7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Heat Transfer Analysis of a Spherical Particle in Rarefied Slip Regime
Vahid Aliakbar a, Behzad Mohajer a, Abouzar Moshfegh a, Mehrzad Shams a
Goodarz Ahmadi b
a Faculty of Mech. Engng., K.N.Toosi University of Tech., Pardis St., Vanak Sq., Tehran 193951999, Iran
b Department of Mech. and Aeronautical Engng., Clarkson University, Potsdam, NY 136995725, USA
Vahid.Aliakbar@gmail.com
Keywords: 3D CFD simulation; spherical particle; slip flow regime; Nusselt number, Temperature jump
Abstract
In order to investigate how far the velocity slip and temperatures jump influence the heat flux, a 3D simulation study for an
incompressible slip flow around a spherical aerosol particle was performed. The full NavierStokes and energy equations
were solved simultaneously and the slip velocity and temperature jump at the gasparticle interface were considered.
GAMBIT, the companion package of commercial software FLUENT was used to build the model geometry and
computational grids. A multiblock structured and bodyfitted grid was considered for analyzing the flow over a single
spherical particle. Incompressible ideal gas model was used to simulate density changes caused by temperature gradient near
the particle surface. Gas properties such as viscosity, conductivity and mean free path were also taken variable with
temperature. Nusselt number values in continuum flow regime were used as a benchmark for code verification, and
reasonable agreement was achieved.
The effect of Knudsen number on spherical particle Nusselt number was taken into account. It was concluded that velocity
slip and temperature jump affect the heat transfer in opposite ways: a large slip on the wall increases the convection along the
surface. On the other hand, a large temperature jump decreases the heat transfer by reducing the temperature gradient at the
wall. Therefore, neglecting temperature jump will result in the overestimation of the heat transfer coefficient.
Introduction
GasSolid twophase flows are encountered frequently in
many environmental applications and in assessing human
exposure to environmental pollutants. Many industrial
applications such as dry powder flows, fluidized bed,
settling chambers, cyclones, electrostatic and
electromagnetic precipitators, and inertial dust collectors
involve dispersed twophase flows. Aerosols in the nature
typically have the size range from a few nanometers to
several micrometers.
When the characteristic size of the particle decreases down
to a value comparable to the mean free path of the
molecules, the rarefaction effects become important and
significantly affect the flow properties, wall shear stress,
aerodynamic drag force and heat transfer. In this case, the
continuum assumption fails and the NavierStokes equations
with noslip and notemperature jump boundary conditions
cannot be applied. In such situations, intermolecular
collisions play a prominent role and the flow properties will
be affected by the Knudsen number, which is a
dimensionless measure of the relative magnitudes of the gas
mean free path and flow characteristic length.
Kn = (1)
7
Regarding the definition of Knudsen number, the flow
lengthscale, 1, is typically evaluated as the ratio of a flow
thermodynamic property (such as temperature or density)
to its spatial gradient. Alternatively, 1 may be estimated as
a length scale of the flow geometry. For the incompressible
flow is simulated here, we used the particle radius (d/2) as
the reference lengthscale of the flow, which is consistent
with the definition used in aerosol communities.
Schaaf and Chambre (1961) have proposed the following
ranges to determine the degree of rarefaction flow regime
based on the local Knudsen number, Kn. For Kn < 0.01,
the continuum hypothesis holds and fluid is in local
thermodynamic equilibrium. Thus, the NavierStokes
equation in conjunction with noslip and notemperature
jump boundary condition describes the flow behaviours.
The rarefaction effects become noticeable when the
Knudsen number is in the range of 0.01 and 0.1. This range
is referred to as the slip flow regime, where the continuum
hypothesis is still valid, but the local thermodynamic
equilibrium of nearwall gas is violated. That is, the
conventional noslip and notemperature jump boundary
condition imposed at the gassolid interface begins to
break down but the linear stressstrain relationship is still
valid (GadelHak, 1999). In this range, the NavierStokes
equations can be utilized, but the slip and temperature
jump boundary condition is used to account for the non
equilibrium effects at the gassolid interface. The range of
Kn > 10 represents the free molecular regime where the
collisions among the molecules are negligible and
individual molecules colloid with the particle (Crowe,
2006). Finally, in the transitional regime (0.1< Kn <10) the
size of particle is comparable to the gas mean free path,
and continuum assumption breaks down but the flow
cannot be regarded as free molecular regime.
Nomenclature
Speed of sound (m.s 1)
Diameter (m)
Enthalpy (J)
Boltzmann constant
Knudsen number
Length scale (m)
Mach number
Nusselt number
Pressure (Pa)
Peclet number
Prandtl number
Gases constant (J.kg .'K)
Reynolds number
Temperature (K)
Velocity (m.s 1)
Greek letters
p Dynamic viscosity (Pa.s)
8 Kronecker delta
y Specific heat ratio
Tangential momentum accommodation
Cv coefficient
CT Thermal accommodation coefficient
T Stress tensor (N.m2)
2 Mean free path (m)
p Density (kg.m3)
Subscripts
Surface
Wall
Diameter
Superscripts
Fluid
Particle
Literature review
Using the method of matched asymptotic expansions,
Proudman and Pearson (1957) determined the velocity
field up to O (Re log Re), where Re
number based on freestream velocity. This solution
contains the classical Stokes solution as the leading term.
Taking this leading term only for the velocity field and
adopting a similar procedure, Acrivos and Taylor (1962)
obtained an analytical expression for the temperature field
in terms of Peclet number Pe, up to O(Pe3 log Pe). The
analysis is valid for the case Re<
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
no restriction on the Prandtl number Pr. Taking into
account the extra terms of O (Re) in the velocity field
predicted by Proudman and Pearson (1957), Rimmer
(1968, 1969) obtained an improved expression for the
mean Nusselt number Nu describing the rate of heat
transfer from the solid surface. His results are valid for
Re
with the numerical results of Dennis et al. (1973). Takhar
and Whitelaw (1977) extended this study to the case of a
rotating sphere taking the velocity field given by Whitelaw
(1974), which was similar to that of Rubinow and Keller
(1961), the difference being simply the orientation of the
axis of rotation. They observed that rotation enhances heat
transfer. Vasudeviah and Patturaj (1992) studied a similar
problem with minor difference, subjecting the axial spin to
be symmetric with the incident flow, Ranger (1971). It
may be recalled that the velocity fields employed in the
above investigations were all obtained under the
conventional noslip boundary conditions.
Thomas D. Taylor (1963) investigated the effect of
velocity slip and temperature jump on the heat transfer
from a sphere in a low Reynolds number flow theoretically
for large and small value of Peclet number. Using the
method of matched asymptotic expansions, Vasudeviah
and Balamurugan (1997) obtained analytical expression for
the mean Nusselt number up to O (Pr2Re2), where Re
and Pr of O (1). They considered velocity slip boundary
condition in their calculations but temperature jump was
neglected. Although, velocity slip and temperature jump
affect the heat transfer in opposite ways: a large slip on the
wall increases the convection along the surface. On the
other hand, a large temperature jump decreases the heat
transfer by reducing the temperature gradient at the wall.
Therefore, neglecting temperature jump will result in the
overestimation of the heat transfer coefficient (S.Kakac et
al. 2004).
In the research works cited above, all investigators tried to
consider the effect of rarefaction on the heat transfer from
a sphere in low Reynolds number flow (creeping flow,
Re
the industry, environment and biomedical applications, the
gasparticle relative Reynolds number may become more
than one (Re>l). Therefore, the proposed investigation
cannot predict the rarefaction effects in this range of
Reynolds number. Thus, availability of a more wide range
of Reynolds number on the heat transfer from a sphere in
slip flow regime would be helpful. In addition, they used
constant properties in their investigations, but the
temperature difference between the particle and ambient
leads to change in properties which have important
influence on the results. There are a lot of articles on the
effect of variable properties in rarefied gas dynamics and
micronano flows.
In the present research, the incompressible slip flow
regime past an unconfined, stationary, impermeable, solid
and spherical particle is simulated. The simulations are
implemented over a range of Reynolds numbers from the
Stokes regime up to the threshold of compressibility where
the flow stays roughly steady (Taneda, 1956; Sakamoto
and Haniu, 1990; Chomaz et al., 1993; Constantinescu and
Squires, 2000) (for the certain gas considered here). Due to
changes of temperature during the calculations, the gas
properties are considered variable. Incompressible ideal
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
gas model has been used for density approximation in
order to consider the temperature effect in the density.
Dynamic viscosity and conductivity vary with temperature
too.
The main goal of the study is to provide a better
understanding of thermal effects originating from the slip
regime for a typical aerosol particle. In particular, we
explore the effects of three variable, Reynolds number,
Knudsen number and temperature difference between
particle and gas, on heat transfer from a sphere in slip flow
regime.
Flow governing equations
For the steady and incompressible flow of a Newtonian
fluid which is in local thermodynamic equilibrium, the
NavierStokes and energy equations governing in
continuum flow regime can be written as
Duk
 0 (2)
dXk
O(ukui)
Oxk
Op +Tik
OXi OXk
Oh Op 02T dui
pUk = i + k + Tik
axi ax ax OXk
where x is the position in coordinate system, u is the
velocity, p is the pressure, p is the fluid density, h is the
enthalpy, T is the temperature and Tik is the stress tensor.
For a Newtonian and isotropic fluid the stress tensor is
given by
/OUi uk\ (Ou
Tik = IXk + (5)
dXk JXi j
Here, 1 is the fluid dynamic viscosity, is the coefficient
of bulk viscosity and 6ik is the Kronecker delta function.
For monatomic gases, the Stokes' hypothesis relates the
coefficients 1 and through the following expression
2
+ = 0 (6)
3
Slip boundary condition on the particle surface
To account for the effects of local nonequilibrium in the
slip flow regime, the conventional noslip and no
temperature jump boundary conditions must be replaced
appropriately. As postulated first by Navier in 1827
(White, 2006), the tangential slip velocity is proportional
to the tangential wall shear. That is,
utlwall a CTtlwall (7)
Here, a is a constant. Later, Maxwell in 1879 (White,
2006) showed from the kinetic theory of gases that a = .
For gases, the mean free path X is the average distance
traveled by molecules between collisions. For a perfect
gas, the mean free path is given as
Here, k is the Boltzmann constant, T is the gas
temperature, and dm is the collision diameter of the
molecules. To account for the portion of molecules
reflected diffusely from the solid surface in the slip
velocity formulation, Schaaf and Chambre (1961)
suggested that the constant a in Eq. (7) must be multiplied
by a factor of 2 ) where av is the tangential
ov
momentum accommodation coefficient (TMAC).
Consequently, the slip velocity can be written as
(2 ca) X
Ut wall Tt wall
Cv 1^
Equation (9) relates the tangential slip velocity to the
tangential shear stress via the TMAC, gas mean free path
and gas dynamic viscosity.
Von Smoluchowski modeled temperature jump effect as
follow (Kennard, 1983)
S(2 T) 2y A dT
UT 7 + 1 Pr On
where TT is thermal accommodation coefficient.
Relationship between Reynolds, Knudsen and
Mach numbers
Rarefied gas flows in low pressure environments and
microflows are dynamically similar, if the two flow
conditions have similar geometries and identical Kn, Re
and Ma numbers (Eckert & Drake, 1972). The particle
relative Reynolds and Mach numbers are defined by
puf uPl1
Re =
Au uf uP
Ma= 
c RT
where Au is the local relative velocity between gas and
particle, c is the speed of sound in the gas, uf is the gas
local velocity at the location of the particle, up is the
particle velocity, y is specific heat ratio, R is the constant
of gases and T is the temperature of the gas surrounding
the particle. For the physical consistency between the
inertial and viscous forces and the slip rarefaction effects,
same lengthscale (l=d /2) is also used for the Re number.
It is well known that the Knudsen number, the Reynolds
number and the Mach number are related. Kinetic theory
of gases relates the gas mean free path to viscosity as
follows
1 1 8
2 Vm = 2 
Here, vm is the mean molecular speed, which is somewhat
higher than the speed of sound (c).
, \
Combining the definition of Kn, Re and Ma numbers and
Eq. (13), it follows that (GadelHak, 1999)
Ma = Kn.Re
2
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
following the results of gridindependency study reported
in the Lee (2000). Therefore, in the xy view, the beginning
length (LB), the wake length (Lw) and the downstream
length (LD) are, respectively, equal to 1.5d, 3.5d and 5d.
This relation reveals the interconnection between gas
compressibility and the rarefaction effects. For a definite
value of Mach number, the Kn and Re numbers are
inversely proportional. Thus, Reynolds number cannot
change arbitrarily for a given Ma in the slip flow regime
stands (0.01< Kn < 0.1).
It is well known (Kawahara et al., 2002; Christlieb et al.,
2004; Gimelshein et al., 2005; White, 2006; Shui et al.,
2007) that the flow of a compressible fluid like can be
regarded as incompressible in the range of Ma < 0.3. It
should be noted here that this criterion is necessary but not
sufficient. There are flow conditions for which Mach
number is smaller that 0.3, but the compressibility effects
may be a prominent feature of the physics of the flow. For
example, wall heating or cooling may cause the density to
change significantly, and the compressibility may become
important even at low Mach numbers. Another example is
for flows in microdevices (especially the confined micro
flows), where the pressure may change sharply due to
viscous effects (GadelHak, 1999).
Here, we are concerned with an unconfined incompressible
slip flow past small spheres. The cases with Ma < 0.3 were
studied and the variations Nusselt number with Kn and Re
were analyzed. The low Mach number flows are assumed
to be incompressible.
Numerical simulation
Computational domain
GAMBIT version 2.0.0, the companion package of
commercial software FLUENTTM, was used to build the
model geometries and the computational grids. A multi
block structured and bodyfitted grid was considered for
analyzing the flow over a single spherical particle with the
diameter of d. The layout of multi block domain
decomposition and tridimensional grid distribution are
illustrated in Fig. 1. Domain decomposition around the
particle to capture the slip velocity and temperature jump
at the gassolid interface is shown in Fig. 1 (a). The view
of plane zy and the grid geometry adjacent to the particle
surface are depicted in Fig. 1 (b). The view of plane xy
and the grid geometry on the particle surface are also
shown in Fig. l(c). Such topology of computational
domain was used earlier by Moshfegh (2008) for
parameterization of spherical aerosols drags in the slip
Flow regimes.
In the zy view, the sides of the cube surrounding the
particle (Lc) and the marginal lengths (LM) were
respectively set to 3d and 2d. These values yield a side
length of 7d for the square cross section of the
computational domain. To minimize the outflow effects on
the flow, the outflow was placed 10 d farther from the
particle center, Id more than what was taken for the
turbulent flow study by Jones and Clarke (2003). To ensure
that the sphere does not affect the uniform inflow, we used
the upstream distance from the particle center of 3d
Figure l(a): Domain decomposition around the spherical particle
to capture the velocity slip and temperature jump.
LM Le LM
Figure l(b): Multi block grid distribution around the particle,
view of plane zy appended with the grid geometry near the
particle surface.
I I I I HIM11 1 111111111111111111I I~ g
~HR ~~;IAAFFFFFFF ~ w~TOi
L L, Lw L
Figure l(c): Multi block grid distribution around the particle,
schematic view of plane xy appended with the grid geometry on
the particle surface.
I
.............................
A.
lp..i'k&.fC'i
Four different grid resolutions (Cases I, II, III and IV) were
picked to perform the gridindependency study by
measuring the total Nusselt number on the particle at Re=l
and Kn = 0.05. The results are given in the Table 1. From
the values given in the relative error column, it could be
concluded that the Case II is the best choice among others
to achieve a gridindependent solution.
Relative error
Total cell Total Nusselte
from the case II
number number
Case I 329539 2.278 0.17
Case II 425150 2.282 0.0
Case III 571612 2.289 0.31
Case IV 743427 2.299 0.75
ladle 1: Lne results or
Re= 1 and Kn = 0.05.
gridindependency study carried out at
Numerical Scheme
The commercial CFD software FLUENTTM v6.3.26 was
adopted to solve the NavierStokes and energy equations
using the finitevolume method (FVM). The pressure
based segregated algorithm was employed to discretize the
3D, incompressible, steady, laminar and NavierStokes
equations (Eqs. (26)). Modified Quadratic Upstream
Interpolation for Convective Kinematics (QUICK) scheme,
which can be implemented on the multidimensional, non
uniform and unstructured grids (Leonard, 1979; Davis et
al., 1984; Freitas et al., 1985; Ji and Wang, 1990; Leonard
and Mokhtari, 1990; Ji and Wang, 1991), was selected to
discretize the convective and diffusive fluxes in the
equations. PressureVelocity coupling was treated by the
SIMPLE Consistent (SIMPLEC) algorithm on a collocated
grid. A secondorder pressure interpolation method was
also used to calculate the pressure at cell faces from the
neighboring nodes. Gradient reconstruction was
performed by GreenGauss nodebased formulation. In
order to consider density variation with temperature,
incompressible ideal gas model was applied. Variation of
fluid properties such as viscosity, conductivity and mean
free path with temperature was performed by the User
Defined Functions (UDF) (FLUENTTM v6.3.26 User's
Guide).
FLUENTTM solves the linearized scalar system of
equations for the dependent variables in each cell using a
point implicit (GaussSeidel) block matrix linear equation
solver in conjunction with an algebraic multigrid (AMG)
method. Flexible multi grid cycle was chosen for the three
momentum equations, while the Vcycle was considered
for continuity equation. To achieve a stable solution
through the iterations, appropriate underrelaxation factors
were devised for pressure, three components of velocity
and energy equation. Solution was assumed to converge
when all of the relative errors in consecutive iteration were
less than 105 for momentum equations and less than 106
for energy equation. The capability of foregoing set of
schemes to simulate the slip flow regimes has been
examined as reported by some researchers (Giors et al.,
2005; Jain and Lin, 2006; Ogedengbe et al., 2006; Zhuo et
al., 2007).
The implementation of velocity slip and temperature jump
boundary conditions in FLUENTTM was performed with
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
the User Defined Functions (UDF). The boundary
conditions are discretized in a way that yields the wall
tangential shear stress ( Twali) as a function of the
tangential velocity of each cell adjacent to the wall and
normal heat flux to the wall (qn) as a function of
temperature difference between the wall and the adjacent
cell. The effects of the surface curvature are considered
automatically through this method. This approach comes
from the strategy we followed in the finite volume method
(FVM) where the computational molecules are designed to
incorporate the dependent variables of cell nodes. The
UDF is called once per iteration at the beginning of the
solution main loop. The relevant coefficients to the
velocity slip and temperature jump boundary condition are
then introduced into the blockmatrix and linearized scalar
system of equations for the dependent variables. The
resultant system of equations is solved implicitly, and
finally the velocity and the temperature of near wall
computational cells are calculated per iteration. Additional
details of this approach may be found in the works of
Barber and Emerson (2001b) and Moriftigo et al. (2007).
Solution verification
Whitaker (1972) recommends the correlation
Nu = 2 + (0.4 Re'/2 + 0.06Re/3)Pr0O4 (1)1/4 (15)
[Is
This correlation is accurate in continuum regime in the
range of: 0.71 < Pr < 380, 3.5 < ReD < 7.6 x 104 and
1.0 < [/[s < 3.2. All properties except 1s are evaluated at
the ambient temperature. Here, the solution verification is
performed by comparison against the numerical results in
continuum regime to above correlation. The results are
given in the Table 2.
ReD NUExpenment NUNumen Relative error (%)
4 2.75 2.76 0.25
5 2.84 2.87 1
6 2.93 2.98 1.7
7 3.01 3.08 2.3
8 3.08 3.17 2.9
9 3.15 3.26 3.5
10 3.22 3.35 4
Table 2: Comparison against the numerical and experimental
results in continuum regime.
In order to verify the UDF code, Slipflow and heat
transfer in rectangular microchannels with constant wall
temperature was solved and good agreement was achieved
by numerical results from Renksizbulut et al. (2006).
Results and Discussion
The point of physical interest in the present work is to
investigate how far the velocity slip and temperatures jump
influence the heat flux. This convective heat transport from
the body to the ambient medium is best described in terms
of Nusselt number with the diameter of the sphere chosen
as the characteristic length. We explore the effects of three
variables, Reynolds number, Knudsen number and
temperature difference between particle and gas, on heat
transfer from a sphere in slip flow regime. In this work, it
is assumed that the particle is traveling in the dry air at
standard ambient temperature and pressure (SATP),
respectively, at 298.15 K and 101.3 kPa. The
corresponding air mean free path, dynamic viscosity,
conductivity and density vary with temperature in the field
of solution.
In figure (2), Nusselt number versus Knudsen number for
different Reynolds number has been shown. The
temperature difference between gas particle and ambient
medium is 200 K. It can e seen that the rarefaction effect
enhances the heat transfer when the temperature difference
between particle and ambient is small. It can be
remembered that slip velocity near the wall increases the
convection which leads to higher Nusselt number.
In figure (3), Nusselt number versus Knudsen number for
the temperature difference equals 400 K has been shown.
This figure shows an interesting effect in rarefaction gas
dynamic. For low Reynolds number, heat transfer
decreases with increasing Knudsen but for higher
Reynolds, Knudsen number has opposite effect.
Velocity slip and temperature jump affect the heat transfer
in opposite ways: a large slip on the wall increases the
convection along the surface. On the other hand, a large
temperature jump decreases the heat transfer by reducing
the temperature gradient at the wall. When the temperature
difference between wall and ambient increases the
temperature jump grows up and the heat transfer decreases.
On the other hand, increasing Reynolds number causes
more slip at the wall and thus more convection. Thomas D.
Taylor (1963) has mentioned this effect in low Reynolds
number slip flow. Therefore, neglecting temperature jump
will result in the overestimation of the heat transfer
coefficient. (S.Kakac, et al.2004).
In figure (4), Nusselt number versus Knudsen number for
the temperature difference equals 600 K has been shown.
In this case, the temperature jump has a greater influence
on heat transfer rate than velocity slip, thus the heat
transfer decreases with increasing rarefaction.
It can be seen from all the figures that Nusselt number
increases by increasing Reynolds number.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
3 Re0.25
 Re=0.5
Re1=l
28 0 Re=2
0 Re=3
26
S0o
24
22
2
18
16
002 004 006 008 01
Kn
Figure: Mean Nusselt number vs. Knudsen number, AT=200 K
3
 Re=0.25
 Re=0.5
Re 1
28 0 Re2
2 Re=2
SRe=3
26
24 0 0 0 0
22
2
18
16
002 004 006 008 01
Kn
Figure: Mean Nusselt number vs. Knudsen number, AT=400 K
r Re=0.25
SRe=0.5
2 8
0 Re=2
< Re=3
24
2
18
16
002 004 006 008 01
Kn
Figure4: Mean Nusselt number vs. Knudsen number, AT=600 K
Conclusions
In the present work, the incompressible slip flow regime
past an unconfined, stationary, impermeable, solid and
spherical aerosol particle with variable properties was
simulated. Gas slip on the particle surface was treated
numerically by the imposition of velocity slip and
temperature jump boundary conditions. The simulations
were implemented over a range of Reynolds numbers from
the Stokesian regime up to threshold of compressibility
where the flow stays roughly steady. We explore the effects
of three variables, Reynolds number, Knudsen number and
temperature difference between particle and gas, on heat
transfer from a sphere in slip flow regime. It was
concluded that velocity slip and temperature jump affect
the heat transfer in opposite ways: a large slip on the wall
increases the convection along the surface. On the other
hand, a large temperature jump decreases the heat transfer
by reducing the temperature gradient at the wall.
Therefore, neglecting temperature jump will result in the
overestimation of the heat transfer coefficient. When the
temperature difference between particle and ambient
medium is small, the velocity slip has a greater influence
on heat transfer rate than temperature jump and thus
Nusselt number increases with increasing Knudsen number
but, when the temperature difference increases the
temperature jump has greater influence and heat transfer
decreases with increasing Knudsen number.
References
Barber, R.W. and Emerson, D.R. Analytic solution of low
Reynolds number slip flow past a sphere. Centre
for Microfluidics, Department of Computational Science
and Engineering, CLRC Daresbury Laboratory,
Daresbury, Warrington, WA4 4AD, 2001a.
Chomaz, J.M., Bonneton, P and Hopfinger, E.J. (1993) The
structure of the near wake of a sphere moving horizontally
in a stratified fluid. J. Fluid Mech., 254, pp. 121.
Christlieb, A.J., Nicholas, W., Hitchon, G., Boyd, I.D. and
Sun, Q. (2004) Kinetic description of flow past a
microplate. J. Computational Physics, 195, pp. 508527.
Constantinescu, G.S. and Squires, K.D. (2000) LES and
DES investigations of turbulent flow over a sphere. AIAA
paper 20000540.
Cunningham, E. (1910) On the Velocity of Steady Fall of
Spherical Particles through Fluid Medium. Proc. Royal
Soc. (London) A, 83, pp. 357365.
Davis, R.W., Noye, J. and Fletcher, C. (1984) Finite
difference methods for fluid flow, Computational
Techniques and Applications., Eds., Elsevier, pp. 5169.
Eckert, E.R.G. & R.M. Drake Jr., 1972 Analysis of Heat &
Mass Transfer, New York: McGrawHill Inc., pp. 467486.
Fluent, Inc., 2001. Fluent 6.0.12 User's Guide. Fluent, Inc.,
Centerra Resource Park, 10 Cavendish Court, Lebanon,
NH 03766.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Freitas, C.J., Street, R.L., Findikakis, A.N. and Koseff, J.R.
(1985) Numerical simulation of threedimensional flow
in a cavity. Int. J. Numer. Methods Fluids, 5, pp. 561575.
Fremerey, J.K. (1982) Spinning rotor vacuum gauges.
Vacuum, 32, pp. 685690.
GadelHak, M. (1999) The fluid mechanics of
microdevices The Freeman Scholar Lecture. J. of Fluids
Engineering, 121, pp. 533.
Giors, S., Subba, F. and Zanino, R. (2005) NavierStokes
modeling of a Gaede pump stage in the viscous and
transitional flow regimes using slipflow boundary
conditions. J. Vac. Sci. Technol. A, Vol. 23, No. 2, pp. 336
346.
Jain, V and Lin, C.X. (2006) Numerical modeling of three
dimensional compressible gas flow in microchannels. J.
Micromech. Microeng., 16, pp. 292302.
Ji, W. and Wang, P.K. (1990) Numerical simulation of three
dimensional unsteady viscous flow past fixed hexagonal ice
crystals in the airPreliminary results. Atmos. Res., 25, pp.
539557.
Ji, W. and Wang, P.K. (1991) Numerical simulation of three
dimensional unsteady viscous flow past finite cylinders in
an unbounded fluid at low intermediate Reynolds numbers.
Theor. Comput. Fluid Dyn., 3, pp. 4359.
Jones, D.A. and Clarke, D.B. (2003) An Evaluation of the
FIDAP Computational Fluid Dynamics Code for the
Calculation of Hydrodynamic Forces on Underwater
Platforms. Maritime Platforms Division, Platforms Sciences
Laboratory, DSTOTR1494.
Kawahara, A., Chung, PM.Y and Kawaji, M. (2002)
Investigation of twophase flow pattern, void fraction and
pressure drop in a microchannel. International Journal of
Multiphase Flow, 28, pp. 14111435.
Knudsen, M. and Weber, S. (1911)Ann. D. Phys., 36, 981.
Lee, S. (2000) A numerical study of the unsteady wake
behind a sphere in a uniform flow at moderate Reynolds
numbers. J. Computers & Fluids, 29, pp. 639667.
Leonard, B. P and Mokhtari, S. (1990) ULTRASHARP
Nonoscillatory Convection Schemes for HighSpeed
Steady Multidimensional Flow. NASA Lewis Research
Center, NASA TM 12568 (ICOMP9012).
Leonard, B.P (1979) Adjusted quadratic upstream
algorithms for transient incompressible convection. A
Collection of Technical Papers. AIAA Computational Fluid
Dynamics Conference, AIAA Paper 791469.
Liu, HC. F., Beskok, A., Gatsonis, N. and Kamiadakis,
G.E. (1998) Flow past a microsphere in a pipe: effects of
rarefaction, DSCVol. 66, MicroElectroMechanical
Systems (MEMS). ASME, pp. 445452.
Morifiigo, J.A., Quesada, J.H. and Requena, F.C. (2007)
Slipmodel Performance for Underexpanded Microscale
Rocket Nozzle Flows. 8th International Symposium on
Experimental and Computational Aerothermodynamics of
Internal Flows Lyon,. Paper reference : ISAIF80026.
Moshfegh, A. (2009), A novel surfaceslip correction for
microparticles motion, Colloids and Surfaces A:
Physicochem. Eng. Aspects, Vol. 345, pp. 112120.
Multiphase Flow Handbook (2006), Ird Edition. Taylor and
Francis, Florida (Edited by Crowe, C.T.).
Ogedengbe, E.O.B., Naterer, G.F. and Rosen, M.A. (2006)
Slipflow irreversibility of dissipative kinetic and
internal energy exchange in microchannels. J. Micromech.
Microeng., 16, pp. 21672176.
Oseen, C.W. (1910) Uber die Stokes'sche formel und Uber
eine verwandte aufgabe in der hydrodynamic. Ark. f.
Math. Astron. och. Fys., 6, no.29.
Peters, T.M. and Leith, D. (2004) Particle Deposition in
Industrial Duct Bends. Ann. occup. Hyg., pp. 18.
Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery,
B.P (1993) Numerical Recipes in C, The Art of Scientific
Computing Cambridge University Press, Cambridge,
England.
Press, W.H. and William, H. et al. (1988) Numerical Recipes
in C; The Art of Scientific Computing Cambridge
University Press, Cambridge, UK. Press, W.H., Teukolsky,
S.A., Vetterling, W.T. and Flannery,
B.P (2002) Numerical Recipes in C++; The Art of Scientific
Computing Cambridge University Press, Cambridge, UK, p.
691.
Rader, D.J. (1990) Momentum slip correction factor for
small particles in nine common gases. J. Aerosol Sci., 21,
pp. 161168.
Reich, G. (1982) Spinning rotor viscosity gauge: A transfer
standard for the laboratory or an accurate gauge for vacuum
process control. J. Vacuum Science and Technology, 20, pp.
11481152.
Sakamato, H. and Haniu, H. (1990) A study of vortex
shedding from spheres in an uniform flow. J. Fluid
Engineering, 112, pp. 386392.
Schaaf, S.A. & Chambre, PL. (1961) Flow of Rarefied
Gases Princeton University Press.
Shui, L., Eijkel, Jan C.T. and Albert van den Berg (2007)
Multiphase flow in micro and nanochannels. Sensors and
Actuators, 121, pp. 263276.
Soltani, M., Ahmadi, G., Ounis, H. and McLaughlin, B.
(1998) Direct simulation of charged particle deposition in a
turbulent flow. Int. J. Multiphase flow, 24, pp. 7792.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Taneda, S. (1956) Experimental Investigations of the Wake
Behind a Sphere at Low Reynolds Numbers. Journal of
Physics, Japan, Vol. 11, No. 10, pp. 11041108.
Thomas, L.B. and Lord, R.G. (1974) Comparative
measurements of tangential momentum and thermal
accommodations on polished and on roughened steel
spheres. Rarefied Gas Dynamics 8th, pp. 405412, ed. K.
Karamcheti, Academic Press, New York.
UDF (User Defined Functions) Manual, FLUENT 6.0.12,
Fluent Inc., November 2001.
Wang, X., Gidwani, A., Girshick, S.L. and McMurry, PH.
(2005) Aerodynamic Focusing of Nanoparticles: II.
Numerical Simulation of Particle Motion Through
Aerodynamic Lenses. J. Aerosol Science and Technology,
39, pp. 624636.
Zhang, Z., Kleinstreuer, C., Donohue, J.F. and Kim, C.S.
(2005) Comparison of micro and nanosize particle
depositions in a human upper airway model. J. Aerosol Sci.,
36,pp. 211233.
Li, Z., He, YL., Tang, GH. and Tao, WQ. (2007)
Experimental and numerical studies of liquid flow and heat
transfer in microtubes. Int. J. Heat and Mass Transfer, 50,
pp. 34473460.
Zuppardi, G. et al. (2007) Quantifying the Effects of
Rarefaction in High Velocity, SlipFlow Regime. Rarefied
Gas Dynamics: 25th International Symposium, Russia.
