EFFECT OF MEDIUM ON THE THERMO-, PHOTO-AND DIFFUSIOPHORESIS OF SPHEROIDAL SOLID PARTICLE

90 ÍÀÓ×ÍÛÅ ÂÅÄÎÌÎÑÒÈ Ñåðèÿ: Ìàòåìàòèêà. Ôèçèêà. 2013. �26(169). Âûï. 33

MSC 76B10

EFFECT OF MEDIUM ON THE THERMO-, PHOTO-

AND DIFFUSIOPHORESIS OF SPHEROIDAL SOLID PARTICLE

N.V. Malay, N.N. Mironova

Belgorod State University,

Pobedy St., 85, Belgorod, 308015 Russia, e-mail: malay�bsu.edu.ru; e-mail: mironovanadya�mail.ru

Abstra t. Steady motion of spheroidal aerosol parti le with inner nonuniformly distributed

heat sour es that are pla ed in some external temperature and on entration gradients is studied

in the Stokes approximation. The mean temperature of the parti le surfa e is assumed to be

di�ered slightly from that of gaseous environment. The analyti expression of the for e and the

rate of thermo-, photo- and di�usiophoresis are found by solving gas-dynami equations in view of

environment evolution.

Keywords: spheroid; thermophoresis; photophoresis; di�usiophoresis.

1. Introdu tion. In modern te hniques di�erent types of multiphase mixtures are often

used espe ially in su h spheres as hemi al te hnology, hydrometeorology, environmental

prote tion et . From this viewpoint, the study of dispersion mixtures onsisted of two phases

one of whi h is parti les, and the other is vis ous liquid spheres are of greatest interest. Gas

(liquid) with weighted parti les in it is alled aerosol (hydrosol) and those parti les is alled

aerosol (hydrosol). Hydro- and aerosol parti les an in�uen e su� iently on physi al and

hemi al pro esses of di�erent types in dispersion system (e.g. in mass and heat hanging

pro esses). The parti les' sizes of dispersion phase an vary from ma ros opi (500 mkm)

s ales to mole ular (10 nm) ones as well as the on entration of the parti les an hange

from 1 parti le to multi on entrated systems (> 1010 ñì

−3). In today's nanote hnology,

the usage of ultra dispersion nanomaterials, Nan parti les is important demand, e.g. in su h

spheres as Nan ele troni s, Nan me hani s, et .

Parti les of dispersion systems an be e�e ted by many ways. It leads to regulated

movement relative of their inertia enters. At this way, their sedimentation happens in gravity

�eld. In gaseous spheres with heterogeneously spread temperature, there an happen regular

parti les' movement in�uen ed by some for es of mole ular origin. This pro ess is the result

of the transmission of un ompensated impulse to parti les by gas mole ules. In this ase

the movement of parti les in�uen ed by outward temperature and on entration gradient is

alled thermophoresis and di�usiophoresis, orrespondingly [1℄. If the movement is in�uen ed

by the inner heat sour e heterogeneously spreaded in parti les volume, so this movement is

alled photophoreti al [2, 3℄.

The average distan e between aerosol parti les in aerodispersial systems is mu h longer

than the usual parti le size. In su h systems the sto k of aerosol in�uen e on physi al pro ess

an be taken in a ount basing on dynami laws, heat-and-mass hanging of environment.

Without appropriate knowledge of su h a behavior, the mathemati al simulation of aerosol

ÍÀÓ×ÍÛÅ ÂÅÄÎÌÎÑÒÈ Ñåðèÿ: Ìàòåìàòèêà. Ôèçèêà. 2013. �26(169). Âûï. 33 91

systems evolution is impossible as well as the solution of su h a problem with orre t a ount

of their in�uen e on aerosols.

Some types of parti les whi h are met in industrial arrangements and nature have the

nonspheri al form, e.g. spheroid (ellipsoid). Therefore, the question of movements of parti les

with di�erent appropriate forms in gaseous (liquid) being both homogeneous and heteroge-

neous is very a tual whi h lays theoreti al and pra ti al interest.

2. Problem formulation.We examine enormous solid parti le of spheroid form weighted

in binomial gaseous mixture with the temperature T∞, the density ρg and the vis ousness µg.

The parti le motion is aused by outside sour es of the little temperature gradient ∇T and

the on entration gradient ∇C1 in this binary gaseous mixture. In this ase C1 + C2 = 1,C1 = n1/ng, C2 = n2/ng, ng = n1 + n2, ρg = ρ1 + ρ2, ρ1 = m1n1, ρ2 = m2n2, m1, n1

and m2, n2 are mass and on entration of �rst and se ond omponents, orrespondingly, in

binary gaseous mixture. In the works published till now under the theory of thermo-, photo-

and di�usiophoresis of large solid aerosol parti les having spheroidal form at small relative

temperature drops [4℄ were not studied in�uen e of environment evolution (i.e. onve tive

terms in the heat ondu tion equation and the di�usion one) and the surfa e heating on

thermo-, photo- and di�usiophoresis simultaneously, that represents theoreti al and pra ti al

interest. In this work the estimate of this in�uen e is resulted.

In theoreti al des ription of thermo-, photo-, di�usiophoresis we suppose that the heat

pro ess and the mass movement is quasistati in the system. It is due to smallness of heat and

di�usion relaxation. The movement happens with small Pe let and Reynolds numbers and

the temperature drop in parti le neighborhood is assumed to be small, i.e. (Ts−T∞)/T∞ ≪ 1where Ts is the mean temperature of spheroid surfa e and T∞ is the gas temperature far

away of parti le. In this ase, the thermal ondu tivity as well as the dynami and kinemati

vis osities an be onsidered as onstants [11℄. The problem is solved by hydrodynami

method. So, the hydrodynami equations with appropriate bordering onditions should be

solved. It is onsidered that there are no some phase transformations and parti les are

homogenous in its stru ture.

We suppose that the �at mono hromium wave falls on the parti le with the intensity

I0 during a time interval. The energy of ele tromagneti radiation having been absorbed inparti le volume transforms into heat energy. The heat spreads heterogeneously in volume,

the lo al distribution of appeared heat sour es an be des ribed by the fun tion qp alledthe volume density of inner heat sour es [10℄.

We des ribe the thermo-, photo- and di�usiophoresis of the parti le in the spheroidal

oordinate system (ε, η, ϕ)with the origin at spheroid enter; i.e. the origin of �xed oordinatesystem oin ides with the instantaneous position of parti le enter. The urvilinear oordi-

nates ε, η, ϕ are related to the Cartesian oordinates by the relations [5℄:

x = c ch ε sin η cosϕ , y = c ch ε sin η sinϕ , z = c sh ε cos η , (1)

x = c sh ε sin η cosϕ , y = c sh ε sin η sinϕ , z = c ch ε cos η (2)

where c =√a2 − b2 when the spheroid is oblate (a > b, Eq.(1)) or c =

√b2 − a2 when it

is prolate (a < b, Eq.(2)); a and b are spheroid semiaxis. The OZ axis of the Cartesian

oordinate system oin ides with the spheroid symmetry axes.


In the frame of formulated theory, spreadings of the velo ity Ug, the pressure Pg, tem-

peratures Tg, Tp and the on entration of �rst omponent C1 in binary gaseous mixture are

des ribed in the following equation system [6℄:

∇Pg = µg△Ug, divUg = 0 ,

ρgcpg(Ug∇)Tg = λg∆Tg, ∆Tp = − qpλp, (3)

(Ug∇)C1 = D12∆C1 .

The equation system (3) is solved with boundary onditions [4℄

ε = ε0 : Uε = −c U ch ε

Hε

cos η,

Uη =c U sh ε

Hε

sin η−KTSνgTg

(∇Tg· eη)−KDSD12(∇C1· eη) ,

Tg = Tp, λg∂Tg∂ε

= λp∂Tp∂ε

,∂C1

∂ε= 0 . (4)

ε→ ∞ : Tg → T∞ + |∇Tg|∞c sh ε cos η,

C1 → C1∞ + |∇C1|∞c sh ε cos η ,Pg → P∞; Ug → 0, (5)

ε→ 0 : Tp 6= ∞ .(6)

Here, eη, eε are unit ve tors of spheroidal oordinate system; Uε, Uη are omponents

of the mass velo ity Ug; U = |U|; λg, λp are thermal ondu tivities of gas and parti les,

orrespondingly, νg, µg are kinemati al and dynami vis osities; Hε = c√ch2 ε− sin2 η is the

Lame oe� ient; KTS, KDS are thermal and di�usion reep oe� ients whi h are al ulated

from kineti theory of gases, the gas kineti oe� ient KTS ≈ 1, 152 when a ommodation

oe� ients of tangential momentum and energy equal to unity (in the ase of spheroid

parti le) [1,7℄; KDS ≈ 0, 3; ε = ε0 is the oordinate surfa e orresponding to the parti le

surfa e.

3. Temperature and on entration distributions. Let us study temperature and

on entration distributions inside and outside of the parti le. We transform equations (3)

and boundary onditions (4)-(6) into dimensionless form by introdu ing dimensionless values:

tk = Tk/T∞, Vg = Ug/U, (k = g, p).In this problem, besides Reynolds and Pe let dimensionless numbers, there are two

ontrollable small parameters ξ1 = a|∇Tg|∞/T∞ ≪ 1 and ξ2 = a|∇C1|∞ ≪ 1 hara -

terizing relative overfall of temperature and on entration. Chara teristi velo ity U ∼(µg/ρgT∞)|∇Tg|∞ on the size order for the purely thermophoresis and U ∼ D12|∇C1|∞ for

the purely di�usiophoresis. Therefore, we look for the solution of the boundary-value problem


(3)-(6) in the form of expansion orresponding physi al sizes in powers of ξ1. Small parameterξ2 is expressed through ξ1 and the Reynolds number al ulated on the pure thermophoresis hara teristi velo ity oin ides with ξ1.

Vg = Vg0 + ξ1Vg1 + ... , pg = pg0 + ξ1pg1 + ... ,

t = t0 + ξ1t1 + ... , C1 = C10 + ξ1C11 + ... . (7)

We restri t out our onsideration to the �rst order terms on ξ1 when al ulating the

for e a ting on parti le and the velo ity of its thermo-, photo-, and di�usiophoreti motion.

In order to �nd these quantities, one has to know the distributions of velo ity, pressure,

temperature and on entration both outside and inside the spheroid. Substituting (7) into

(3), leaving terms ξ1 and solving sets of equations found by the method of separation of

variables, we �nally �nd at zero and �rst approximations

tg0(λ) = 1 + γλ0arcctg λ , (8)

tp0(λ) = 1+ (1− δ)γλ0arcctg λ0 + δγλ0arcctg λ+

∫ λ

λ0

f0arcctg λdλ− arcctg λ

∫ λ

λ0

f0dλ , (9)

C10 = C1∞ ; (10)

tg1 = cos η{cλa

+ Γ(λ arcctg λ− 1) + Pr∞γλ0ac

{A2

(arcctg λ− λ

2arcctg2λ

)+

+A1

2

(arcctg λ− λarcctg2 λ

)}}, (11)

tp1 = cos η{Bλ+

3(1− λ arcctg λ)

4πc2λpT∞

∫

V

qpzdV − λ

∫ λ

λ0

f1(λ arcctg λ− 1)dλ+

+(λ arcctg λ− 1)

∫ λ

λ0

f1λdλ}, (12)

C11 = cos η{cλa

− c(1 + λ20)

((1 + λ20)arcctg λ0 − λ0)a

(λ arcctg λ− 1

)}(13)

where λ = sh ε, λ0 = sh ε0 and δ = λg/λp, γ = ts − 1 is the dimensionless parameter

hara terizing the temperature on spheroid surfa e; ts = Ts/T∞,

TsT∞

= 1 +1

4πcλ0λgT∞

∫

V

qpdV . (14)

In (14) the integral is taken over the total entire parti le volume,

fn = −2n + 1

2λpT∞

∫ 1

−1

c2qp(λ2 + x2)Pn(x)dx ,

x = cos η, Pn(x) � Legendre's polynomials.


Constants in expressions (11) and (12) of temperature �elds inside and outside the parti le

are onne ted with orresponding boundary onditions on the spheroid surfa e. Sin e an

expression for the oe� ient Γ is of interest, we write it in the expli it form:

Γ = −c(1− δ)

a∆+

3

4πc2λpλ0T∞(1 + λ20)∆

∫

V

qpzdV+

+Pr∞γλ0ac

{A2

(arcctg λ0 −

1− δ

2△ arcctg2λ0 −δ

(1 + λ20)∆

)+

+A1

2

(arcctg λ0 +

δ(λ0arcctg λ0 − 1)

(1 + λ20)∆

)},

(15)

∆ = (1− δ)arcctg λ0 +δλ0

1 + λ20− 1

λ0.

4. For e and velo ity of thermo-, photo- and di�usiophoresis. The general

solution of the hydrodynami equations in the spheroidal oordinate system has form [5℄:

Uε(ε, η) =U

c ch εHεcos η

{λA2 +

[λ− (1 + λ2)arcctg λ

]A1 + c2(1 + λ2)

},

Uη(ε, η) = − U

cHε

sin η{A2

λ+[1− λ arcctg λ

]A1 + c2λ

}, (16)

Pg(ε, η) = P∞ + cµgU

H4ε

x(x2 + λ2)A2 .

Integration onstants A1, A2 are found from boundary onditions on the spheroid surfa e:

A2 = − 2c2

λ0 + (1− λ20)arcctg λ0−

−2KTSc2νgδ

Uts· |∇Tg|∞T∞(1 + λ20)∆

· λ0 − (1 + λ20)arcctg λ0λ0 + (1− λ20)arcctg λ0

(1 +

3a(1− λ0arcctg λ0)

4πc3λ0λgT∞

∫

V

qpzdV−

− Pr∞γλ02(λ0 + (1− λ20)arcctg λ0)

[2arcctg2λ0 + 4(λ0arcctg λ0 − 1) + (1− λ20)(λ0arcctg λ0 − 1)2

])−

−2KDSD12c2|∇C1|∞

U(λ0 + (1− λ20)arcctg λ0). (17)

The resultant for e a ting on spheroidal parti le is de�ned by integrating the stress tensor

over the aerosol parti le surfa e [5, 6℄ and has the form:

Fz = 4πµgU

cA2 . (18)


In view of the expli it form of oe� ient A2, we �nd the general expression of for e

a ting on spheroidal solid aerosol parti le. This for e is the sum of the vis ous for e Fµ, the

thermophoreti for e Fth, the photophoreti for e Fph proportional to the dipole moment

of heat sour es nonuniformly distributed over parti le volume, the for e onne ted with the

medium and the di�usiophoreti for e Fdh

Fz = Fµ + Fth + Fph + Fdh , (19)

Fµ = −8πµgUc

λ0 + (1− λ20)arcctg λ0,

Fth = −8πKTSµgνgts

· |∇Tg|∞T∞


· cδ

(1 + λ20)∆×

×(1− Pr∞γλ0

2(λ0 + (1− λ20)arcctg λ0)

[arcctg2λ0 − λ20(λ0arcctg λ0 − 1)2

])

Fph = −8πKTSµgνgts

· |∇Tg|∞T∞


· cδ

(1 + λ20)∆×

×(3a(1− λ0arcctg λ0)

4πc3λ0λgT∞

∫

V

qpzdV −

− Pr∞γλ02(λ0 + (1− λ20)arcctg λ0)

[arcctg2λ0 + 4(λ0arcctg λ0 − 1) + (λ0arcctg λ0 − 1)2

]),

Fdh = −8πKDSD12µgc|∇C1|∞

λ0 + (1− λ20)arcctg λ0.

Equating the resultant for e F to zero, we arrive to the general expression of the drift

(thermo-, photo- and di�usiophoreti ) velo ity of solid oblate spheroidal parti le in external

temperature and on entration gradient �elds:

U = − b

aKTS · νgδ

ts· |∇Tg|∞

T∞

(1− (λ0 + λ0−1)arcctg λ0)√

1 + λ20∆

(1 +

3a(1− λ0arcctg λ0)

4πc3λ0λgT∞

∫

V

qpzdV−

− Pr∞γλ02(λ0 + (1− λ20)arcctg λ0)

[2arcctg2λ0 + 4(λ0arcctg λ0 − 1) + (1− λ20)(λ0arcctg λ0 − 1)2

])

−KDSD12|∇C1|∞ . (20)

In order to �nd the rate of thermo-, photo- and di�usiophoresis for the ase of prolate

spheroid, one has to substitute iλ for λ and −ic for c (i is the imaginary unit) in (19), (20).Thus, formulas (19) and (20) have the most general form and make it possible to estimate

the resultant for e a ting on a solid spheroidal aerosol parti le and its drift velo ity in the

external temperature and on entration gradient �elds for the ase when heat sour es (sinks)


are nonuniformly distributed inside the parti le. In this approa h, the environment evolution

is taken into a ount for small temperature di�eren es in the vi inity of parti le.

5. Results and dis ussion. From formulas (18), (19), it is visible that environment

evolution does not bring the ontribution in di�usiophoresis. Conve tive terms in the di�usion

equation do not in�uen e on movement of solid aerosol parti le in the �eld of on entration

gradient. It an be of great importan e in pra ti al appendi es, for example, at the des ription

of aerosol parti le movement in di�usiophoresis �elds. That onve tive terms in�uen e on

the di�usiophoresis, it is ne essary to hange boundary onditions of di�usion part. It an

be made by di�erent ways. For example, we an onsider movement not solid parti le taking

into a ount the evaporating drop.

Movement of environment brings the ontribution as in thermophoresis and in fotophoresis.

These ontributions are proportional to produ t of Prandtl's number on average temperature

of parti le surfa e. Prandtl's number for the most of gases is about of one. As the problem

dared at small temperature di�eren es, the movement ontribution an make no more than

10-12 per ents.

It is interesting to examine the spe ial ases of movement of spheroidal parti les. If one

does not take into a ount the motion of environment and internal heat sour es, (20) is

redu ed to the expression of the purely thermo- and di�usiophoresis velo ity of spheroidal

parti le:

U = − b

aKTS

νgts

· |∇Tg|∞T∞

· δ(1− (λ0 + λ0−1)arcctg λ0)√

1 + λ20∆−KDSD12|∇C1|∞

whi h oin ides with formula (9) in [4℄.

In the spheri al ase, (20) turns into the expression for the thermo-, photo- and di�usio-

phoreti velo ity of solid spheri al parti le of radius R that in ludes the �ow of environment

and internal heat sour es:

Ua=b=R = −KTSνgts· |∇Tg|∞T∞

· 2δ

1 + 2δ

(1 +

1

4πR2λgT∞

∫

V

qpzdV − Pr∞5γ

24

)−KDSD12|∇C1|∞ .

(21)

Disregarding the environment �ow and internal heat sour es yields the onventional formula

for the thermophoreti velo ity of large spheri al parti le [8, 9℄.

U |a=b=R = −KTSνgts

· |∇Tg|∞T∞

· 2δ

1 + 2δ. (22)

In order to estimate how the environment motion e�e ts on the thermo- and photophoreti

velo ity of spheroidal parti le, one has to spe ify the nature of heat sour es nonuniformly

distributed over its volume. As an example, let us onsider the simplest ase when the parti le

absorbs radiation as the bla k body. In this ase, the radiation is absorbed in the thin layer

of depth δε≪ ε0 that is adja ent to the heated parti le surfa e. The density of heat sour esinside the layer of depth δε is equal to [10, 11℄


qp(ε, η) =

− ch ε cos ηc(ch2 ε− sin2 η)δε

I0,π2 ≤ η ≤ π, ε0 − δε ≤ ε ≤ ε0 ;

0, 0 ≤ η ≤ π2

(23)

where I0 is the intensity of in ident radiation.

Integrals

∫V

qpdV and

∫V

qpzdV appear in the expression for the thermo- and photophoreti

velo ity. Substituting into (23) these integrals in view of the fa t that δε≪ ε0 and performingintegration, we �nd

∫

V

qpzdV = −2

3πc3I0λ

30

(1 + λ−2

0

), (24)

∫

V

qpdV = πc2I0λ20

(1 + λ−2

0

). (25)

In view of (24), (25) the expression (20) takes the form

U = − b

aKTS

νgts

· |∇Tg|∞T∞

· δ(1− (λ0 + λ0−1)arcctg λ0)√

1 + λ20∆

(1 +

aI0(1 + λ20)

2λgT∞

)×

[λ0arcctg λ0 − 1− Pr∞

2arcctg2λ0 + 4(λ0arcctg λ0 − 1) + (1− λ20)(λ0arcctg λ0 − 1)2

4√1 + λ20(λ0 + (1− λ20)arcctg λ0)

]

−KDSD12|∇C1|∞ . (26)

In the ase of sphere, (26) is re ast as

U(a = b = R) = −KTSνgts

· |∇Tg|∞T∞

· 2δ

1 + 2δ

(1− I0R

6λgT∞(1 +

5Pr∞16

))−KDSD12|∇C1|∞ .

In order to illustrate ontributions of the form-fa tor (ratio of spheroid semiaxes), the

environment �ow and the internal heat release (nonuniform distribution of heat sour es

over the parti le volume) to the thermo- and photophoreti velo ity (26), in drawing list

urves orresponding to values f = (f ∗t.ph/f

∗∗t.ph)|T∞=300K with intensity of in ident radiation

of borated graphite (λp = 55W/(mK)) parti les with spheroidal (the urve 1 taking into

a ount movement of environment, the urve 2 is the same without movement) and spheri al

(the urve 3) forms of surfa es suspended in air at T∞ = 300K and Pg = 105Pa for variousrelations of spheroid semiaxes.


1) 2) 3)

Fig. 1. Dependen e of fun tion f on the intensity of an in ident radiation at the relation

of semiaxes b/a = 0, 2 � 1); b/a = 0, 5 � 2); b/a = 0, 7 � 3). I0 −W/sm2.

Numeri al analysis showed that, at the given ratio between semiaxes, the relative ontribution

of other fa tors (�ow of environment, internal heat release) leads to monotonous redu tion

of velo ity thermo-, photo- and di�usiophoresis with in reasing in ident radiation intensity.

This e�e t depends signi� antly on the equatorial radius a of spheroid.Quantitative resear h of the dis ussed phenomen for �rm hearted parti les represents

quite real experimental problem.

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6. Landau L.D., Lifshitz E.M. Fluid Me hani s / Gostekhizdat: Mos ow, 1958; (Pergamon:

Oxford, 1959).

7. Yalamov Yu.I., Poddoskin A.B., Yushkanov A.A. // Dokl. Akad. Nauk SSSR. � 1980. � 254. �

P.1047.

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Documents

EFFECT OF MEDIUM ON THE THERMO-, PHOTO-AND DIFFUSIOPHORESIS OF SPHEROIDAL SOLID PARTICLE