Thermophoretic interaction of heat releasing particles

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 9, NUMBER 7 1 APRIL 200 Thermophoretic interaction of heat reeasing partices Yu Doinsky a) and T Eperin b) Department of Mechanica Engineering, The Pearstone Center for Aeronautica Engineering Studies, Ben-Gurion University of the Negev, POB 65, Beer-Sheva 84105, Israe Received 5 November 2002; accepted 7 January 200 This study investigates thermophoretic force acting at heat reeasing absorbing partices near the interface between two media with different therma conductivities This force is caused by the induced temperature gradient which is proportiona to the rate of heat reease absorption by the partice Therefore the magnitude of the thermophoretic force is proportiona to the rate of heat reease absorption by the partice, and its direction depends upon the sign of the parameter 1 2, where 1 is therma conductivity of a host medium and 2 is therma conductivity of the adjacent medium The obtained resuts impy that a heat reeasing absorbing partice is attracted repeed to the interface when therma conductivity of a host medium is ess than therma conductivity of the adjacent medium Thus, eg, growing in air by condensation partice is attracted to a meta surface whie an evaporating in air partice is repeed from a meta surface The change of temperature distribution caused by heat reeasing partices resuts in the additiona thermophoretic interaction of these partices We determined a condition for mutua attraction of two spherica heat reeasing partices and derived an expression for the thermophoretic force acting at the partices The magnitudes of the considered thermophoretic forces are compared with the cassic thermophoretic force 200 American Institute of Physics DOI: 10106/ I INTRODUCTION a Eectronic mai: yui@menixbguaci b Author to whom correspondence shoud be addressed; eectronic mai: eperin@menixbguaci Behavior of micropartices inside a host medium attracted considerabe attention because of the emergence of new technoogica processes invoving micropartices, their significance in atmospheric physics and due to a arge variety of mechanisms determining the behavior of micropartices see Refs 1 and references therein One of the mechanisms determining the behavior of sma partices in fuids is associated with thermophoretic forces These forces arise due to the kinetic sip of a fuid fow near the partice in the presence of the temperature gradient aong its surface,4 When temperature gradient is caused by an externa source then the magnitude of the effect depends upon therma conductivity of a host medium 1 and of the partice p In the imiting case when p this effect disappears Different situation occurs when a partice is ocated near the boundary between two media and when the partice itsef is a heat source or sink The atter can occur, eg, in the case of a chemicay active partice when heat is reeased or absorbed due to a chemica reaction In this situation therma conductivity of a partice is of esser importance since the induced at the partice temperature gradient due to the presence of the interface between two media is determined primariy by therma characteristics of the host and adjacent media In Sec II of this study we investigated a force acting at the partice due to the induced temperature gradient caused by the presence of the interface between two media with different therma conductivities Athough the effect of thermophoresis was discovered more than one hundred years ago see, eg, Ref 4 and references therein the feasibiity of the thermophoretic sef-action of a heat reeasing partice near the interface was not discussed before The obtained resuts impy that a heat reeasing absorbing partice is attracted repeed to the interface when therma conductivity of a host medium is ess than therma conductivity of the adjacent medium Thus, eg, growing in air by condensation partice is attracted to a meta surface whie an evaporating in air partice is repeed from a meta surface Using the resuts obtained in Sec II of this study in Sec III we determined a condition for mutua attraction of heat reeasing partices and derived an expression for the thermophoretic force acting at the partice II FORCES ACTING AT A HEAT RELEASING PARTICLE LOCATED NEAR THE BOUNDARY BETWEEN TWO MEDIA WITH DIFFERENT THERMAL CONDUCTIVITIES Consider a pane surface z0 separating two media with therma conductivities 1 and 2, correspondingy Assume that a spherica heat reeasing partice with a center at c having a radius a and a heat reease rate q is ocated inside a host fuid with therma conductivity 1 see Fig 1 The host fuid is considered to be incompressibe and Newtonian, and we negect temperature dependence of the physica properties Assume sma Knudsen and Reynods numbers, Kn1 and Re 1, so that inertia effects can be negected The force acting at a partice in a stationary regime is determined by the foowing system of equations: 2 Tq0, p 2 u, u /200/9(7)/421/7/$ American Institute of Physics

2 422 J App Phys, Vo 9, No 7, 1 Apri 200 Yu Doinsky and T Eperin to the temperature gradient induced by a heat source when a heat source is ocated near the edges of this surface Here we consider a situation when a heat source is ocated far from the edges and the surface can be considered as an infinite The third boundary condition in Eq 4 corresponds to a rigid partice, and the ast condition in Eqs 4 impies a sip of fuid at the partice due to the induced by the surface temperature gradient Hereafter we assume that a heat reeasing partice is ocated in the region z0 The Green s function G(r,r) which satisfies an equation 2 Gr,rrr and boundary conditions with respect to variabe r is we known see, eg, Ref 6, Chap 2, Sec 7 Under the condition that a point r is ocated in a region 1, G(r,r) G 1 (r,r) and FIG 1 Location of a partice with a radius a near the interface between two media with therma conductivities 1 and 2 G 1 r,r rr rrr 5 Here T is temperature, u is a veocity of a host fuid with an imbedded heat source partice q, p is fuid pressure, and is a dynamic viscosity Equation 1 must be suppemented with boundary conditions for temperature and veocity When veocity u is found then the force acting at the partice is determined as foows see, eg, Refs 1 and 5: f ˆ ds, ˆ pê u u T, 2 where ˆ is a stress tensor, and ê is a unit tensor and integration is performed over the surface of a partice Boundary conditions for temperature read T S 0, n T S 0, where n is a unit externa norma vector, and S S S, S are vaues of at the externa and interna surfaces separating two media, respectivey Hereafter subscript S denotes that a corresponding variabe is determined at the surface separating between two media, whie subscript p impies that variabe is determined at the surface of the rigid partice Boundary conditions for the veocity are as foows: u 0, u S 0 n u p 0, u p T p, 4 where is a unit tangentia vector, ( ) denotes the magnitude far from the partice, and ( ) p denotes the vaue at the partice s surface A coefficient in Eq 4 is considered to be constant in order to inearize the probem, and its dependence upon the parameters of the probem is discussed beow The first condition in the boundary conditions 4 impies that a partice is at rest in the aboratory frame of reference Thus, it is assumed that an externa force is appied at the partice in order to compensate a force determined by Eq 2 The second condition in the boundary conditions 4 impies the absence of sip at the surface separating between the fuid and an externa soid medium Sip may occur due Here R is an operator of specuar refection with respect to a pane z0, ie, Rr(x,y,z) If a point r is ocated in region 2, then G(r,r) G 2 (r,r), and G 2 r,r rr 6 Then the soution for the temperature reads Tr 1 1 Gr,rqrdr Expression 5 for the Green function impies that temperature distribution T(r) is given by a sum of two terms The first term describes temperature distribution due to the heat source, and the second term is associated with the presence of the interface The atter term is responsibe for the effect of sef induced thermophoresis, whie the contribution of the first term is zero due to boundary conditions 4 The formua for the second term reads T 1 r ,m Y m n m rrc 1, where m a Y m * Rna 2 qa,nda dn, n rrc rrc, rc n rc, arc, and Y m (n) are spherica functions see, eg, Ref 7 If coefficients m have azimutha symmetry with respect to the axis connecting two points c and Rc, then T 1 can be written as foows: T 1 r P n rrc 1, 1 2,

3 J App Phys, Vo 9, No 7, 1 Apri 200 Yu Doinsky and T Eperin a 1 P qa,a 2 da d, 0 and P () are Legendre poynomias 7 Keeping ony the first nonvanishing terms with respect to a parameter a/, where is a distance between the center of the partice and the interface between the two media, we obtain the foowing expression for T 1 : T 1 k , 9 where k crc crc Thus, in the first nonvanishing order with respect to the parameter a/, the induced temperature gradient fied is homogeneous, and its magnitude is determined by the geometry of a heat source q In order to sove a hydrodynamic part of the probem one can use either bispherica coordinates 8 or empoy a method of refections see Ref 2, p 189 In the atter case a veocity fied is determined by the foowing expansion: uu 0 u 1 u 2, where each term is determined by boundary conditions 4 and a boundary condition which is satisfied by a preceding term Thus if u 0 is determined by boundary conditions 4 for a partice immersed into an infinite medium, then u 1 is determined by a boundary condition u 1 S u 0 S, the veocity u 2 is determined by a boundary condition u 2 p u 1 p, and so on In the first nonvanishing order with respect to the parameter a/, the expression for fuid veocity u 0 reads see Ref 9, Chap 1, Sec 15 u 0 Aŝ k, 10 where the tensor ŝ is as foows: ŝ 1 a rc 2 2 ênnênn, rc n rc rc Coefficient A is determined from the ast expression in the boundary conditions 4: 0 A a The force F acting at the partice is determined by the foowing expression see Ref 9, Chap 1, Sec 15: F8Ak 11 In particuar cases when heat sources are distributed uniformy over the surface, qq S (ra), or over the voume, qq V of the partice, (x) is Dirac s deta function Fk a Q, where Qa 2 q S for a surface heat source, and Q (a /)q V for a voumetric heat source Equation 12 accounts for the temperature gradient induced at the partice embedded into the host medium when therma conductivities of a host medium and an adjacent medium are different If 1 2, the system is homogeneous, and spatia temperature distribution is determined by spatia distribution of heat sources In this study we consider an isotropic distribution of interna heat sources, and F0 When 1 2, Eqs 5 and 7 impy the existence of the additiona heat source with the same sign as a rea heat source Thus, a heat reeasing partice is repeed from the surface, and heat absorbing partice is attracted to the surface Indeed, the ast in Eqs 4 shows that the veocity of the fuid has the same sign as an induced temperature gradient Since the direction of the partice veocity is the same as that of the surrounding fuid, the partice wi move in the direction of the induced temperature gradient Simiary, when 1 2, Eqs 5 and 7 impy the existence of the additiona heat source with the sign opposite to the sign of a rea heat source Thus, eg, in the case of a heat reeasing partice there occurs the induced heat absorption source ocated at Rc Therefore the induced temperature gradient, Tk, and heat reeasing partice is attracted to the surface The situation is reversed in the case of a heat absorbing partice Equation 12 impies that a heat reeasing partice is attracted to the interface when therma conductivity of the host medium is ess than therma conductivity of the adjacent medium Thus, eg, growing in air by the condensation partice is attracted to a meta surface whie an evaporating in air partice is repeed from a meta surface Certainy the resut expressed by Eq 12 remains ony quaitative without determining a dependence of the coefficient upon the parameters of the probem Kinetic theory see Ref 4 yieds the foowing formua for this coefficient: 4 T S, 1 where / is a kinematic viscosity When a partice itsef is a heat source, a temperature at the partice surface T S depends upon the power of a heat source, and upon the therma conductivity of the partice, 0 Temperature distribution in a stationary regime is determined by the first equation in Eq 1 where boundary conditions at the interface must be suppemented with boundary conditions at the surface of the partice Consider a case when a heat source has a spherica symmetry, ie, q(r)q V (arc)q S (arc), where q V is the density of a uniform voumetric heat source, q S is the density of a surface heat source, and (x) is the Heaviside step function Then boundary conditions at the surface of the partice read T p 0, n T p q S 14 Soving this boundary vaue probem by the method of refections as described above we arrive at the foowing expression for temperature T:

4 424 J App Phys, Vo 9, No 7, 1 Apri 200 Yu Doinsky and T Eperin TT 0 TT 15 The term T 0 in Eq 15 is determined by Eq 1 and boundary condition 14 The term T in Eq 15 is determined by Eq 1 with q0 and with the foowing boundary conditions: n T S 2 1 n T 0 S, 16 T S 0, T 0 The term T in Eq 15 is determined by Eq 1 with q 0 and with the foowing boundary conditions: n T p 0 1 n T p, 17 T p 0, T 0 Denote by subscript 0 temperature inside a partice and by subscript 1 temperature of a host fuid Then T 0 0 q Va r2 a 2 q Sa 1 T, rrc, 18 T 0 1 a2 Q 1 rc T, Q q S 1 aq V, where T is the initia temperature before introducing a heat source The temperature at the partice s surface T S Q a T 19 1 and T Q a 2 1, 1 rrˆ c T rc, 20 T a rc rc, where Qa 2 k 1 2 2, k c Equations 4, 10, 11, 1, and 20 yied the foowing formua for the force F: F 4 a 2 2 aq S 1 2 f T Q a 1 k, f In the case of a reguar thermophoresis associated with the externa temperature gradient ( T) ext a thermophoretic force is determined by the foowing formua see, eg, Ref 9, Chap 1, Sec 14: F ext f a2 T T ext 22 Let L be a characteristic ength scae of variation of the externa temperature Then ( T) ext T /L and F 1 F ext 4 al 2 F 1 F ext 4 al, 2 aq 1 T, aq if aq if 1 T 1 1 T 1, 2 Is must be noted that athough in this study we considered boundary conditions for rigid partices see Eq 4 the obtained resuts are vaid at east quaitativey for sma iquid dropets with arge viscosity in comparison with that of a host medium In view of the atter remark et us consider the magnitude of the investigated phenomenon using an exampe of an evaporating dropet of acoho in air at temperature T 7 K (Q q 0 ȧ, where q 0 is a atent heat of evaporation aq / 1 T 01 When externa temperature gradient is sma, ie, L 2 /a, the sef-induced thermophoretic force can be arger than a thermophoretic force caused by an externa temperature gradient Note that a thermophoretic force given by Eq 18 F ext f (a/l) 2, where 0 f /2 and the ratio a/l can be varied in the experiments The size of a partice when a force F 2 exceeds the gravitationa force is of the order of a( 2 /g) 1/ (/ p ) 1/, where g is the acceeration of gravity and p is a materia density of the partice Thus, eg, in air under norma conditions this size is on the order of a00 cm Athough the obtained resuts were derived under the assumption that a, they are essentiay vaid aso for a/1 It is known that a thermophoretic force increases indefinitey when a partice approaches a surface, 8 and the smaer is a Knudsen number Kn/a, where is a free path ength of gas moecues, the steeper the increase of the thermophoretic force In the present study it is assumed that Kn1 The reason for the increase of a thermophoretic force is the increase of a temperature gradient when a partice approaches a surface When a heat reeasing partice approaches a surface, a temperature gradient aso increases Anaysis of the behavior of a thermophoretic force acting on a heat reeasing partice near the surface when a/1 requires numerica cacuations and was not performed before III THERMOPHORETIC INTERACTION OF HEAT RELEASING PARTICLES One of the direct consequences of the above considered phenomenon of thermophoretic sef action of a heat reeasing partice is the existence of the effect of attraction in a system consisting of a sma partice with the radius R a and interna heat source q a and a arge partice with the radius R b and interna heat source q b see Fig 2 Denote therma conductivities of the partices a and b, and assume that b 1, where 1 is therma conductivity of a host medium Temperature gradient induced by a partice a at the surface of a partice b, T 0 a (b), resuts at the fow sip at the surface of the partice b If the partice b is immobiized, then there is a force F 0 b acting at the partice due to the viscosity of a host medium Simiary, a force F 0 a acts at the partice a due to an

5 J App Phys, Vo 9, No 7, 1 Apri 200 Yu Doinsky and T Eperin 425 T D R 5 k 26 In expression 26 we took into account that R R Hereafter we assume that R a R b A gradient of the temperature induced by a heat source b in the vicinity of a partice a can be determined using expression 18, and in the eading order approximation FIG 2 Locations of partices a and b in a host medium with therma conductivity 1 R a, R b and a, b are radii and therma conductivities of partices a and b, respectivey induced temperature gradient T 0 b (a) However, due to the thermophoretic sef action which is caused by the induced temperatures T i a (a) and T i b (b), the partices wi be attracted to each other if therma conductivities of partices are arger than therma conductivity of the host medium Since the effect of the induced temperature is the effect of the higher order in the parameter R a / ab and R b / ab, where ab is a distance between the centers of the partices, the tota effect is mutua repusion of partices However, if interna heat sources are such that q b q a, then T i a (a) T 0 b (a) In this case the sef-induced attractive force acting at partice a is arger than a repusive force produced by partice b The resuting effect is that partice a is attracted to partice b Let us determine these forces using a method of refections Due to inearity of the probem temperature T as in the previous case can be represented as TT a T b, where T T 0 T T 24 Temperature T 0, where a,b is determined by Eq 1 and boundary conditions 14 at the surfaces of both partices Thus temperature T 0 is determined by expression 18 where 0, Q, a must be repaced by, Q, R Temperature T is determined by heat conduction equation with boundary conditions 16 where subscript S is repaced by subscript When a then b and vice versa For spherica partices with radii R and R TD 1 1 R 2 1 P rc 1, 25 where k n, k(c c )/, c c, n (r c )/rc, D R 2 Q / 1 Expression 25 is a mutipoes expansion of temperature T reative to a center of a partice This expression foows from Eq 18 for T 1 0 and the known reation for a fied of a point source near a sphere for detais see, eg, Ref 10, Probem 157 In order to determine expansion of temperature T into mutipoes series with respect to a partice one can use Ref 7 Chap 4, Sec 7 In the range rc in the eading order approximation with respect to the parameter rc /1 we find T 0 b a R b 2 Q bk 1 2 D b 2 k In order to determine temperature T a we must use an equation simiar to Eq 17 when we have to take into account temperature gradient T 0 b (a), Thus for T we have the foowing boundary conditions: T 0, n T 1 n T T 0 Denote 1 2R D D 2 1 k 2 Then we obtain T rc, 1 27 T 1 1 R 2 1 rc rc In this approximation the tangentia component of the temperature gradient ( T) is determined by the foowing expression: T f, f Inspection of the expression for Eq 27 yieds the foowing condition when temperature gradient at the partice s surface changes sign: or D 1 R D 2 1 Q R 2 R Q, It is possibe to continue the iteration procedure keeping ony the eading order term at each iteration Such a procedure can be termed a procedure of eading order iterations Since every partice is characterized by its parameters, the effect induced by one partice at the current iteration may exceed the effect induced by another partice at the preceding iteration It is exacty a condition Eq 29 that describes a situation when at the first iteration the effect of partice prevais over the effect of partice at the zeroth iteration Let us determine the forces exerted by fuid at the partices when partices are in rest in the aboratory frame of

6 426 J App Phys, Vo 9, No 7, 1 Apri 200 Yu Doinsky and T Eperin reference As it was noted above this impies that there is a force F which is appied at the partice and keeps it at rest Thermophoretic force that is cacuated in this study, F T F Simiary to Sec II for simpicity we consider the probem in the zeroth order approximation in Knudsen number Kn The veocity fied is determined by Eq 1 with the foowing boundary conditions: u T, n u 0, u 0 0 Here is determined by formua 1 where T S is assumed to be equa to the temperature of the host medium without partices The atter assumption is vaid when the heat reease rates are not high, so that according to Eq 19 T S T Fuid veocity is given by superposition uu a u b, where u u 0 uu Each iteration u n is determined by the equations 2 ( u n )0, u n 0 with corresponding boundary conditions Thus, the term u 0 satisfies boundary conditions 0 at the surface of partice and u n u n1, 1 where n0 and denotes coordinates at the surface of partice Since the partices are kept at rest the ony characteristic direction in the probem is determined by a vector k Expression for a veocity u n can be written as foows see Ref 9, Chap 1, Sec 15 u n 1 rc a n Ŝ 1 b n 2 R rc 2 Ŝ 2 k, 2 where Ŝ 1 ên n, Ŝ 2 ên n Since the probem is a inear one the tota force acting at the partice can be written as F F n, and F n 8a n see Ref 5, Chap 2, Sec 20 In order to transform the mutipoes expansion with respect to a partice into a mutipoes expansion with respect to a partice, we wi use the foowing formuas: n n tk P 1 t, trc /, 0 n n tk P 1 t, trc /, 0 where k n and k n In the first nonvanishing approximation with respect to parameters t and t we arrive at the foowing expression for the veocity: u n1 2 a n1 k Then boundary condition 0 and formua 1 yied a n R 2 a n1, b n 1 R 2 a n1 Thus in the eading order approximation the foowing force acts at partice a: F a F a 0 2 R a F b 0 Here the force F a 0 is caused by the temperature gradient at the surface of partice a and it is given by the foowing expression: F a 0 ( 2 /T ) a R a f a or F a 0 2 T f R a a 2 R b 2 Q b 1 R a 2 Q a 1 k 1 b 2 R b k, b f a 1 a 2 1 The force given by expression 4 comprises two terms The first term describes a force induced by an externa source b whie the second term describes a sef-induced thermophoretic force The second term in expression, F a R a 2 F b 0, is caused by a fuid fow induced by a temperature gradient at partice b If condition 29 where subscript corresponds to a partice b is satisfied, then contribution of a sef-induced temperature gradient into a force F 0 b can be negected (Q b Q a) and F 0 b 2 T f R 2 a Q a br b 1 k 5 2 Expression 5 determines a force acting at a partice b due to the temperature gradient at its surface induced by a partice a Expressions 5 provide a soution of the probem We do not present here a genera formua that can be obtained by substituting Eqs 4 5 into Eq, and consider conditions when a force acting at a partice a is attractive These conditions comprise a condition 29 that can be rewritten as b 1 R 2 a R b Q b2q a b and a condition F a 0 2 R a F b 0 see Eq If condition 6 is satisfied within a wide margin of safety, then the first term in the right-hand side of Eq 4 can be negected eg, Q b0) and the condition for attraction reads f a 2 R b f b b The atter condition corresponds to a case when a sefinduced by a partice a thermophoretic force attracting force is arger than a thermophoretic force acting at partice a due to a fuid fow caused by a temperature gradient at a partice b repeing force Thus, a heat reeasing partice with a ow therma conductivity is attracted to the partice with a high therma conductivity In the imiting case when b 1 and a 1, the condition for attraction Eq 7 can be rewritten as

7 J App Phys, Vo 9, No 7, 1 Apri 200 Yu Doinsky and T Eperin 427 R b 2 2 a 8 4 b It is interesting to note that in the range of the parameters indicated above neither condition 7 nor condition 8 depend upon the size of a partice a and a heat reease rate Since R b is fixed, condition 7 determines the distance between the partices where repusion changes to attraction In contrast to a case of a partice interacting with a pane surface where attraction occurs at any distance, attraction between spherica partices occurs ony within a certain finite distance The reason for this behavior is a fuid fow sip at the partice b which is caused by a heat source Q a The higher the therma conductivity of a partice b, b, the ower the temperature gradient at the surface of a partice b that is induced by a heat source q a and the smaer the repusion force F a R a 2 F b 0 Let us compare the magnitude of a thermophoretic force F atr a which is responsibe for attraction of partice a the second term formua 4 with a cassica thermophoretic force F ext Eq 22 Since in Sec II we have aready compared a thermophoretic force F acting at a partice near the interface Eq 21 with F ext see Eq 2, it is sufficient to compare a force F atr a with F Equations 21 and 4 yied F a atr F 8R b a 1 9 b 2 1 In deriving Eq 9 for consistency of notations we introduced the foowing changes in Eq 21: 2 b, a R a, atr Q Q a In the most interesting case when b 1, F a (8R b / )F IV CONCLUSIONS We considered a thermophoretic force caused by the sef action of a heat reeasing or absorbing partices near the interface between two media with different therma conductivities This force is caused by the induced temperature gradient, which is proportiona to the rate of heat reease absorption by the partice Therefore the magnitude of the thermophoretic force is proportiona to the rate of heat reease absorption by the partice, and its direction depends upon the sign of the parameter 1 2, where 1 is therma conductivity of a host medium and 2 is therma conductivity of the adjacent medium The obtained resuts impy that a heat reeasing partice is attracted to the interface when therma conductivity of a host medium is ess than therma conductivity of the adjacent medium Thus, eg, growing in air by condensation partice is attracted to a meta surface whie an evaporating in air partice is repeed from a meta surface We determined a condition for mutua attraction of heat reeasing partices and derived an expression for the thermophoretic force acting at the partice In contrast to a case of a partice interacting with a pane surface where attraction occurs at any distance, attraction between spherica partices occurs ony within a certain finite distance It must be noted that thermophoretic interaction of partices was considered in earier studies see, eg, Refs and references therein However, a case with heat reeasing partices was not considered before athough it may pay an important roe in various phenomena, eg, capture of condensing partices by a partice with a high therma conductivity, capture of oxidizing partice by a arger partice corrosion, etc ACKNOWLEDGMENTS This study was partiay supported by German Israei Project Cooperation DIP administered by the German Ministry of Education and Research BMBF and by INTAS Grant No J Happe and H Brenner, Low Reynods Number Hydrodynamics Martunus Nihhoff, Dordrecht, S Kim and J Karria, Microhydrodynamics: Principes and Seected Appications Butterworth-Heinemann, Boston, 1991 S K Friedander, Smoke, Dust, and Haze Fundamentas of Aeroso Dynamics Oxford University Press, York, S P Bakanov, B V Deryagin, and V I Rodugin, Sov Phys Usp 22, Usp Fiz Nauk 129, ; S P Bakanov, Sov Phys Usp 5, Usp Fiz Nauk 162, L D Landau and E M Lifshitz, Fuid Mechanics Butterworth- Heinemann, Oxford, L D Landau and E M Lifshitz, Eectrodynamics of Continuous Media Pergamon, Oxford, E W Hobson, The Theory of Spherica and Eipsoida Harmonics Cambridge University Press, Cambridge, Tatsuo Kanki, Heat Transfer-Jpn Res 27, E M Lifshitz and L P Pitaevsky, Physica Kinetics Pergamon, Oxford, V V Batygin and I N Toptygin, Probems in Eectrodynamics Academic, London, H J Keh and S H Chen, Chem Eng Sci 50, S H Chen and H J Keh, J Aeroso Sci 26, Yu Doinsky and T Eperin, Phys Rev E 64, S H Chen, J Aeroso Sci,