31
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 proposes 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 analytical 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 computational micropolar fluid dynamics (CMFD). The discretization scheme in space is demonstrated 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
*Corresponding author: Department of Mechanical and
Aerospace Engineering, The George Washington University,
Washington, DC, 20052, USA
email: jikembo@gwmail.gwu.edu
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 capture 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 formulation of the micropolar theory has additional degrees
of freedom – gyration – to determine the rotation of
the microstructure. Hence, the balance law of angular momentum are given for solving gyration. This
equation introduces a mechanism to take into
account the effect of molecular spin. The micropolar 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
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
32
J Chen, C Liang, and J D Lee
macroscale. The Navier–Stokes description is incapable of explaining the observed effects. The calculated 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 micropolar theory that have focused on numerical analysis of Hagen–Poiseuille flow and its applications on
nano- and microfluidics, including Papautsky [10],
Ye [12], and Hansen [13]. However, all of the studies 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 example, lid-driven cavity flow is simulated by solving
the micropolar equations. Nomenclature can be
found in the Appendix section.
2
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]
xk 5xk ðX ; t Þ; k51;2;3
(1)
jk 5x kK ðX; t ÞK ; K 51;2;3
(2)
Since the material particles are considered to be
geometrical points with mass and inertia, x kK ðX ; t Þ
here represents the three deformable directors
attached to the material particles.
For a material body called a micropolar continuum, the micromotion is further reduced to a rotation. In other words, its directors are orthonormal
and rigid, that is
xkK x kL ¼ dKL
akl ¼ vl;k 1elkm vm ; bkl ¼ vk;l
(3)
For fluid flow, deformation-rate tensors are crucial to the characterization of the viscous resistance.
Deformation-rate tensors may be deduced by
simply calculating the material time-rates of the
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
(4)
vm is the gyration vector, which is the additional 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
r1rv
_
l;l ¼ 0
(5)
Balance of momentum
tkl;k 1rðfl v_ l Þ ¼ 0
(6)
Balance of angular momentum
mkl;k 1elmn tmn 1rðll iv_ l Þ ¼ 0
MICROPOLAR FLUID THEORY
x kK xlK ¼ dkl ;
spatial deformation tensors. For micropolar fluid,
two objective deformation-rate tensors are [5–8]
(7)
Conservation of energy
re_ tkl ðvl;k 1elkr vr Þ mkl vl;k 1 qk;k rh ¼ 0
(8)
Clausius–Duhem inequality
qk
_
_
rðc1h
uÞ1t
u;k 0
kl akl 1mbl blk u
(9)
The linear constitutive equations for Cauchy
stress, moment stress, and heat flux are derived to
be [5, 6]
tkl 5 pdkl 1l trðamn Þdkl 1ðm1kÞakl 1malk
[ pdkl 1D tkl
a
mkl 5 eklm u;m 1a trðbmn Þdkl 1bbkl 1gblk
u
K
qk 5 u;k 1aeklm vm;l
u
(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
r1rv
_
l;l ¼ 0
(11)
Theory and simulation of micropolar fluid dynamics
Balance of momentum
rp1ðl1mÞrr v1ðm1kÞr2 v
1kr3v1rf ¼ rv_
33
D
ðxk;K Þ ¼ vk;l xl;K
Dt
(12)
Balance of angular momentum
_ (13)
ða1bÞrr v1gr2 v1kðr3v 2vÞ1rl ¼ riv
D
ðx Þ ¼ vkl xlK
Dt kK
(16)
If the micromotion equals the macromotion, that
is, xl,K = xlK, this leads to
vk;l ¼ vkl
(17)
Conservation of energy
re_ D t : aT m : b1r q rh ¼ 0
(14)
In micropolar theory, the gyration tensor is antisymmetric, that is
The microinertia is defined as
vkl ¼ eklm vm
i [ h2jk jk i
R 0
r j j dv0
5 2 R k0 k 0
r dv
This leads to
[l
1
vm ¼ elkm vk;l
2
(15)
2
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 activities can possibly be measured experimentally using
Largragian velocities of tracer particles [14,15].
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 similar 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 continuum is considered as a continuous collection of
finite-size particles. The translation of finite-size
fluid particles can be imagined as the earth revolution with, the gyration being similar to the spin of
the earth.
The material time rates of spatial deformation
tensors can be obtained as
(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 r2 v1rf ¼ rv_
3
(18)
(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 developed by Fu and Hodges in 2009 for unsteady advection problems [18]. Here the TCSM is further
extended for incompressible MFD. The incompressible 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 projection method is incorporated with TCSM to update
the pressure gradient term for solving the Poisson
equation. Also, it is noted that the effect of thermomechanical 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
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
34
J Chen, C Liang, and J D Lee
using the momentum equations, ignoring the pressure gradient term
v vn
1v rv
ðt t n Þ
ri
¼
(21)
ða1bÞrr v 1gr2 v 1kðr3v 2v Þ1rl
(26)
where vn is the velocity at the nth time step. In the
next step, the velocity is updated with
5. Neglect the pressure effect and update velocity
from t** to t*** while dealing with the convective
term as v rv = v** rv**
v vn
¼ v n rv n 1mr2 v n
ðt t n Þ
v n11 v 1
¼ rpn11
r
ðt n11 t Þ
(22)
In order to guarantee that vn11 satisfies the continuity equation, taking divergence on both sides of
equation (22) leads to
r v n11 r v ¼ ðt n11 t Þ 2 n11
r p
r
(23)
Thus, a Poisson equation for pn11 is obtained as
r2 pn11 ¼
r
r v
ðt n11 t Þ
v vn
1v n rv ðt t n Þ
¼ ðm1kÞr2 v 1kr3vn
(25)
2. Solve
r2 p ¼
ðt r
r v
tÞ
3. March time from t* to t**and update velocity as
v ¼ v ðt t Þ
rp
r
which guarantees velocity divergence free at t**.
4. Solve gyration v** at t** using velocity field v**
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
¼
(27)
ðm1kÞr2 v 1kr3v
6. Solve
r2 pn11 ¼
r
r v ðt n11 t Þ
7. March time from t*** to tn11 and update velocity
using
v
1. Neglect the pressure effect and update the
velocity from tn to t* while dealing with the convective term as v rv = vn rv*
v v 1v rv ðt t Þ
(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.
r
r
n11
¼v
t n11 t rpn11 :v n11
r
to satisfy the continuity equation at tn11.
8. Solve gyration vn11 at tn11 using the velocity
field nn11
vn11 v
1v n11 rvn11
ri
ðt n11 t Þ
ða1bÞrr vn11 1gr2 vn11 1
¼
(28)
kðr3v n11 2vn11 Þ1rl
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 equations 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
vx ðx11; y; zÞ 2vx ðx; y; zÞ1vx ðx 1; y; zÞ
x2
vx ðx; y11; zÞ 2vx ðx; y; zÞ1vx ðx; y 1; zÞ
1
y 2
vx ðx; y; z11Þ 2vx ðx; y; zÞ1vx ðx; y; z 1Þ
1
z2
r2 v x ¼
(29)
Theory and simulation of micropolar fluid dynamics
vx ðx11; y; zÞ 2vx ðx; y; zÞ1vx ðx 1; y; zÞ
r vx ¼
x2
vx ðx; y11; zÞ 2vx ðx; y; zÞ1vx ðx; y 1; zÞ
1
y 2
vx ðx; y; z11Þ 2vx ðx; y; zÞ1vx ðx; y; z 1Þ
1
z2
2
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 ∂vy vx ðx;y11;zÞ vx ðx;y 1;zÞ
5
2y
∂y
∂x
vy ðx11;y;zÞ vy ðx 1;y;zÞ
2x
∂vx ∂vy vx ðx;y11;zÞ vx ðx;y 1;zÞ
5
2y
∂y
∂x
vy ðx11;y;zÞ vy ðx 1;y;zÞ
2x
(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
vx vj;x
8
v j ðx;y;zÞ vj ðx 1;y;zÞ
>
>
n
>
v
ð
x;y;z
Þ
>
x
>
x
>
>
>
<
if vxn ðx;y;zÞ . 0
)
>
v j ðx11;y;zÞ vj ðx;y;zÞ
>
>
>
vxn ðx;y;zÞ
>
>
x
>
>
:
if vxn ðx;y;zÞ\0
5
(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**
vx vj;x )
8
v ðx; y; zÞ vx ðx 1; y; zÞ
>
>
ðx; y; zÞ x
v >
x
>
>
x
>
>
>
>
<
if vx ðx; y; zÞ . 0
vx vj;x
(35)
ANALYTICAL AND EXACT SOLUTIONS OF
COUETTE 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 xand 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
k
C2 e MyC3 e My 12ðC_21C_3Þy1C4
Mðm1kÞ
m1k
C1
vz ¼ C2 e My 1C3 e My 2m1k
(36)
vx ¼
>
v ðx11; y; zÞ vx ðx; y; zÞ
>
>
>
ðx; y; zÞ x
v x
>
>
x
>
>
>
:
if vx ðx; y; zÞ\0 (32)
where j can be x, y, or z. At steps 4 and 8, the convective terms in the angular momentum equations
are discretized as
vx vj;x
35
8
vj ðx; y; zÞ vj ðx 1; y; zÞ
>
>
>
ðx;
y;
zÞ
v
>
x
>
x
>
>
>
<
if vx ðx; y; zÞ . 0
)
>
vj ðx11; y; zÞ vj ðx; y; zÞ
>
>
>
vx ðx; y; zÞ
>
>
x
>
>
:
if vx ðx; y; zÞ\0 (33)
8
vn11
ðx; y; zÞ vn11
ðx 1; y; zÞ
>
j
j
n11
>
>
v
ðx;
y;
zÞ
>
x
>
x
>
>
>
<
if vxn11 ðx; y; zÞ . 0
)
>
vn11
ðx11; y; zÞ vn11
ðx; y; zÞ
>
j
j
>
n11
>
v
ðx;
y;
zÞ
>
x
>
x
>
>
:
if vxn11 ðx; y; zÞ\0
(34)
where
kð2m1kÞ
gðm1kÞ
2m1k
C1 5
ðC2 1C3 Þ
m1k
U0
C2 5 Mh
Mh
kð1eMh Þ
2 M ðm1kÞ 1 e eMhe1 h
M 25
1 e Mh
C2
e Mh 1
k
C4 5
ðC2 C3 Þ
M ðm1kÞ
C3 5
(37)
The steady solution of Newtonian fluid is
y
U0
h
1 ∂v x
U0
¼
v
z ¼ 2 ∂y
2h
vx ¼
(38)
Taking into account that g is the viscosity coefficient, which tends to stop the rotation of the finite
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
36
J Chen, C Liang, and J D Lee
Fig. 1 The comparison of angular velocity in Navier–
Stokes equations and microgyration in MFD
(Couette flow)
Fig. 3 The comparison of velocity profiles in Navier–
Stokes equations and MFD (Poiseuille flow)
Figure 2 shows the time evolution of the velocity
profile in Couette flow
6
ANALYTICAL AND EXACT SOLUTIONS OF
POISEUILLE 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
2
G
h y2
ð2m1kÞ
kC2 Mh Mh My My 1
e 1e
e 1e
M ðm1kÞ
Gy
1C2 e My e My
vz 5
ð2m1kÞ
vx 5
Fig. 2 Time evolution of velocity in Couette flow. The
arrow indicates the transient process as time
marches on
size particles, one can also define the internal characteristic length l
g m1k
k 2m1k
1
2
l¼
(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 tackled. The fluid is initially at rest. The time step is set
as 2 31023 and the result is output every 100 steps.
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
(40)
where
M2 ¼
kð2m1kÞ
;
gðm1kÞ
C2 ¼
Gh
ðe Mh e Mh Þð2m1kÞ
(41)
The steady solution of Newtonian fluid is
G
ðh2 y 2 Þ
2m
1 ∂v x
Gy
¼ v
z ¼ 2 ∂y
2m
vx ¼
(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 different from the velocity obtained from the Navier–
Stokes equations. The plot of gyration and centre
Theory and simulation of micropolar fluid dynamics
37
compared. Different numbers of grid point (including 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
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
Velocity
Microgyration
x
L1 error
Order
L2 error
Order
L1 error
Order
L2 error
Order
636
11 3 11
21 3 21
0.079 84
0.024 89
1.681 5
0.028 46
1.797 6
0.006 77
1.414 77
0.008 69
1.530 8
1
0.005 46
2.189
0.006 27
2.182
0.002 1
1.688 8
0.002 52
1.786
0.5
0.098 94
0.018 05
0.025 11
2
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 solutions of the flat-plate Hagen–Poiseuille flow are
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 104 ;
k ¼ 93104 ; g ¼ 107 . 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 influence on the numerical method. Figure 6 plots the
centre velocity vector. A big recirculation region can
be clearly seen. Figure 7 shows the pressure distribution. 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 pressure ru2‘ is 0.1.
Figure 8 shows the gyration in this cavity. It is
apparently seen that the fluid particles spin clockwise below the top of the box and they spin counterclockwise 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 conditions. Based on the center velocity, it is straightforward 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 neighbouring fluid molecules but the gyration is the selfspinning of the fluid molecule. The total velocity in
the micropolar theory is defined as
vk ðx; z; tÞ ¼ vk ðx; tÞ1ekml vm zl
(43)
Figure 10 shows the total velocity. It is plotted on
a finer mesh and calculated based on equation (43).
The maximum kjk 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
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
38
J Chen, C Liang, and J D Lee
Fig. 9 Vorticity in cavity flow
Fig. 6 Centre velocity in cavity flow
Fig. 7 Pressure distribution in cavity flow
Fig. 10 Total velocity in cavity flow
of 20 fluid particles, 20323 maxkjk, 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 concept 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
Fig. 8 Gyration in cavity flow
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems
CONCLUDING REMARKS
Microcontinuum field theories provide additional
degrees of freedom to incorporate the microstructure of the continuous medium. In this paper,
the micropolar theory is briefly introduced. Extra
Theory and simulation of micropolar fluid dynamics
rotating degrees of freedom not only widen the
physical background of microfluidics andthe fluid
mechanics at micro- and nanoscales, but also enlarge the capacity to address various features missing 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 completely in the proposed numerical scheme. The
developed discretization schemes in space are demonstrated 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 methods developed in this paper to explore the feasibility
of micropolar fluid dynamics on multiscale fluid
mechanics problems.
Ó Authors 2011
REFERENCES
1 Brody, J. P. and Yager, P. Low Reynolds number
micro-fluidic devices. In the Proceedings of SolidState Sensor and Actuator Workshop, South Carolina, USA, June 1999, pp. 105–108.
2 Holmes, D. B. and Vermeulen, J. R. Velocity profiles 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 water
flows in nanochannels. Bulltn Polish Acad. Sci.:
Tech. Sci., 2009, 57, 55–61.
4 Delhommelle, J. and Evans, J. D. Poiseuille flow of
a micropolar fluid. Molec. Phys., 2002, 100, 2857–
2865.
5 Eringen, A. C. Simple microfluids. Int. J. Engng
Sci., 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 of
micromorphic theory in multiscaling of synthetic
and natural systems with self-adaptive capability,
2010, pp. 81–84 (National Taiwan University of Science and Technology Press).
10 Papautsky, I., Brazzle, J., Ameel, T., and
Frazier, A. B. Laminar fluid behavior in microchannel using micropolar fluid theory. Sensors and
Actuators, 1999, 73, 101–108.
39
11 Gad-El-Hak, M. The fluid mechanics of microdevices. J. Fluid Engng, 1999, 121, 1215–1233.
12 Ye, S., Zhu, K., and Wang, W. Laminar flow of
micropolar fluid in rectangular microchannels.
Acta Mechanica Sinica, 2006, 22, 403–408.
13 Hansen, J. S., Davis, P. J., and Todd, B. D. Molecular spin in nano-confined fluidic flows. Microfluidics 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 fully
developed 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 (John
Wiley & Sons, New Jersey) pp. 271.
18 Fu, S. and Hodges, B. R. Time-center split method
for implicit discretization of unsteady advection
problems. 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
fk
h
i
K/u
ll
mkl
p
qk
tkl
xk,K
internal energy density
body force tensor
energy source density
microinertia
Fourier heat-conduction coefficient
body moment density
coupled stress tensor
pressure
heat vector
Cauchy stress tensor
deformation gradient
a, b, g
dkl, dKL
eklm
h
u
l, m, k
nk
nk,l
vk
r
x kK
c = e – hu
vk
vkl
total velocity
Kronecker delta
permutation symbols
entropy density
absolute temperature
viscosity coefficients for stress
centre velocity
velocity gradient
total velocity
mass density
micromotion
Helmholtz free energy
gyration vector
gyration tensor
Proc. IMechE Vol. 224 Part N: J. Nanoengineering and Nanosystems