Thermophoresis of an aerosol spheroid along its axis of revolution Huan J. Keha) and Yu C. Chang Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, Republic of China
Abstract The thermophoretic motion of a spheroidal particle freely suspended in a gaseous medium prescribed with a uniformly temperature gradient along the axis of revolution of the particle is studied theoretically in the steady limit of small Peclet and Reynolds numbers.
The Knudsen
number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the particle surface.
The general solutions
in prolate and oblate spheroidal coordinates can be expressed in infinite-series forms of separation of variables for the temperature distribution and of semi-separation of variables for the stream function. The jump/slip boundary conditions on the particle surface are applied to these general solutions to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary collocation method or explicit formulas derived analytically. Numerical results for the thermophoretic velocity of the spheroidal particle are obtained in a broad range of its aspect ratio with good convergence behavior for various cases.
For the axisymmetric
thermophoresis of an aerosol spheroid, prolate or oblate, with no temperature jump and frictional slip at its surface, our results agree excellently with the analytical solution obtained previously. The agreement between our results and the available numerical solutions obtained by using a singularity method is also very good.
For most practical cases of a spheroid with a specified
aspect ratio, the thermophoretic mobility of the particle is not a monotonic function of its relative jump/slip coefficients and thermal conductivity.
The axisymmetric thermophoretic mobilities of a
prolate spheroid and of an oblate spheroid with large aspect ratios can be much greater and smaller, respectively, than that of a sphere with the same equatorial radius.
a)
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I. INTRODUCTION Thermophoresis refers to the motion of an aerosol particle in response to a temperature gradient. This phenomenon was first described in 1870 by Tyndall, who observed the removal of dust particles from air in the vicinity of hot surfaces [1].
Being a mechanism for the capture of
aerosol particles on cool surfaces, thermophoresis is of considerable importance in many practical applications, such as sampling of aerosol particles [2], cleaning of air [3], scale formation on surfaces of heat exchangers [4], modified chemical vapor deposition [5], manufacturing of microelectronics [6], nuclear reactor safety [7], and removal of soot particles for combustion exhaust gas systems [8]. The thermophoretic effect can be explained in part by appealing to the kinetic theory of gases [9]. The higher-energy gas molecules in the hot regions impinge on the particle with greater momenta than molecules coming from the cold regions, thus leading to the migration of the particle in the direction opposite to the temperature gradient.
It is convenient to express the
thermophoretic velocity of an isolated spherical particle in a constant temperature gradient T as U 0 M T T ,
(1)
where the negative sign indicates that the particle motion is in the direction of decreasing temperature.
The thermophoretic mobility M T depends on the magnitude of the Knudsen
number, Kn l / b , where l is the mean free path of the gas molecules and b is a characteristic linear dimension of the particle. In the free molecule regime ( l / b 1), the velocity distribution of the incoming gas molecules may be taken to be uninfluenced by the small particle and given by the Maxwell or Chapman-Enskog distributions [10].
Under this assumption, the thermophoretic mobility was
found to be
MT
6 (8 d ) f T0
,
(2)
where is the fluid viscosity, f is the fluid density, and T0 is the bulk-gas absolute temperature at the particle center in the absence of the particle (or the mean gas temperature in the vicinity of the particle).
The theory adopts the usual assumption that a fraction d of the gas
molecules colliding with the particle is reflected diffusely (thermally) with a Maxwellian distribution and the remaining fraction ( 1 d ) is reflected specularly.
The value of d is
usually about 0.9 [2]. Note that f T0 is constant for an ideal gas at constant pressure and the thermophoretic mobility for the “small particle” regime is independent of particle size. 2
In the “large particle” regime ( l / b 1), the fluid flow may be described by a continuum model and the thermophoretic force arises from an induced thermal slip along the particle surface due to the existence of a tangential temperature gradient at the surface.
Utilizing the gas kinetic
theory, Maxwell [11] predicted that a tangential temperature gradient sT at a gas-solid surface would cause a thin layer of gas (known as the Knudsen layer) adjacent to the surface to move (as a thermoosmotic flow), with the relative velocity at the outer edge of the layer being v ( s ) Cs
sT . f T
(3)
The thermal slip coefficient C s was found to be 3/4 by Maxwell on the assumption that the distribution function in the bulk of the gas held all the way to the solid surface.
Note that the
thermal slip velocity v (s ) is directed toward the high temperature side. By using the Maxwellian thermal slip velocity in Eq. (3), which gives the coupling between temperature and velocity fields, as well as the effects of temperature jump and frictional slip at the particle surface and solving the equation of continuum fluid motion incorporating with the heat conduction in the gas and particle at low Reynolds and Peclet numbers, Brock [12] derived an expression for the thermophoretic mobility of a suspended aerosol sphere as
MT
2Cs (k k p C t l / b)
f T0 (1 2C m l / b)(2k k p 2k p C t l / b)
.
(4)
Here k and k p are the thermal conductivities of the gas and particle, respectively, C t and C m are the dimensionless coefficients on the order of unity accounting for the temperature jump and frictional slip, respectively, at the particle surface and must be determined experimentally for each gas-solid system. Satisfactory agreement of the prediction by Eq. (4) with experiments [13,14] has been obtained. Derjaguin et al. [15,16] presented the experimental data of the thermophoretic mobility for a variety of aerosols, which are in good agreement with Eq. (4) with C m 0 , C s 3 / 2 and a suitable selection of the coefficient C t .
A set of kinetic-theory values for complete energy and momentum
accommodations of maxwellian/hard-sphere molecules appear to be Cs 1.17 , C t 2.18 and C m 1.14 [17].
These slip and jump coefficients have also been evaluated for the Lennard-Jones
and n(r ) 6 potentials by using the Chapman-Enskog solutions for the linearized Boltzmann equation and found to depend to various extents on the intermolecular force law [18].
Recently,
kinetic-theory values of the slip and jump coefficients have been obtained accurately under various conditions [19-22].
For example, for the case of complete thermal and momentum 3
accommodations of gas molecules with appropriate interaction potentials, a set of these coefficients are specified by C s 1.02 , C t 2.06 and C m 1.10 [22].
According to Eqs. (1) and (4),
particles with large thermal conductivity and small Knudsen number (say, k p / k 100 and l / b 0.01 ) will migrate by thermophoresis at velocities of 10-50 μm/s in temperature gradients
on the order of 100 K/cm, such gradients are easily attainable in thermal boundary layers. Usually, aerosol particles are not spherical and it is interesting to examine the effect of particle shape on thermophoresis [23,24]. The thermophoretic motion of a long circular cylinder in the direction normal to its axis has been studied with the consideration of the effects of temperature jump, thermal slip, and frictional slip at the particle surface [25,26]. On the other hand, the axisymmetric thermophoresis of a spheroidal particle along its axis of revolution without temperature jump and frictional slip at its surface was also analyzed [27,28]. This analysis has been generalized to a spheroid [29] and a particle departing slightly in shape from a sphere [30] with an arbitrary orientation relative to the imposed temperature gradient.
Recently, the
thermophoresis of a slightly deformed sphere with the effects of temperature jump, thermal slip and frictional slip was investigated, and an explicit expression for the thermophoretic velocity was obtained to the first order in the small parameter characterizing the deformation [31].
Although
the thermophoretic motion of a general axisymmetric particle with slip/jump surface conditions along its axis of revolution was numerically examined to some extent by using a method of internal singularity distributions [32], the problem of thermophoresis of a general spheroidal particle with all the effects of temperature jump, thermal slip, and frictional slip at the particle surface has not been analytically solved yet, mainly due to the fact that, if the temperature jump and/or frictional slip is included, a separable solution of the fluid stream function is not feasible for the prolate and oblate spheroidal coordinate systems [27-29]. In this work, we present an analysis of the steady thermophoresis of an aerosol spheroid along its axis of revolution with the temperature jump, thermal slip and frictional slip at the particle surface.
A separation-of-variable general solution in the form of infinite series in spheroidal
coordinates is used for the temperature distribution and a semi-separable general solution developed by Dassios et al. [33,34] is used for the stream function of the axisymmetric creeping flow. The thermophoretic velocity of the particle as a function of the physical and surface properties of the particle-gas system and the aspect ratio of the spheroid can be expressed in an asymptotic but explicit form and calculated numerically using a boundary-collocation method convergent for a broad range of the aspect ratio.
Our results show excellent agreement with the available analytical
and numerical solutions obtained previously. 4
II. ANALYSIS We consider the steady thermophoresis of a spheroidal particle along its axis of revolution in an unbounded gaseous medium, as shown in Fig. 1.
Here, right-handed circular cylindrical
coordinates ( , , z ) and bifocal coordinates ( , , ) [35] are established such that the surface of the spheroid is represented by 0 or z2 2 1, a2 b2
(5)
where a and b are the half-length along the axis of revolution and the equatorial radius, respectively, of the spheroid.
A linear temperature field T (z ) with a uniform thermal gradient E e z
( T , where e z is the unit vector in the z direction and E is taken to be positive) is prescribed in the surrounding fluid far away from the particle.
It is assumed that the Knudsen
numbers l / a and l / b are small so that the fluid flow is in the continuum regime and the Knudsen layer at the particle surface is thin in comparison with the linear dimensions of the particle. Before determining the thermophoretic velocity of the particle, the fluid velocity field needs to be found.
Because the boundary condition of the velocity field is coupled with the temperature
gradient at the particle surface, it is necessary to determine the temperature distribution first. 2.1. Temperature Distribution The Peclet number of the system of thermophoresis is assumed to be small. Hence, the equation of energy governing the temperature distribution T for the fluid phase of constant thermal conductivity k is the Laplace equation,
2T 0 .
(6a)
For the temperature field Tp inside the particle of constant conductivity k p , one has
2Tp 0 .
(6b)
Owing to the axial symmetry, both T and Tp are functions of coordinates ( , z) or ( , ) only. The boundary conditions at the particle surface require that the normal component of heat flux be continuous and a temperature jump that is proportional to the normal temperature gradient [9,12] occur.
Also, the fluid temperature far from the particle approaches the undisturbed distribution.
Thus,
5
k
Tp T , kp
T Tp C t lh 2 1
T
on 0
(7a,b)
and T T T0 E c
as .
(8)
Here, ω and λ are the variables and c is a constant related to the bifocal-coordinate transformations, h c 1 (2 2 ) 1 / 2 is a metric coefficient in bifocal coordinates, and C t is the temperature jump coefficient about the surface of the particle.
At the temperature range of
300-400 K, k p =0.022-0.024 W m 1 K 1 for silica aerosol and k p =0.9-3 W m 1 K 1 for clay, soil, and stone, whereas k=0.026-0.033 W m 1 K 1 for air [36]. For prolate spheroids (b < a), the coordinate transformation used is
cosh , cos , z c cosh cos , c sinh sin , a c cosh 0 , b c sinh 0 ,
(9)
c ( a 2 b 2 )1 / 2 ,
whereas for oblate spheroids (b > a), the coordinate transformation becomes
i sinh , cos , z ic sinh cos , ic cosh sin , a ic sinh 0 , b ic cosh 0 ,
(10)
c i (b 2 a 2 )1 / 2 .
The origin (midpoint between the foci) of the bifocal coordinates (with 0 and 0 π ) has been set at the center of the spheroid, and the coordinate surface 0 [1 (b / a) 2 ]1 / 2 corresponds to the surface of the spheroid defined by Eq. (5). A general separation-of-variable solution of Eqs. (6a,b) in bifocal coordinates is [28]
T T0 E c d 2 j 1 ( )P2 j 1 ( ) ,
(11a)
j 1
Tp T0 E c q 2 j 1 ( )P2 j 1 ( ) ,
(11b)
j 1
where d n ( ) E n Pn ( ) Fn Qn ( ) q n ( ) E n Pn ( ) F n Qn ( )
for n 1, 3, 5, ....
(12a,b)
In the above equations, Pn and Q n are the Legendre polynomials of the first and second kinds, 6
respectively, of order n, and E n , Fn , E n , and F n are unknown coefficients to be determined. Because the temperature distribution is anti-symmetric about the equatorial plane z 0 , only the odd terms of the expansions in Eq. (11) are retained.
In accordance with Eq. (8) and the
requirement that the temperature is finite at any position inside the particle, we immediately find that E1 1 , E n 0 for n 3 , and F n 0 for all n in Eq. (12). Applying Eq. (11) to boundary condition (7a) at the particle surface, we obtain
[d j 1
2 j 1
(0 ) k * q 2 j 1 (0 )]P2 j 1 ( ) 0 ,
(13)
where k * k p / k , and a prime on d n ( ) and q n ( ) denotes a differentiation with respect to . Because of the orthogonal characteristic of the Legendre polynomials, Eq. (13) is equivalent to d n (0 ) k * q n (0 ) 0
for n 1, 3, 5, ...,
(14)
and the dependence on disappears. The substitution of Eq. (11) into boundary condition (7b) leads to b 20 1 [d 2 j 1 (0 ) q 2 j 1 (0 ) C d 2 j 1 (0 )]P2 j 1 ( ) 0 , c 20 2 j 1
* t
where C t* C t l / b , which is the temperature jump coefficient relative to 1 / Kn b / l .
(15)
The
unknown coefficients Fn and E n in Eq. (12) are to be determined using Eqs. (14) and (15). In the following, we present separately a boundary collocation method to obtain a numerical solution for the unknown coefficients Fn and E n and an analytical method to result in explicit formulas for these coefficients of leading orders. 2.1.1. Boundary collocation method To satisfy boundary conditions (14) and (15) exactly along the entire semi-elliptic generating arc of the spheroid in a meridian plane would require the solution of the entire infinite array of the unknown constants Fn and E n . However, the boundary collocation technique [37,38] enforces the boundary conditions at a finite number of discrete points on the particle’s quarter-elliptic longitudinal arc (from 0 to π / 2 , owing to the symmetry of the system geometry) and truncates the infinite series in Eqs. (11) and (15) into finite ones. The unknown constants in the terms of the finite series permit one to satisfy the exact boundary conditions at the discrete points on the particle surface. Thus, if the boundary is approximated by satisfying condition (15) at M 7
discrete points, then the infinite series are truncated after M terms, resulting in a system of 2 M simultaneous linear algebraic equations [including the first M equations of condition (14)]. This matrix equation can be solved by any of the standard matrix-reduction techniques to yield the 2 M unknown constants Fn and E n required in the truncated Eq. (11) for the temperature field. The accuracy of the truncation technique can be improved to any degree by taking a sufficiently large value of M . Naturally, the truncation error vanishes as M . 2.1.2. Analytical method On the other hand, an analytical solution for the leading unknowns Fn and E n required in Eq. (12) for the temperature distribution can be found. To simplify the boundary condition given by Eq. (15), we first expand the function (20 2 ) 1 / 2 into a Taylor series in exp( 0 ) with respect to 0 (only odd terms are nonzero) and then apply the recurrence relation
2 Pn ( ) a n Pn 2 ( ) bn Pn ( ) cn Pn 2 ( )
for n 1 ,
(16)
with an
n(n 1) , (2n 1)( 2n 1)
(17a)
bn
n 2 (2n 3) (n 1) 2 (2n 1) , (2n 1)( 2n 1)( 2n 3)
(17b)
cn
(n 1)( n 2) , (2n 3)( 2n 1)
(17c)
to result in 20 1 P ( ) t m ( 2 j 1) P2 j 1 ( ) . m 20 2 j 1
(18)
In the above expansion, the coefficients t mn are known functions of exp( 0 ) with the first ones given by Eq. (A.7) in Appendix A. Substituting Eq. (18) into Eq. (15) and using the orthogonal property of the Legendre polynomials, we obtain
d n (0 ) q n (0 ) C t*
b t (2 j 1)n d n (0 ) c j 1
for n= 1, 3, 5, …,
(19)
in which the dependence on disappears. If Eq. (11) for the temperature field is truncated after M terms, the first M equations of each of Eqs. (14) and (19) can be used to solve the 2 M unknown constants Fn and E n . Again, the truncation error disappears as M . 8
In Appendix A, the four algebraic equations
required to solve the unknown coefficients F1 , E 1 , F3 , and E 3 and their explicit solutions are given for a specific case that Eq. (11) is truncated after two terms ( M 2 ). Although we have also obtained the explicit solutions for the unknown coefficients Fn and E n for the more accurate cases of M 3 and M 4 , their formulas are not presented here for conciseness. 2.2. Fluid Velocity Distribution Having obtained the solution for the external temperature distribution on the particle surface which drives the thermophoretic migration, we can now proceed to find the fluid flow field. fluid is assumed to be incompressible and Newtonian.
The
Owing to the low Reynolds number
encountered in thermophoresis, the fluid motion is governed by the quasisteady fourth-order differential equation for viscous axisymmetric creeping flows, E 2 ( E 2Ψ ) 0 ,
(20)
in which the Stokes stream function Ψ ( , ) is related to the velocity components (with v 0 ) in bifocal coordinates by [34]
v v
1 c
2
2
2
1 c 2 2 2
Ψ , 1
(21a)
Ψ , 1 2
(21b)
2
and the Stokes operator E 2 has the form
1 2 2 2 2 E 2 2 [( 1) 2 (1 ) 2 ] . c ( 2 ) 2
(22)
There exist thermal and frictional slip velocities along the particle surface and the fluid flow vanishes far from the particle.
v 0 v
Cml
Hence, the boundary conditions for the fluid velocity are [12,31]
Cs 2 T h 1 f T0
1 Ψ Uc 2 (2 1)(1 2 ) 2
on 0 ,
as ,
(23a,b)
(24)
where is the fluid shear stress expressed as
[ 2 1
(hv ) 1 2 (hv )] ,
(25)
C s is the thermal slip coefficient defined in Eq. (3), Cm is the frictional slip coefficient, and U is
the thermophoretic velocity of the particle to be determined. The tangential temperature gradient 9
in Eq. (23b) can be obtained from Eq. (11a).
Note that Eqs. (23) and (24) take a reference frame
that the particle is at rest and the velocity of the fluid at infinity is the particle velocity in the opposite direction. A general solution of Eq. (20) in bifocal coordinates has been obtained by Dassios et al. [33,34,39] in a series expansion of semi-separable form as
Ψ ( , ) Uc 2 g 2 j ( )G2 j ( ) ,
(26)
j 1
where g 2 ( ) A1G1 ( ) C 2 G2 ( ) D2 H 2 ( ) A4 G4 ( ) B4 H 4 ( ) ,
(27a)
g n ( ) An Gn 2 ( ) Bn H n 2 ( ) C n Gn ( ) Dn H n ( ) An 2 Gn 2 ( ) Bn 2 H n 2 ( )
for n= 4, 6, 8, … ,
(27b)
G n ( ) and H n ( ) denote the Gegenbauer functions of the first and second kinds, respectively, of
order n and degree 1 / 2 , and An , Bn , C n , and Dn are unknown coefficients to be determined. Note that each individual term in Eq. (26) is not a solution of Eq. (20), whereas the entire expansion, which is not a complete separation of the variables ω and λ , is.
Because the stream function is
symmetric about the equatorial plane z 0 , only the even terms of the expansion in Eq. (26) are Using Eq. (24), one obtains C 2 2 and An C n 0 for n 4 in Eq. (27).
retained.
The application of Eqs. (21a) and (26) to boundary condition (23a) at the particle surface leads to
g j 1
2j
(0 ) P2 j 1 ( ) 0 .
(28)
From the orthogonal characteristic of the Legendre polynomials, Eq. (28) is equivalent to g n (0 ) 0
for n= 2, 4, 6, …,
(29)
and the dependence on disappears. The substitution of Eqs. (11a), (21), (25)-(27) and (29) into boundary condition (23b) results in
[Uh2 j (0 , )G2 j ( ) j 1
CsE ( 2 2 ) 3 / 2 (1 2 ) 0 d 2 j 1 (0 ) P2 j 1 ( )] 0 , f T0 20 1
(30)
where
hn ( , ) C m*
( 2 2 ) 3 / 2 b 2 [( 2 ) g n ( ) 2g n ( )] g n ( ) for n=2, 4, 6, …, (31) c 2 1 10
C m* C m l / b , a prime on Pn ( ) or g n ( ) denotes a differentiation with respect to its argument,
and d n ( ) defined by Eq. (12a) have been determined in the previous subsection. The unknown coefficients A1 , D2 , B4 , D4 , B6 , etc. in Eq. (27) are to be determined using Eqs. (29) and (30). Analogous to the solution for the temperature field presented in the previous subsection, both the boundary collocation method to obtain a numerical solution for the unknown coefficients A1 , D 2 , B 4 , D 4 , B6 , etc. and the analytical method to obtain explicit formulas for these coefficients
of leading orders are described below. 2.2.1. Boundary collocation method The boundary-collocation technique enforces boundary conditions (29) and (30) at a finite number of discrete points on the particle’s quarter-elliptic longitudinal arc (from 0 to π / 2 ) and truncates the infinite series in Eqs. (26) and (30) into finite ones.
If the boundary is
approximated by satisfying condition (30) at N discrete points, then the infinite series are truncated after N terms, leading to a system of 2 N simultaneous linear algebraic equations [including first N equations of condition (29)]. This matrix equation can be solved to yield the 2 N unknown constants A1 , D 2 , B 4 , D 4 , B6 , etc. required in the truncated Eq. (26) for the flow
field.
Actually, the system resulting from the boundary collocation has 2 N 1 unknown
constants but only 2 N algebraic equations. unknown constant B2 N 2 is negligible.
Even so, this problem can be solved if the last
Of course, the more terms are retained in Eq. (26), the
less error will be caused by omitting the coefficient B2 N 2 .
The truncation (including omission of
the coefficient B2 N 2 ) error vanishes as N . 2.2.2. Analytical method To simplify the boundary condition given by Eqs. (30) and (31), we expand the function (20 2 ) 3 / 2 into a Taylor series in exp( 0 ) with respect to 0 (only odd terms are
nonzero) and then apply the recurrence relation [33,34]
2 Gn ( ) n Gn2 ( ) n Gn ( ) n Gn2 ( )
for n 2 ,
(32)
with (n 3)( n 2) , (2n 3)( 2n 1) (n 1)( n 2) n , (2n 1)( 2n 1)
n
n
(33a) (33b)
(2n 2 2n 3) , (2n 3)( 2n 1)
(33c) 11
to result in (20 2 ) 3 / 2
20 1
Gm ( ) s m ( 2 j ) G2 j ( ) .
(34)
j 1
Here, the coefficients smn are known functions of exp( 0 ) with the first ones given by Eq. (B.6) in Appendix B.
Substituting the relation
(1 2 ) Pm ( ) m(m 1)Gm1 ( )
for m =1, 3, 5, …
(35)
and Eq. (34) into Eqs. (30) and (31), applying Eq. (32) again, and using the orthogonal property of the Gegenbauer polynomials, we obtain b UC m* [(20 n ) g n (0 ) 20 g n (0 ) n 2 g n 2 (0 ) n 2 g n 2 (0 )] c
s( 2 j ) n [Ug 2 j (0 ) j 1
CsE (2 j 1)( 2 j )d 2 j 1 (0 )] f T0
for n =2, 4, 6, …,
(36)
in which the dependence on disappears. If Eq. (26) for the fluid flow is truncated after N terms, the first N equations of each of Eqs. (29) and (36) can be used to solve the 2 N unknown constants A1 , D 2 , B 4 , D 4 , B6 , etc. Again, the constant B2 N 2 is neglected in the solution and the accuracy will be acceptable when the value of N is sufficiently large.
In Appendix B, the four algebraic equations required to
solve the unknown coefficients A1 , D2 , B4 , and D4 and their explicit solutions are given for a specific case that Eq. (26) is truncated after two terms ( N 2 ).
Although we have also obtained
the explicit solutions for the unknown coefficients for the more accurate case of N 3 and N 4 , their formulas are not presented here.
2.3. Derivation of the particle velocity The hydrodynamic force Fe z acting on the particle can be determined from [35,40] F 8πc lim (
Ψ
2
U ). 2
(37)
The substitution of Eqs. (26) and (27) into Eq. (37) gives F 4 πcUA1 .
(38)
This expression shows that only the lowest-order coefficient A1 contributes to the drag force exerted on the particle by the fluid.
This leading coefficient normally is the most accurate result
obtainable from the boundary-collocation/truncation technique. Since the particle is freely suspended in the surrounding fluid, the net force acting on the particle must vanish.
Applying this constraint to Eq. (38), one obtains 12
A1 0 .
(39)
To determine the thermophoretic velocity U of the particle, Eq. (39) and the 2 N algebraic equations resulting from Eqs. (29) and (30) for the boundary collocation scheme or from Eqs. (29) and (36) for the analytical solution are to be solved simultaneously.
Similar to the thermophoretic
velocity of an isolated sphere given by Eqs. (1) and (4), the value of U is proportional to the quantity CsE / f T0 and dependent on the dimensionless parameters k * , C t* , and C m* (in addition to the aspect ratio a/b). If the particle velocity in Eq. (24) is disabled (i.e., U=0 is set), then the force obtained from Eq. (38) can be taken as the thermophoretic force exerted on the particle due to the imposed temperature gradient T . This force can be expressed as F 6bU 0 F * ,
(40)
where U 0 is the corresponding thermophoretic velocity of a spherical particle of radius b given by Eqs. (1) and (4) and F * is the normalized magnitude of the thermophoretic force.
The value of
F * also equals f *U /U 0 , where f * is the dimensionless Stokes resistance coefficient of the slip spheroid migrating along its axis of revolution driven by a body force in the absence of the temperature gradient [39] and U is the thermophoretic velocity of the particle obtained from Eq. (39).
III. RESULTS AND DISCUSSION In this section, we first present the numerical results of the thermophoretic velocity of a spheroidal particle along its axis of revolution obtained by using the boundary collocation method described in Subsections 2.1.1 and 2.2.1 for the temperature and velocity distributions, respectively. Then, the leading-order asymptotic solutions of the thermophoretic velocity resulting from the analytical method introduced in Subsection 2.1.2 and 2.2.2 for the temperature and velocity fields, will be given and compared with the convergent collocation solutions. 3.1. Boundary collocation solutions The system of linear algebraic equations to be solved for the coefficients Fn and E n is constructed from Eqs. (14) and (15), whereas that for the coefficients A1 , D2 , B 4 , D4 , B6 , etc. is composed of Eqs. (29) and (30). When selecting the points along the quarter-elliptic generating arc of the spheroid where the boundary conditions (15) and (30) are exactly satisfied, the first point that should be chosen is π 2 (or 0 ), since this point defines the projected area of the particle normal to the direction of motion.
In addition, the point 0 (or 1 ) is also 13
important.
However, an examination of the systems of linear algebraic equations in the truncated
form of Eqs. (15) and (30) shows that the matrix equations become singular if these points are used. To overcome this difficulty, these points are replaced by closely adjacent points, i.e., and π 2 [37].
Additional points along the boundary are selected to divide the quarter-elliptic arc
of the spheroid into segments with equal angles in . The optimum value of in this work is found to be 0.01 , with which the numerical results of the thermophoretic velocity of the particle converge satisfactorily. In our continuum-with-slippage analysis given in the previous section, the Knudsen number l / b of the system should be smaller than about 0.1.
As mentioned in the first section, a set of
well adapted values for the temperature jump and frictional slip coefficients under the condition of complete energy and momentum accommodations are 2.18 (or 2.06) and 1.14 (or 1.10), respectively. Consequently, the normalized coefficients C t* and C m* (proportional to l / b ) must be restricted to be less than unity.
For convenience we will use the ratio C t* / C m* 2 (a rounded value to
2.18/1.14=1.91 or 2.06/1.10=1.87) throughout the article to ease the parametric study, without loss in reality or generality.
On the other hand, the thermal conductivity of an aerosol particle is
usually greater than that of the ambient gas.
Thus, the value of the relative conductivity k * will
exceed unity under most practical situations. In Tables 1 and 2, the boundary collocation results of the axisymmetric thermophoretic mobility U /U 0 of a prolate spheroid and an oblate spheroid, respectively, normalized by that of a spherical particle with radius equal to the maximum cross-sectional radius of the spheroid are presented for several representative cases of the axial-to-radial aspect ratio a / b with various values of the parameters k * and C m* . All of the results were obtained by increasing the numbers of collocation points M and N until the convergence of four or more significant digits is achieved. The exact analytical solutions for the axisymmetric thermophoresis of a spheroid with no temperature jump and frictional slip ( C t* C m* 0 ) [29] are also given in these tables for comparison.
It can be seen that our results from the boundary collocation method agree
excellently with the exact solutions in this limit.
In general, the convergence behavior of the
collocation method is quite good, even for the relatively difficult case of large or small values of a/b.
Recently, we investigated the problem of thermophoresis of an axisymmetric particle with temperature jump, thermal slip, and frictional slip at the surface along its axis of revolution using a method of internal singularity distribution [32]. 14
The numerical results of the normalized
thermophoretic velocity U /U 0 for a spheroid obtained in that work (only available for the aspect ratio in the range 0.5 a / b 2 ) are also listed in Tables 1 and 2 for comparison. As one can see in these tables, our collocation results agree very well with the corresponding values resulting from the singularity method. Note that, when the singularity method is used, the numerical solutions only converge in a narrow range of the numbers of the retained terms in the infinite-series general solution and of the divided segments for the distributed singularities. However, this convergence difficulty can be avoided if we adopt the current method with Eq. (11) for the general solution of the temperature distribution in an expansion form of separation of variables and Eq. (26) for the general solution of the fluid flow field in an expansion form of semi-separation of variables. Our numerical results of the normalized thermophoretic velocity U /U 0 of a prolate spheroid and an oblate spheroid with C t* 2C m* along their axes of revolution as functions of the aspect ratio a / b for several different values of the thermal conductivity ratio k * and the slip parameter C m* are plotted in Figs. 2 and 3, respectively.
The cases of k * 1 , which are not likely to exist in
practice, are considered here for the sake of numerical comparison.
For specified values of k * ,
C t* , and C m* , the value of the normalized axisymmetric thermophoretic velocity U /U 0 equals
unity at a / b 1 as expected and in general increases with an increase in a / b .
This behavior is
understood since the fraction of the particle surface with the thermal slip of the fluid in the axial direction, which drives the movement of the particle, increases with the increase of a / b .
When
C m* is small but finite and k * is large (e.g., C m* 0.01 and k * 100 , as illustrated in Fig. 3a),
interestingly, U /U 0 for an oblate spheroid may not be a monotonic function of a / b and its value can be greater than unity.
Note that the values of U /U 0 vanishes for an oblate spheroid with
a / b 0 and can be much greater than unity for a prolate spheroid if both of the values of a / b
and k * are large. Figures 4a and 4b show the effects of variation in k * on the normalized axisymmetric thermophoretic velocity U /U 0 of a prolate spheroid and an oblate spheroid, respectively, for the case of C t* 2C m* 0.02 .
For a prolate spheroid with finite values of C t* , C m* , and a / b , the
value of U /U 0 increases with an increase in k * when k * is small, reaches a maximum at some moderate value of k * , and then decreases with a further increase in k * .
On the contrary, for an
oblate spheroid with finite values of C t* , C m* , and a / b , the value of U /U 0 decreases with an increase in k * when k * is small, reaches a minimum at some moderate value of k * , and then 15
increases with a further increase in k * . This maximum or minimum occurs at a higher value of
k * if the value of C m* ( C t* / 2) is smaller or the value of a / b is greater. When C m* is small but finite and k * is very large (e.g., C m* 0.01 and k * 1000 ), U /U 0 for a prolate spheroid may not be a monotonic function of a / b and its value can be less than unity, and again, U /U 0 for an oblate spheroid can increase with a decrease in a / b and its value can be greater than unity. ( C t* / 2) on the normalized axisymmetric
The effects of varying the parameter C m*
thermophoretic velocity U /U 0 of a prolate spheroid and an oblate spheroid are illustrated in Figs. 5a and 5b, respectively, for the practical case of k * 100 .
For a prolate spheroid with a fixed
value of a / b , the value of U /U 0 decreases with an increase in C m* when C m* is small, reaches a minimum at some moderate value of C m* , and then increases with a further increase in C m* .
On
the contrary, for an oblate spheroid with a specified value of a / b , the value of U /U 0 increases with an increase in C m* when C m* is small, reaches a maximum at some moderate value of C m* , and then decreases with a further increase in C m* .
This minimum or maximum occurs at a higher
value of C m* if the value of k * or a / b is greater.
Since the effects of the four parameters k * ,
C t* , Cm* , and a / b on the thermophoretic velocity of a spheroid interact one another in a quite
complicated manner, it would be difficult to provide a concise physical analysis for the above observations from Figs. 2-5. 3.2. Asymptotic analytical solutions The thermophoretic velocity of an aerosol spheroid along its axis of revolution can also be solved explicitly using Eqs. (14) and (19) for the temperature distribution and Eqs. (29) and (36) for the velocity field, as discussed in sections 2.1.2 and 2.2.2. The infinite series in Eqs. (19) and (36) are truncated into finite M and N terms, respectively.
Although the truncation error disappears as
M and N , only the approximate analytical solutions with small values of M and N can be obtained in practice.
In Tables 3 and 4, we present the asymptotic results of the normalized
axisymmetric thermophoretic velocity U /U 0 of a prolate spheroid and an oblate spheroid with
M N 2 , 3, and 4 for several representative cases of the aspect ratio a / b with various values of the parameters k * and C m* ( C t* / 2) , whereas only the formulas leading to this velocity for the case of M N 2 are given in Appendixes A and B for conciseness.
The relevant boundary
collocation solutions obtained in the previous subsection are also listed in Tables 3 and 4 for comparison.
One can see that the larger the values of N and M are, the more accurate the results 16
will be.
If M=N=4 is chosen, the analytical solution agrees very well with the exact numerical
solutions for cases of a moderate aspect ratio in the range 0.5 a / b 2 .
When the aspect ratio
a / b deviates much from unity, the agreement between the approximate values of U /U 0 with
M=N=4 and the convergent collocation results becomes not as good as the cases with a / b close to unity.
For the limiting case of C m* C t* 0 , only the first term of each infinite series in Eqs. (11)
and (26) is nonzero, and a closed-form expression for the thermophoretic velocity U can be derived exactly from Eqs. (14), (19), (29), and (36) [29].
IV. CONCLUDING REMARKS In this paper, the boundary-collocation numerical solutions and asymptotic analytical solutions for the thermophoresis of a rigid spheroidal particle with a temperature jump, a thermal slip, and a frictional slip at its surface along its axis of revolution in a viscous fluid (e.g., a slightly rarefied gas) are obtained.
The general solution for the temperature distribution is derived as two
infinite series expansions by using separation of variables in spheroidal coordinates, whereas the general solution for the Stokes stream function is expressed as another infinite series expansion in a semi-separable form, in which the unknown coefficients can be determined analytically to their leading orders and numerically with excellent convergence for a wide range of the particle’s axial-to-radial aspect ratio a / b .
It is found that the thermophoretic mobility of a prolate or oblate
spheroid along its axis of revolution normalized by that of a sphere with identical equatorial radius, U /U 0 , in general increases with an increase in the value of a / b ; the exceptions may occur when
the relative slip/jump coefficients at the particle surface, C m* and C t* , are small but finite and the relative thermal conductivity of the particle, k * , is large.
The value of U /U 0 approaches zero
for an oblate spheroid with a / b 0 and can be much greater than unity for a prolate spheroid with a large value of a / b .
For most practical cases of a spheroid with a fixed value of a / b , the
value of U /U 0 is not a monotonic function of the parameters k * , C t* and C m* . The results indicate that the shape and the relative physical and surface properties of a spheroidal particle can have significant effects on its thermophoretic behavior. It is worth repeating that an exact analytical solution for a problem of thermophoretic motion of a spheroid with slip/jump conditions at its surface in a viscous fluid is not feasible mainly because a complete separation-of-variable general solution for the Stokes flow does not exist in prolate or oblate spheroidal coordinates.
This difficulty has been resolved to a great extent in the
present study through the use of a method of semi-separation of variables for the stream function (it 17
turns out that the first term of the series solution for the stream function is sufficient if C t* C m* 0 ), and analytical solutions for the thermophoretic motion of a spheroid along its axis of
revolution are obtained with very good accuracy for all values of the parameters k * , C t* , and C m* within the aspect-ratio range 0.5 a / b 2 .
In addition, this method of semi-separation of
variables incorporated with a boundary collocation technique can be used to obtain numerical solutions for the same motion convergent and correct for arbitrary practical values of k * , C t* , C m* , and a / b . These collocation solutions are much superior to those resulting from a method of distributed internal singularities [32], in which the convergence behavior is relatively poor and no solution could be obtained outside the aspect-ratio range 0.5 a / b 2 .
ACKNOWLEDGMENT Part of this research was supported by the National Science Council of the Republic of China.
Appendix A. Analytical solution for the coefficients in Eq. (11) truncated after two terms When Eq. (11) for the temperature distribution is truncated after two terms ( M 2 ), T T0 E c[d1 ( ) P1 ( ) d 3 ( ) P3 ( )] ,
(A.1)
Tp T0 E c[q1 ( ) P1 ( ) q3 ( ) P3 ( )] ,
(A.2)
where d1 ( ) , q1 ( ) , d 3 ( ) and q3 ( ) are expressed by Eq. (12) with E1 1 and
E3 F 1 F 3 0 , the boundary conditions (14) and (19) for n 1 and 3 become F1Q1 (0 ) E 1k * 1,
(A.3)
F3Q3 (0 ) E 3 k * P3(0 ) 0 , b F1Q1 (0 ) ( E 1 1)0 C t* {t11 [ F1Q1 (0 ) 1] t 31 F3 Q3 (0 )} , c b F3 Q3 (0 ) E 3 P3 (0 ) C t* {t13 [ F1Q1 (0 ) 1] t 33 F3 Q3 (0 )} . c Here a prime on Pn ( ) or Qn ( ) denotes a differentiation with respect to , and 2 3 0 2 5 0 2 7 0 2 9 0 2 11 0 e e e e e ...) , 5 35 105 231 429 8 8 16 7 0 16 90 8 11 0 20 1( e 30 e 50 e e e ...) , 5 15 165 429 429 24 8 16 7 0 16 90 8 11 0 20 1( e 30 e 50 e e e ...) , 35 35 385 1001 1001 2 98 5 0 722 70 98 90 22 11 0 20 1(2e 0 e 30 e e e e ...) . 45 165 2145 1287 663 t mn are convergent for a positive value of 0 .
(A.4) (A.5) (A.6)
t11 20 1(2e 0
(A.7a)
t13
(A.7b)
t 31 t 33
Note that all
18
(A.7c) (A.7d)
Equations (A.3)-(A.6) are used to determine the coefficients F1 , E 1 , F3 , and E 3 , and their result in explicit forms is b F1 [ z1 z 2 (C t* ) 2 t13 t 31Q3 (0 )] Γ t , c E1
(A.8a)
1 [ F1Q1 (0 ) 1] , k*
(A.8b)
b F3 C t* t13 [0 Q1 (0 ) Q1 (0 )] Γ t , c
E3
(A.8c)
F3Q3 (0 ) , k * P3(0 )
(A.8d)
where b Γ t {z 2 [Q1 (0 )( z1 0 ) Q1 (0 )] (C t* ) 2 t13 t 31Q1 (0 )Q3 (0 )}1 , c 1 b z1 (1 * )0 C t* t11 , c k P ( ) b z 2 Q3 (0 ) Q3 (0 )[C t* t 33 * 3 0 ] . c k P3(0 )
(A.9) (A.10a) (A.10b)
Appendix B. Analytical solution for the coefficients in Eq. (26) truncated after two terms When Eq. (26) for the Stokes stream function is truncated after two terms ( N 2 ), Ψ Uc 2 [ g 2 ( )G2 ( ) g 4 ( )G4 ( )] ,
(B.1)
where g 2 ( ) and g 4 ( ) are expressed by Eq. (27) with C 2 2 and A4 C 4 A6 0 , the boundary conditions (29) and (36) for n 2 and 4 become g 2 (0 ) A1G1 (0 ) 2G2 (0 ) D2 H 2 (0 ) B4 H 4 (0 ) 0 ,
(B.2)
g 4 (0 ) B4 H 2 (0 ) D4 H 4 (0 ) B6 H 6 (0 ) 0 , b UC m* {(20 2 )[ 2 D2 H 2 (0 ) B4 H 4 (0 )] 20 [ A1 20 D2 H 2 (0 ) c B4 H 4 (0 )] 4 [ B4 H 2 (0 ) D4 H 4 (0 ) B6 H 6(0 )]}
(B.3)
CsE 2[ F1Q1 (0 ) 0 ]} f T0 C E s 42 {U [ B4 H 2 (0 ) D4 H 4 (0 ) B6 H 6 (0 )] s 12 F3Q3 (0 )} , f T0
s 22 {U [ A1 20 D2 H 2 (0 ) B4 H 4 (0 )]
b UC m* {(20 4 )[ B4 H 2 (0 ) D4 H 4 (0 ) B6 H 6(0 )] 20 [ B4 H 2 (0 ) c D4 H 4 (0 ) B6 H 6 (0 )] 2 [2 D2 H 2 (0 ) B4 H 4(0 )]} 19
(B.4)
CsE 2[ F1Q1 (0 ) 0 ]} f T0 C E (B.5) s 44 {U [ B4 H 2 (0 ) D4 H 4 (0 ) B6 H 6 (0 )] s 12 F3Q3 (0 )]} . f T0 Here a prime on H n ( ) denotes a differentiation with respect to . The coefficients F1 and s 24 {U [ A1 20 D2 H 2 (0 ) B4 H 4 (0 )]
F3 have been determined in Appendix A, and
s 22 s 24 s 42 s 44
1
1 9 81 0 71 30 6 50 18 7 0 ( e 3 0 e 0 e e e e ...) , 40 280 840 385 5005 20 1 8 1
3 1 12 36 50 4 7 0 ( e 0 e 0 e 3 0 e e ...) , 5 55 715 715 1 5 2 0
1
20 1
(
3 0 1 0 6 3 0 18 5 0 2 7 0 e e e e e ...) , 70 70 385 5005 5005
(B.6a) (B.6b) (B.6c)
1
1 1 117 0 31 3 0 16 5 0 48 7 0 ( e 3 0 e 0 e e e e ...) . (B.6d) 40 440 17160 715 12155 1 8 2 0
Note that all smn are convergent for a positive value of 0 . Equations (B.2)-(B.5) with the terms involving the coefficient B6 being neglected along with A1 0 [from Eq. (39)] are used to determine the thermophoretic velocity U, and its result in
explicit form is
U
CsE [ f c k 23 H 4 (0 ) k c f 34 H 2 (0 ) f c k 4 H 22 (0 ) f 2 k c H 42 (0 )] Γ , f T0
(B.7)
where Γ {( f1k 23 f 2 k13 ) H 4 (0 ) k 4 H 2 (0 )[ f1 H 2 (0 ) f 2 (1 20 )] f 34 k12 }1 , f 34 f 3 H 4 (0 ) f 4 H 2 (0 ) ,
C m* b 2 (0 2 )] , c C m* b 2 s 22 H 2 (0 ) [(0 2 ) H 2 (0 ) 20 H 2 (0 )] , c C* b s 42 H 2 (0 ) s22 H 4 (0 ) m [(20 2 ) H 4(0 ) 20 H 4 (0 ) 4 H 2(0 )] , c * C b s 42 H 4 (0 ) m 4 H 4(0 ) , c 2{s 22 [ F1Q1 (0 ) 0 ] 6s 42 F3Q3 (0 )} ,
(B.8) (B.9a)
f1 2[ s 22 0
(B.9b)
f2
(B.9c)
f3 f4 fc
(B.9d) (B.9e) (B.9f)
k12 k1 H 2 (0 ) k 2 (1 20 ) ,
(B.9g)
k13 k1 H 4 (0 ) k 3 (1 ) ,
(B.9h)
k 23 k 2 H 4 (0 ) k 3 H 2 (0 ) ,
(B.9i)
2 0
20
C m* b (B.9j) k1 2( s 24 0 2 ) , c C* b (B.9k) k 2 s 24 H 2 (0 ) m 2 H 2(0 ) , c C* b (B.9l) k 3 s 44 H 2 (0 ) s 24 H 4 (0 ) m [(20 4 ) H 2(0 ) 20 H 2 (0 ) 2 H 4(0 )] , c C m* b 2 (B.9m) k 4 s 44 H 4 (0 ) [(0 4 ) H 4(0 ) 20 H 4 (0 )] , c k c 2{s 24 [ F1Q1 (0 ) 0 ] 6 s 44 F3 Q3 (0 )} . (B.9n) In the limiting case of a / b ( 0 1 and b / c 0 ), F1 0 and F3 0 from Eqs.
(A.8a,c), k c k1 , f c f1 , and Eq. (B.7) (and the corresponding expressions for cases of N 2 ) predicts that U CsE / f T0 .
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[12] J. R. Brock, “On the theory of thermal forces acting on aerosol particles,” J. Colloid Sci. 17, 768 (1962). [13] C. F. Schadt and R. D. Cadle, “Thermal forces on aerosol particles,” J. Phys. Chem. 65, 1689 (1961). [14] W. Li and E. J. Davis, “Measurement of the thermophoretic force by electrodynamic levitation: Microspheres in air,” J. Aerosol Sci. 26, 1063 (1995). [15] B. V. Derjaguin, A. I. Storozhilova, and Ya. I. Rabinovich, “Experimental verification of the theory of thermophoresis of aerosol particles,” J. Colloid Interface Sci. 21, 35 (1966). [16] B. V. Derjaguin, Ya. I. Rabinovich, A. I. Storozhilova, and G. I. Shcherbina, “Measurement of the coefficient of thermal slip of gases and the thermophoresis velocity of large-size aerosol particles,” J. Colloid Interface Sci. 57, 451 (1976). [17] L. Talbot, R. K. Cheng, R. W. Schefer, D. R. Willis, “Thermophoresis of particles in heated boundary layer,” J. Fluid Mech. 101, 737 (1980). [18] S. K. Loyalka, “Slip and jump coefficients for rarified gas flows: variational results for Lennard-Jones and n(r)-6 potentials,” Physica A 163, 813 (1990). [19] F. Sharipov and D. Kalempa, “Velocity slip and temperature jump coefficients for gaseous mixtures. I. Viscous slip coefficient,” Phys. Fluids 15, 1800 (2003). [20] F. Sharipov and D. Kalempa, “Velocity slip and temperature jump coefficients for gaseous mixtures. II. Thermal slip coefficient,” Phys. Fluids 16, 759 (2004). [21] N. J. McCormick, “Gas-surface accommodation coefficients from viscous slip and temperature jump coefficients,” Phys. Fluids 17, 107104 (2005). [22] I. N. Ivchenko, S. K. Loyalka, and R. V. Tompson, Analytical Methods for Problems of Molecular Transport (Springer, Dordrecht, The Netherlands, 2007). [23] B. E. Dahneke, “Slip correction factors for nonspherical bodies: Introduction and continuum flow,” J. Aerosol Sci. 4, 139 (1973). [24] M. L. Laucks, G. Roll, G. Schweiger, and E. J. Davis, “Physical and chemical (Raman) characterization of bioaerosols: Pollen,” J. Aerosol Sci. 31, 307 (2000). [25] L. D. Reed, “A continuum slip flow analysis of steady and transient thermophoresis,” M.S. thesis, University of Illinois, Urbana-Champaign, Illinois, (1971). [26] H. J. Keh and H.J. Tu, “Thermophoresis and photophoresis of cylindrical particles,” Colloids Surf. A 176, 213 (2001). [27] K. H. Leong, “Thermophoresis and diffusiophoresis of large aerosol particles of different shapes,” J. Aerosol Sci. 15, 511 (1984). [28] M. M. R. Williams, “Thermophoretic forces acting on a spheroid,” J. Phys. D 19, 1631 (1986). 22
[29] H. J. Keh and C. L. Ou, “Thermophoresis of aerosol spheroids,” Aerosol Sci. Technol. 38, 675 (2004). [30] A. Mohan and H. Brenner, “Thermophoretic motion of a slightly deformed sphere through a viscous fluid,” AIAM J. Appl. Math. 66, 787 (2006). [31] S. Senchenko and H. J. Keh, “Thermophoresis of a slightly deformed aerosol sphere,” Phys. Fluid 19, 033102 (2007). [32] Y. C. Chang and H. J. Keh, “Thermophoresis of axisymmetric aerosol particles along their axes of revolution,” AIChE J. 55, 35 (2009).. [33] G. Dassios, M. Hadjinicolaou, and A. C. Payatakes, “Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates,” Quart. Appl. Math. 52, 157 (1994). [34] G. Dassios, M. Hadjinicolaou, F. A. Coutelieris, and A. C. Payatakes, “Stokes flow in spheroidal particle-in-cell models with Happel and Kuwabara boundary conditions,” Int. J. Eng. Sci. 10, 1465 (1995). [35] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Martinus Nijhoff, The Netherlands, 1983). [36] J. H. Lienhard, A Heat Transfer Textbook, second ed. (Prentice-Hall, Englewood Cliffs, New Jersey, 1987). [37] M. J. Gluckman, R. Pfeffer, and S. Weinbaum, “A new technique for treating multi-particle slow viscous flow: axisymmetric flow past spheres and spheroids,” J. Fluid Mech. 50, 705 (1971). [38] H. J. Keh and W. J. Li, “A study of bipolar spheroids in an electrolytic cell,” J. Electrochem. Soc. 144, 1323 (1997). [39] H. J. Keh and Y. C. Chang, “Slow motion of a slip spheroid along its axis of revolution,” Int. J. Multiphase Flow 34, 713 (2008). [40] L. E. Payne and W. H. Pell, “The Stokes flow problem for a class of axially symmetric bodies,” J. Fluid Mech. 7, 529 (1960).
23
Figure captions FIG. 1. Geometrical sketch for the thermophoresis of an aerosol spheroid along its axis of revolution.
FIG. 2. Plots of the normalized thermophoretic velocity U /U 0 of a prolate spheroid along its axis of revolution versus its aspect ratio a / b for various values of k * : (a) with C t* 2C m* 0.02 ; (b) with C t* 2C m* 0.2 .
FIG. 3. Plots of the normalized thermophoretic velocity U /U 0 of an oblate spheroid along its axis of revolution versus its aspect ratio a / b for various values of k * : (a) with C t* 2C m* 0.02 ; (b) with C t* 2C m* 0.2 .
FIG. 4. Plots of the normalized thermophoretic velocity U /U 0 of an aerosol spheroid with C t* 2C m* 0.02 along its axis of revolution versus its relative thermal conductivity k * for
various values of its aspect ratio a / b : (a) prolate spheroid; (b) oblate spheroid.
FIG. 5. Plots of the normalized thermophoretic velocity U /U 0 of an aerosol spheroid with
k * 100 along its axis of revolution versus its normalized frictional slip coefficient C m* ( C t* / 2 ) for various values of its aspect ratio a / b : (a) prolate spheroid; (b) oblate spheroid.
24
Table 1 Boundary collocation results of the normalized thermophoretic velocity of a prolate spheroid with C t* 2C m* along its axis of revolution for various values of the aspect ratio a / b and the parameters k * and C m*
k* 100
100
100
1
1
1
C m*
0
0.01
0.1
0
0.01
0.1
U /U 0
M=N a / b 1.1
a/b 2
a/b 5
a / b 10
2 3
1.1192 1.1192
2.3180 2.3180
7.3783 7.3783
16.6092 16.6092
4 Exact solution 22 23
1.1192
2.3180
7.3783
16.6092
1.1192
2.3180
7.3783
16.6092
1.0299 1.0299
1.4115 1.4115
3.2019 3.2019
6.4282 6.4281
24
1.0299
1.4115
3.2019
6.4281
17 18 19
1.0127 1.0127 1.0127
1.0911 1.0911 1.0911
1.3862 1.3862 1.3862
1.9338 1.9337 1.9337
1.0127
1.0911
1.0376 1.0376 1.0376
1.2397 1.2397 1.2397
1.4163 1.4163 1.4163
1.4696 1.4696 1.4696
1.0376
1.2397
1.4163
1.4696
1.0388 1.0388
1.2481 1.2481
1.4317 1.4317
1.4876 1.4875
14
1.0388
1.2481
1.4317
1.4875
19 20 21
1.0489 1.0489 1.0489
1.3208 1.3208 1.3208
1.5706 1.5706 1.5706
1.6498 1.6497 1.6497
Singularity solution
1.0489
1.3208
Singularity solution 2 3 4 Exact Solution 12 13
Exact solutions and singularity solutions are obtained from Keh and Ou [29] and Chang and Keh [32], respectively.
25
Table 2 Boundary collocation results of the normalized thermophoretic velocity of an oblate spheroid with C t* 2C m* along its axis of revolution for various values of the aspect ratio a / b and the parameters k * and C m*
k* 100
100
100
1
1
1
C m*
0
0.01
0.1
0
0.01
0.1
U /U 0
M=N a / b 0.9
a / b 0.5
a / b 0.2
a / b 0.1
2 3
0.8839 0.8839
0.4533 0.4533
0.1690 0.1690
0.08234 0.08234
4 Exact solution 31 32
0.8839
0.4533
0.1690
0.08234
0.8839
0.4533
0.1690
0.08234
0.9744 0.9744
0.9438 0.9438
1.0366 1.0366
0.9109 0.9110
33
0.9744
0.9438
1.0366
0.9110
30 31 32
0.9844 0.9844 0.9844
0.8473 0.8473 0.8473
0.4881 0.4881 0.4881
0.2616 0.2617 0.2617
0.9844
0.8474
0.9573 0.9573 0.9573
0.7092 0.7092 0.7092
0.3743 0.3743 0.3743
0.2088 0.2088 0.2088
0.9573
0.7092
0.3743
0.2088
0.9559 0.9559
0.6999 0.6999
0.3513 0.3513
0.1781 0.1782
23
0.9559
0.6999
0.3513
0.1782
19 20 21
0.9450 0.9450 0.9450
0.6389 0.6389 0.6389
0.2730 0.2730 0.2730
0.1283 0.1284 0.1284
Singularity solution
0.9450
0.6389
Singularity solution 2 3 4 Exact Solution 21 22
Exact solutions and singularity solutions are obtained from Keh and Ou [29] and Chang and Keh [32], respectively.
26
Table 3 Leading-order asymptotic results of the normalized thermophoretic velocity of a prolate spheroid with C t* 2C m* along its axis of revolution for various values of the aspect ratio a / b and the parameters k * and C m*
k* 100
100
1
1
C m*
0.01
0.1
0.01
0.1
U /U 0
M=N a / b 1.1
a / b 1.5
a/b 2
a/b 5
2
1.0299
1.1794
1.4053
3.0845
3 4
1.0299 1.0299
1.1801 1.1801
1.4111 1.4115
3.1551 3.1855
Collocation solution
1.0299
1.1801
1.4115
3.2019
2
1.0127
1.0516
1.0959
1.3707
3 4
1.0127 1.0127
1.0506 1.0506
1.0924 1.0913
1.4086 1.4079
Collocation solution
1.0127
1.0506
1.0911
1.3862
1.0388 1.0388 1.0388
1.1558 1.1557 1.1557
1.2485 1.2482 1.2481
1.4330 1.4326 1.4323
1.0388
1.1557
1.2481
1.4317
1.0489 1.0489 1.0489
1.1996 1.1990 1.1990
1.3238 1.3213 1.3209
1.5819 1.5772 1.5743
1.0489
1.1990
1.3208
1.5706
2 3 4 Collocation solution 2 3 4 Collocation solution
27
Table 4 Leading-order asymptotic results of the normalized thermophoretic velocity of an oblate spheroid with C t* 2C m* along its axis of revolution for various values of the aspect ratio a / b and the parameters k * and C m*
k* 100
100
1
1
C m*
0.01
0.1
0.01
0.1
U /U 0
M=N a / b 0.9
a / b 0.7
a / b 0.5
a / b 0.2
2
0.9744
0.9406
0.9421
1.1069
3 4
0.9744 0.9744
0.9409 0.9409
0.9446 0.9440
1.1618 1.1387
Collocation solution
0.9744
0.9409
0.9438
1.0366
2
0.9844
0.9390
0.8672
0.8644
3 4
0.9844 0.9844
0.9377 0.9377
0.8500 0.8476
0.6520 0.5514
Collocation solution
0.9844
0.9377
0.8473
0.4881
0.9559 0.9559 0.9559
0.8473 0.8472 0.8472
0.7022 0.6832 0.6999
0.3992 0.3793 0.3699
0.9559
0.8472
0.6999
0.3513
0.9450 0.9450 0.9450
0.8130 0.8123 0.8123
0.6494 0.6402 0.6391
0.4500 0.3461 0.3012
0.9450
0.8123
0.6389
0.2730
2 3 4 Collocation solution 2 3 4 Collocation solution
28
Fig. 1.
z
Ue z
T a
0 ( 0 )
ρ
b
29
Fig. 2a.
6
5
4 U U0
k*=10 3
100
2
1 0.1 0
1 0.0
0.2
0.4
0.6
(a/b)
-1
30
0.8
1.0
Fig. 2b.
4
3
k*=10
U U0
100
2
1 0.1 0
1 0.0
0.2
0.4
0.6 (a/b)
31
-1
0.8
1.0
Fig. 3a.
1.2 k*=100
0 0.8 0.1 U U0
10
0.4
1
0.0 0.0
0.2
0.4
0.6 a/b
32
0.8
1.0
Fig. 3b.
1.0
0.8
k*=100 0
0.6
10 0.1
U U0
0.4
1
0.2
0.0 0.0
0.2
0.4
0.6 a/b
33
0.8
1.0
Fig. 4a.
3 2.5
2.0 2
U U0
1.5
1.5 a/b=1.1
1.0
0.1
1
10 k*
100
34
1000
Fig. 4b.
1.6
1.2 U U0
0.9
0.8
0.7 0.5 0.4 a/b=0.2 0.0 0.1
1
10 k*
35
100
1000
Fig. 5a.
3.5
3.0 3 2.5 U U0
2.0 2 1.5
1.5 a/b=1.1
1.0 1E-3
0.01
C*m
36
0.1
1
Fig. 5b.
1.2
0.9 0.8 0.7 U U0
0.5
0.4
a/b=0.2 0.0 1E-3
0.01
C*m
37
0.1
1
