � v�
ðtn11 � t�Þ ¼ �1
rrpn11 (22)
In order to guarantee that vn11 satisfies the conti-
nuity equation, taking divergence on both sides of
equation (22) leads to
r � vn11 �r � v� ¼ � ðtn11 � t�Þ
rr2pn11 (23)
Thus, a Poisson equation for pn11 is obtained as
r2pn11 ¼ r
ðtn11 � t�Þr � v�
(24)
A distinguished feature of Chorin’s projection
method is that the velocity field is forced to satisfy
the continuity equation at the end of each time
step.
In this paper, a new method is proposed. It
incorporates TCSM into Chorin’s projection method
for MFD equations and enforces the continuity
equation to be satisfied in the middle and at the
end of each time step. The procedures of this
method are listed as follows.
1. Neglect the pressure effect and update the
velocity from tn to t* while dealing with the con-
vective term as v � rv = vn � rv*
rv� � vn
t� � tnð Þ1vn � rv�� �
¼ m1kð Þr2v�1kr3vn
(25)
2. Solve
r2p�� ¼ r
ðt�� � t�Þr � v�
3. March time from t* to t**and update velocity as
v�� ¼ v� � ðt�� � t�Þ
rrp��
which guarantees velocity divergence free at t**.
4. Solve gyration v** at t** using velocity field v**
riv�� �vn
t�� � tnð Þ1v�� � rv��� �
¼
a1bð Þrr �v��1gr2v��1k r3v�� � 2v��ð Þ1rl(26)
5. Neglect the pressure effect and update velocity
from t** to t*** while dealing with the convective
term as v � rv = v** � rv**
rv��� � v��
t��� � t��ð Þ1v��� � rv��� �
¼
ðm1kÞr2v���1kr3v��(27)
6. Solve
r2pn11 ¼ r
tn11 � t���ð Þr � v���
7. March time from t*** to tn11 and update velocity
using
vn11 ¼ v��� �tn11 � t���� �
rrpn11:vn11
to satisfy the continuity equation at tn11.
8. Solve gyration vn11 at tn11 using the velocity
field nn11
rivn11 �v��
tn11 � t��ð Þ1vn11 � rvn11
� �¼
ða1bÞrr �vn111gr2vn111
kðr3vn11 � 2vn11Þ1rl
(28)
The procedures from step 1 to step 8 complete
a physical step of time marching. The advantage of
this algorithm is to avoid the non-linear terms in
the equations and to provide a set of linear equa-
tions with a second-order accuracy in time evolving
[18].
For the viscous terms of velocity and gyration,
they are discretized using the central difference
method, for example
r2vx ¼vxðx11; y; zÞ � 2vxðx; y; zÞ1vxðx � 1; y; zÞ
�x2
� �
1vxðx; y11; zÞ � 2vxðx; y; zÞ1vxðx; y � 1; zÞ
�y2
� �
1vxðx; y; z11Þ � 2vxðx; y; zÞ1vxðx; y; z � 1Þ
�z2
� �
(29)
34 J Chen, C Liang, and J D Lee
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
r2vx ¼vxðx11; y; zÞ � 2vxðx; y; zÞ1vxðx � 1; y; zÞ
�x2
� �
1vxðx; y11; zÞ � 2vxðx; y; zÞ1vxðx; y � 1; zÞ
�y2
� �
1vxðx; y; z11Þ � 2vxðx; y; zÞ1vxðx; y; z � 1Þ
�z2
� �
(30)
For the convective terms, v � rv and v � rv, an
upwind scheme is adopted due to the stability
issue. For example, at step 1 in the time marching
algorithm, the convective terms in momentum
equations can be discretized as
vxvj;x )
vnx x;y;zð Þ
v�j x;y;zð Þ � v�j x � 1;y;zð Þ�x
if vnx x;y;zð Þ. 0
vnx x;y;zð Þ
v�j x11;y;zð Þ � v�j x;y;zð Þ�x
if vnx x;y;zð Þ\0
8>>>>>>>><>>>>>>>>:
(31)
where j can be x, y, or z. However at step 5, v*** is
unknown, so the upwind scheme is chosen based
on v**
vxvj;x )
v���x ðx; y; zÞv��x ðx; y; zÞ � v��x ðx � 1; y; zÞ
�x
if v��x ðx; y; zÞ. 0
v���x ðx; y; zÞv��x ðx11; y; zÞ � v��x ðx; y; zÞ
�x
if v��x ðx; y; zÞ\0
8>>>>>>>>><>>>>>>>>>:
(32)
where j can be x, y, or z. At steps 4 and 8, the con-
vective terms in the angular momentum equations
are discretized as
vxvj;x )
v�xðx; y; zÞv�j ðx; y; zÞ � v�j ðx � 1; y; zÞ
�x
if v�xðx; y; zÞ. 0
v�xðx; y; zÞv�j ðx11; y; zÞ � v�j ðx; y; zÞ
�x
if v�xðx; y; zÞ\0
8>>>>>>>><>>>>>>>>:
(33)
vxvj;x )
vn11x ðx; y; zÞ
vn11j ðx; y; zÞ � vn11
j ðx � 1; y; zÞ�x
if vn11x ðx; y; zÞ. 0
vn11x ðx; y; zÞ
vn11j ðx11; y; zÞ � vn11
j ðx; y; zÞ�x
if vn11x ðx; y; zÞ\0
8>>>>>>>><>>>>>>>>:
(34)
where j can be x or y or z. The central difference
method is also employed to discretize the curl of
velocity and gyration.
∂vx
∂y� ∂vy
∂x5
vx x;y11;zð Þ � vx x;y � 1;zð Þ2�y
� vy x11;y;zð Þ � vy x � 1;y;zð Þ2�x
∂vx
∂y� ∂vy
∂x5
vx x;y11;zð Þ � vx x;y � 1;zð Þ2�y
� vy x11;y;zð Þ � vy x � 1;y;zð Þ2�x (35)
5 ANALYTICAL AND EXACT SOLUTIONS OFCOUETTE FLOW
Consider an incompressible fluid with a top plate in
height h moving with a velocity U0, a bottom plate
fixed, and the following assumptions: (a) no velocity
in the y- and z-directions, (b) no gyration in the x-
and y-directions, (c) fully developed flow (i.e. both
x-direction velocity and z-direction gyration are
functions of y only), and (d) no body force.
The steady solution for micropolar fluid is
vx ¼�k
Mðm1kÞ C2eMy�C3e�My� �
12ðC_21C_3Þy1C4
vz ¼ C2eMy1C3e�My � m1k
2m1kC1
(36)
where
M25k 2m1kð Þg m1kð Þ
C152m1k
m1kC21C3ð Þ
C25U0
2 k 1�eMhð ÞM m1kð Þ 1 e�Mh�eMh
e�Mh�1
� �h
� �
C351� eMh
e�Mh � 1C2
C45k
M m1kð Þ C2 � C3ð Þ(37)
The steady solution of Newtonian fluid is
vx ¼y
hU0
�vz ¼ �1
2
∂vx
∂y¼ �U0
2h
(38)
Taking into account that g is the viscosity coeffi-
cient, which tends to stop the rotation of the finite
Theory and simulation of micropolar fluid dynamics 35
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
size particles, one can also define the internal char-
acteristic length l
l ¼ g
k� m1k
2m1k
� �12
(39)
Note that g = 0 leads to l = 0. Figure 1 shows the
gyration plot with the change of l. It can be
observed that as g decreases, the gyration effect
intensifies. It should be mentioned that the gyration
is normalized by the angular velocity solved from
Navier–Stokes equations.
Utilizing the proposed numerical method, the
transient process of Couette flow can now be tack-
led. The fluid is initially at rest. The time step is set
as 2 31023 and the result is output every 100 steps.
Figure 2 shows the time evolution of the velocity
profile in Couette flow
6 ANALYTICAL AND EXACT SOLUTIONS OFPOISEUILLE FLOW
Consider an incompressible fluid in a channel with
a uniform pressure gradient 2G and half channel
height h. The steady state solution for micropolar
fluid is
vx5G
2m1kð Þ h2 � y2� �
1kC2
M m1kð Þ eMh1e�Mh� �
� eMy1e�My� ��
vz5Gy
2m1kð Þ1C2 eMy � e�My� �
(40)
where
M2 ¼ kð2m1kÞgðm1kÞ ; C2 ¼
�Gh
ðeMh � e�MhÞð2m1kÞ (41)
The steady solution of Newtonian fluid is
vx ¼G
2m�ðh2 � y2Þ
�vz ¼ �1
2
∂vx
∂y¼ Gy
2m�
(42)
It can be observed that k is the connection
between velocity and gyration, which indicates the
strength of the coupling effect. In Figs 3 and 4, m1k
keeps constant while k is changing. It is obvious that
when k is dominating in m1k, the coupling effect is
so strong that the centre velocity of MFD is quite dif-
ferent from the velocity obtained from the Navier–
Stokes equations. The plot of gyration and centre
Fig. 1 The comparison of angular velocity in Navier–Stokes equations and microgyration in MFD(Couette flow)
Fig. 2 Time evolution of velocity in Couette flow. Thearrow indicates the transient process as timemarches on
Fig. 3 The comparison of velocity profiles in Navier–Stokes equations and MFD (Poiseuille flow)
36 J Chen, C Liang, and J D Lee
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
velocity are in Figs 3 and 4, respectively, while the
centre velocity and the gyration are normalized by
the velocity and angular velocity, respectively, from
the Navier–Stokes equations.
7 NUMERICAL ACCURACY STUDY
Uniform mesh is utilized to analyse numerical
accuracy, while the numerical and analytical solu-
tions of the flat-plate Hagen–Poiseuille flow are
compared. Different numbers of grid point (includ-
ing 6 3 6, 11 3 11, and 21 3 21) are tested. Figure
5 plots the L1 and L2 error for the centre velocity
and gyration. The errors all decay as grid points
increase. The comparison is also listed in Table 1.
The average orders of L1 error of velocity are 1.94
and 1.55 for velocity and gyration, respectively. The
average order of L2 error of velocity is 1.55, and
gyration is 1.658. The accuracies of both velocity
and gyration are nearly second-order in space.
8 LAMINAR LID-DRIVEN CAVITY FLOW
The cavity is a square box with side length d = 0.1.
The velocity on the top of the box, uN, is 1.
The material constants are set as follows: m ¼ 10�4;
k ¼ 9310�4; g ¼ 10�7. Consequently, the Reynolds
number, Re = ( ruN d/( m 1 k), is 10. The mesh
number is 20 3 20. The time step is set as 0.005,
while the Courant condition requires that �t ��tmax = �x/uN = 0.005. Note that this proposed
numerical scheme is semi-implicit so that the
Courant condition does not have a significant influ-
ence on the numerical method. Figure 6 plots the
centre velocity vector. A big recirculation region can
be clearly seen. Figure 7 shows the pressure distri-
bution. The boundary condition of pressure is set
under Neumann boundary conditions, exactly on
the wall, while the reference point is set in the
centre of the box. In addition, the normalized pres-
sure ru2‘ is 0.1.
Figure 8 shows the gyration in this cavity. It is
apparently seen that the fluid particles spin clock-
wise below the top of the box and they spin coun-
terclockwise at both sides of the box. However, the
maximum of both spinning directions (clockwise
and counterclockwise) does not occur on the sides
because gyration is set to zero under boundary con-
ditions. Based on the center velocity, it is straight-
forward to calculate the vorticity as shown in Fig. 9.
It should be emphasized that the vorticity is the
rotation of the fluid molecule relative to its neigh-
bouring fluid molecules but the gyration is the self-
spinning of the fluid molecule. The total velocity in
the micropolar theory is defined as
�vkðx; z; tÞ ¼ vkðx; tÞ1ekmlvmzl (43)
Figure 10 shows the total velocity. It is plotted on
a finer mesh and calculated based on equation (43).
The maximum jk k of the finite-size particle is set to
be the half diagonal of the element. Therefore, the
size of the element is as large as the fluid particle.
Hence the size of simulated cavity is as large as that
Fig. 4 The comparison of angular velocity in Navier–Stokes equations and microgyration in MFD(Poiseuille flow)
Fig. 5 Error analysis of the numerical scheme
Table 1 Error analysis of Poiseuille flow
6 3 6 11 3 11 21 3 21
Velocity L1 error 0.079 84 0.024 89 0.005 46Order 1.681 5 2.189L2 error 0.098 94 0.028 46 0.006 27Order 1.797 6 2.182
Microgyration L1 error 0.018 05 0.006 77 0.002 1Order 1.414 77 1.688 8L2 error 0.025 11 0.008 69 0.002 52Order 1.530 8 1.786
�x 2 1 0.5
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
REFERENCES
1 Brody, J. P. and Yager, P. Low Reynolds numbermicro-fluidic devices. In the Proceedings of Solid-State Sensor and Actuator Workshop, South Caro-lina, USA, June 1999, pp. 105–108.
2 Holmes, D. B. and Vermeulen, J. R. Velocity pro-files in ducts with rectangular cross sections.Chem. Engng Sci., 1968, 23, 717–722.
3 Kucaba-Pietal, A., Walenta, Z., and Peradzynski, Z.Molecular dynamics computer simulation of waterflows in nanochannels. Bulltn Polish Acad. Sci.:Tech. Sci., 2009, 57, 55–61.
4 Delhommelle, J. and Evans, J. D. Poiseuille flow ofa micropolar fluid. Molec. Phys., 2002, 100, 2857–2865.
5 Eringen, A. C. Simple microfluids. Int. J. EngngSci., 1964, 2, 205–217.
6 Eringen, A. C. Theory of micropolar fluids. J. Appl.Math. Mech., 1966, 16, 1–8.
7 Eringen, A. C. Microcontinuum field theories I:foundations and solid, 1999 (Springer, New York)pp. 1–56.
8 Eringen, A. C. Microcontinuum field theories II:fluent media, 2001 (Springer, New York) pp. 1–80.
9 Lee, J. D., Wang, X., and Chen, J. An overview ofmicromorphic theory in multiscaling of syntheticand natural systems with self-adaptive capability,2010, pp. 81–84 (National Taiwan University of Sci-ence and Technology Press).
10 Papautsky, I., Brazzle, J., Ameel, T., andFrazier, A. B. Laminar fluid behavior in micro-channel using micropolar fluid theory. Sensors andActuators, 1999, 73, 101–108.
11 Gad-El-Hak, M. The fluid mechanics of micro-devices. J. Fluid Engng, 1999, 121, 1215–1233.
12 Ye, S., Zhu, K., and Wang, W. Laminar flow ofmicropolar fluid in rectangular microchannels.Acta Mechanica Sinica, 2006, 22, 403–408.
13 Hansen, J. S., Davis, P. J., and Todd, B. D. Molec-ular spin in nano-confined fluidic flows. Microflui-dics and Nanofluidics, 2009, 6, 785–795.
14 Heinloo, J. Formulation of turbulence mechanics.Phys. Rev. E, 2004, 69, 056317.
15 Mordant, N., Metz, P., Michel, O., and Pinton, J.-F. Measurement of Lagrangian velocity in fullydeveloped turbulence. Phys. Rev. Lett., 2001, 87,214501.
16 Batchelor, G. K. An introduction to fluid dynamics,1967 (Cambridge University Press, Cambridge) pp.79–84.
17 Panton, R. L. Incompressible flow, 2005 (JohnWiley & Sons, New Jersey) pp. 271.
18 Fu, S. and Hodges, B. R. Time-center split methodfor implicit discretization of unsteady advectionproblems. J. Engng Mech, 2009, 135, 256–264.
19 Chorin, A. J. Numerical solution of the Navier–Stokes equations. Math. Comput., 1968, 22, 745–762.
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