Simulation of the motion of a sphere through a viscous fluid

Transcription

1 ENSEÑANZA REVISTA MEXICANA DE FÍSICA 49 () ABRIL 003 Simulaion of he moion of a sphere hrough a viscous fluid R.M. Valladares a, P. Goldsein b, C. Sern c, and A. Calles d Deparameno de Física, Faculad de Ciencias,Universidad Nacional Auónoma de México, Aparado Posal 04510, México, D. F. a [email protected] b [email protected] c [email protected] d [email protected] Recibido el 11 de mayo de 001; acepado el 0 de junio de 00 We have designed a friendly program o help sudens, in he firs courses of physics and engineering, o undersand he moion of objecs hrough fluids. In his paper we presen a simulaion of he dynamics of a sphere of arbirary bu relaively small radius hrough an incompressible viscous fluid. The exernal forces acing on he sphere are graviy, fricion, a sochasic force ha simulaes microscopic ineracions and buoyancy. The Reynolds numbers are small enough o assure unseparaed and symmerical flow around he sphere. The numerical analysis is carried ou by solving he equaion of moion using he Verle algorihm. Besides he numerical resuls, he program includes an ineracive animaion of he physical phenomenon. Alhough originally conceived for eaching, he program may be used in research o invesigae, among oher hings, he moion of raindrops or polluans in he amosphere. Keywords: Fluid dinamics; Sokes law; sochaics forces; polluans. Hemos diseñado un programa ineracivo y amigable que permie a los esudianes de los primeros cursos de física e ingeniería enender el movimieno de objeos a ravés de fluidos. En ese rabajo presenamos una simulación de la dinámica de una esfera de radio arbirario, pero relaivamene pequeño, a ravés de un fluido incompresible y viscoso. Las fuerzas exernas que acúan sobre la esfera son la gravedad, la fuerza de arrasre viscoso, el empuje y una fuerza esocásica mediane la cual se simulan las ineracciones microscópicas del medio sobre la esfera. Los números de Reynolds considerados son lo suficienemene pequeños para asegurar un flujo no separado y simérico alrededor de la esfera. Se resuelve la ecuación de movimieno numéricamene uilizando el algorimo de Verle. Adicionalmene a los resulados numéricos, el programa incluye una animación ineraciva del fenómeno físico. Aunque ese programa fue concebido inicialmene para ser uilizado con fines de enseñanza, puede ser uilizado para realizar invesigación en problemas ales como la dinámica de goas de lluvia o de conaminanes en la amósfera. Descripores: Dinámica de fluidos; ley de Sokes; fuerzas esocásicas; conaminanes. PACS: j; x; j 1. Inroducion In his paper we presen a numerical soluion and a simulaion of he moion of a sphere hrough an incompressible and viscous fluid. The aim is o find and illusrae soluions o hydrodynamic and hermodynamic problems using compuer simulaions. The moion of a sphere hrough a viscous fluid can be sudied heoreically and experimenally from he firs undergraduae courses in Physics, and has diverse applicaions. The mehod used o solve he equaion of moion is he posiion Verle algorihm [1] wih an adjusmen for velociies ha depend linearly on forces. This algorihm is successfully used in molecular dynamics and, given conservaive forces, guaranees he conservaion of energy. The program can be used wih various objecives in mind since i is quie versaile. On one hand we are able o develop a eaching aid which will be useful in boh experimenal and heoreical courses. On he oher, we show ha he Verle algorihm, frequenly used in molecular dynamics, may be used as well in his oher conex. This code has also been applied o problems involving drops of waer or polluans falling in air which are of ineres in research.. The dynamics We consider a sphere of radius R ha moves in a viscous fluid. Acing on he sphere are he force of graviy, buoyancy and viscous drag. As we shall furher discuss we may add o hese hree forces a sochasic force (see Fig. 1). FIGURE 1. Schemaics of he exernal forces acing on he sphere.

2 SIMULATIONOFTHEMOTIONOFASPHERE THROUGHAVISCOUSFLUID 167 Buoyancy, given by Archimedes principle assers ha a body in a fluid experiences an apparen reducion in weigh equal o he weigh of he displaced fluid. The drag force associaes all forces ha resis he moion of he sphere due o fricion (viscosiy) or o he moion of he fluid close o he sphere. In general, he drag force canno be represened in a simple mahemaical way since, besides he effec due o viscosiy (viscous drag), for Reynolds numbers larger han five, he form of he wake (form drag) is deerminan in describing he behavior of he sphere. We recall ha he Reynolds number is an adimensional parameer ha characerizes he flow and is given by R e = Rvρ, (1) η where v is he uniform speed of he flow, η is he viscosiy and ρ he densiy of he fluid. For small Reynolds numbers, he flow around he sphere is symmerical and does no separae, and he drag can be considered o be due only o fricion forces over he surface of he sphere. Sokes [] calculaed he drag force for a saic sphere in a seady, uniform, infinie viscous and incompressible flow (or a sphere ha moves hrough a saic fluid) for sufficienly small Reynolds numbers, F a = cv = (6πηR)v. () Parameer c is of geomeric origin, and has been calculaed for geomeries oher han spheres. Comparison of Eq. () wih experimens show ha for Re< 0.5 he expression is very good and accurae o abou 10% for Re close o 1 [3]. For Reynolds numbers larger han five he wake behind he sphere ges more and more complicaed as he speed increases. This form drag can be reduced designing aerodynamic objecs ha diminish he wake. The value of various geomerical facors as well as he evoluion of he wake wih Reynolds numbers can be found in any inroducory book of fluid mechanics [, 4]. For solid paricles falling in air, he geomery correcion is negligible. In his paper we limi our analysis o small Reynolds numbers in which he drag is due only o viscosiy, and a simple mahemaical form can be assumed. If we consider a paricle subjec o graviy, buoyancy and drag as menioned above, he balance of he hree forces will deermine wheher he moion will be uniform or acceleraed. In he examples shown below, he iniial moion is acceleraed and, under cerain condiions, as he drag force increases wih speed he forces balance ou and he paricle aains erminal (consan) speed. Equaion () was obained for seady condiions so in he acceleraion phase addiional erms mus be considered. One of hose erms is relaed o he resisance of he nonviscous fluid o he moion of he paricle. I can be obained by adding a cerain mass of he fluid o he rue mass of he sphere and is called virual or added mass. Added mass effecs are usually associaed wih bodies immersed in liquids because heir densiy is larger han in gases [3]. In he case of small solid paricles or raindrops falling in air he added mass effec is of he order of 10 3 or smaller. The oher erm, proposed originally by Basse, is relaed o he diffusion of voriciy. I is also more imporan in liquids han in gases and for regimes wih high Reynolds numbers. The erm can be negleced in he examples ha we show below. When he moion of fluid spheres in anoher fluid is considered he recirculaion of he fluid a he inerface migh play an imporan role. This is rue however only for gas bubbles in a liquid. In he case of raindrops i has been shown ha hey behave like solid paricles: when hey are very small Eq. () is valid and for higher Reynolds numbers he drag depends on he complex flow ha surrounds he sphere. In addiion o he hree forces menioned above we may propose he exisence of a sochasic force ha depends on he emperaure and which allows he sudy of cases in which Brownian moion migh be of ineres. The form of his sochasic force [5] is F e = G RAN [ ckb T ] 1, (3) where G RAN is a random number wih a Gaussian disribuion cenered a zero and sandard deviaion 1, k B is Bolzmann s consan, T is he absolue emperaure, c is he coefficien in Sokes formula [Eq. ()]. In molecular dynamics i is common o use sochasic moion equaions o carry ou he sampling of a canonical ensemble. The dynamic equaion for he sphere in one dimension is herefore given by he following Langevin ype equaion [4]: m dv d = md x d = cv + F T, (4) where he oal force F T is given by F T = mg V ρg + F e, (5) and c = 6πηR, V he volume displaced by he sphere and m is he sphere s mass. 3. The Verle algorihm In order o solve Eq. (4) numerically we urn o he Verle algorihm so frequenly used in molecular dynamics [1]. We now make a brief descripion of he Verle algorihm wih a modificaion o accoun for forces ha have a linear dependence on velociy such as Sokes drag force. Le us call he ime sep of he Verle algorihm h= where h is assumed very small compared o he normal ime scale of he problem. To calculae he posiion and velociy we use a Taylor series expansion for x( + h): x( + h) = x() + ( dx d ) h + 1 ( d x d ) h +..., (6)

3 168 R.M.VALLADARES,P.GOLDSTEIN,C.STERN,ANDA.CALLES and x( h) = x() ( ) dx h + 1 d ( d ) x d h... (7) Adding Eqs. (6) and (7) up o second order we find x( + h) = x() x( h) + a()h. (8) On he oher hand subracing Eq. (7) from Eq. (6) we find ( ) dx x( h) x( + h) = h, (9) d so he expression for he velociy urns ou o be: ( ) dx x( + h) x( h) v() = =. (10) d h Since he force depends on he velociy, he posiion a ime + h will depend on he velociy a ime, and his velociy requires he expression for x( + h). If we wrie where and a() = F () m = b c v(), { b = g 1 [ V ρg + G RAN ckb T m c = c m and subsiue i ino Eq.(8) we have ] 1 }, x( + h) = x() x( h) + [b c v()] h. (11) Subsiuing eq.(10) and facorizing, ( ) ( ) x(+h) 1+ c h =x() x( h) 1+ c h +bh, (1) we have finally ha he uncoupled equaions o solve by ieraions are ( ) x() x( h) 1 + c h + bh x( + h) = ( ) (13) 1 + c h and x( + h) x( h) v() =, (14) h which consiue he modified Verle algorihm for forces ha have a linear dependence on velociy. For = 0 we have x(0) = 0, v(0) = v 0, a(0) = a 0 = g 1 m ( ) 1 ckb ρv g + c v 0 G ran T, o calculae x(h) we need x(-h) which we obain from Eq. (7) o second order: x( h)=x(0) v(0)h+ 1 a(0)h =x 0 v 0 h+ 1 a 0h. (15) Wih hese values for he iniial posiion and velociy, ieraing we can generae all he previous values for boh posiion and velociy a ime inervals of h. This algorihm, aside from being fas and reliable, is quie compac and easy o program [1]. 4. Applicaions In his paper wo differen applicaions of he sofware, ha we hink are of ineres o sudens and researchers, are considered. The resuls are presened in he nex secion. The firs phenomenon sudied is he precipiaion of raindrops of differen radii (from 1x10 5 o 1x10 3 m.). Each radius is kep consan hroughou he fall. However, an opion for a variable radius exiss in he program, and more sophisicaed models can be easily inroduced. The moion of each drop depends on he size. The smalles drops remain floaing in he air, he middle one acquires very soon erminal speed, and mos of he rajecory has consan velociy. The bigger drops remain in acceleraed moion. The second applicaion refers o he moion of polluans in he amosphere. Five differen maerials in wo ypical sizes are considered. From his very simple model i can be seen ha he smalles polluans can remain floaing in he air ha we breahe for long periods of ime. The sochasic force always has a small bu noiceable effec in he moion of he polluans. 5. Resuls Using he previous equaions and iniial condiions, an ineracive program in C language was wrien o calculae posiion and velociy as a funcion of ime. The program was developed for PC in MS-DOS environmen, and shows an animaion of he moion. The parameers and iniial condiions can be changed easily using special keys. An example of a screen ha will appear o he suden is shown in Fig. (see he appendix for deails). The program can do up o five numerical experimens simulaneously. Spheres are dropped inside ubes filled wih a paricular fluid a a given emperaure. The use of he ubes is only for visual effecs. All hypoheses consider he moion of a sphere in an infinie medium. The program has a daabase of en differen fluids and weny-four possible maerials for he spheres. Table I has he values of he densiies and viscosiies of he fluids in SI unis. Table II has he densiies of he maerials for he spheres. Figure shows he simulaion of he fall of waer drops wih radii varying from 1x10 5 o 1x10 3 m. The ime sep used in he inegraion is 0.004s. The saring poin is in he middle of he screen. The smalles drops do no seem o

4 SIMULATIONOFTHEMOTIONOFASPHERE THROUGHAVISCOUSFLUID 169 FIGURE. Simulaion of he fall of waer drops wih radii varying from 1x10-5 o 1x10-3 m. TABLE I. Fluid Densiy (kg/m) 3 Viscosiy (kg/ms) Vacuum 0 0 Air Waer Oil (10) Oil (0) Oil (30) Oil (40) Oil (50) Glycerine Helium move. They floa wihou falling in he ime scale considered. The reason is ha here is a minimum size for waer drops o precipiae. The wo drops on he righ are much bigger and ouch he boom very soon: 0.464s and 0.456s as shown on op of heir respecive ubes. Their Reynolds number was already high afer 0.4s, and he assumpion of symmerical flow around hem was probably no valid in he las par of heir rajecory. Figures 3 and 4 show he posiion and speed of he drops as a funcion of ime. Paricle 1 moves very lile wih speed close o zero, paricles and 3 aain erminal velociy very soon, and paricles 4 and 5 remain in acceleraed moion. The simulaion is a close approximaion only for he firs wo raindrops whose moion remains a very small Reynolds numbers. For drop number hree he ype of moion is correcly described even hough he acual values of he speed and posiion are no accurae. The Reynolds numbers of he las wo raindrops are oo big o be described by his heory. The sochasic force due o emperaure is exremely small in hese experimens, is effec can be negleced. FIGURE 3. Posiion of he waer drops as a funcion of ime. For he wo smalles paricles he graphs are sraigh lines almos from he beginning because hey soon aain erminal velociy, and move a consan speed. The hird paricle sars wih acceleraed moion (curved line) bu evenually aains erminal velociy. The wo bigger paricles fall wih acceleraed moion all he ime.

5 170 R.M.VALLADARES,P.GOLDSTEIN,C.STERN,ANDA.CALLES TABLE II. Maerial Densiy (kg/m) 3 Seel 7850 Asphal 100 Sulphur 1960 Clay (1) 1800 Clay () 600 Sand (dry) 1600 Asbesos 500 Coal (M) 100 Coal (V) 300 Wax 960 Concree 400 Cork 00 Glass fiber 100 Chalk 1800 Graphie 100 Ice 960 Soo 1600 Brick 1400 Wood 500 Paper 700 Peroleum 800 Lead Helium 0.18 Waer 1000 FIGURE 4. Velociy of he waer drops as a funcion of ime. The speed of he firs drop remains almos zero. The speed of he second is consan mos of he ime. The hird shows a clear change of behavior. Drops 4 and 5 move wih consan acceleraion Figure 5 shows he fall in air from a heigh of one meer of five differen conaminans: lead, soo, sulfur, clay and coal. All spheres have a radius of 5x10 6 m. These are ypical sizes for PM-10 polluans in he air. The paricles fall very slowly so he inegraion ime has been increased (0.04s). The program can be sopped eiher when he firs or when he las paricle ouches he ground. In his experimen he lead paricle ouched he boom a 8.9s as can be seen a he op FIGURE 5. Simulaion of he fall in air from a heigh of one meer of five differen PM-10 (radius of 5x10-6 m) conaminans: lead, soo, sulfur, clay and coal.

6 SIMULATIONOFTHEMOTIONOFASPHERE THROUGHAVISCOUSFLUID 171 of he corresponding ube. The oher four paricles are sill in he air. The insan a which hey ouch he ground will appear above he posiions and velociies a he op of each ube. Their Reynolds number appears a he boom. In he case of variable radius, he size would be shown in his space also. As can be seen, all Reynolds numbers correspond o he range where our hypohesis is valid. The parameers of he experimen appear below each ube. The oal ime up o he insan when (S) was yped (31.640s) appears a he op righ corner of he screen. Figure 6 shows he graph of speed as a funcion of ime for each of he paricles. All paricles sar downwards wih acceleraed moion. Evenually, he drag force increases wih speed, and soon he sum of he Sokes and he buoyancy forces balances he weigh. The paricles seem o aain erminal speed and coninue a consan speed. However, if he proper scale is ploed like in Fig. 7, i can be seen ha here is always a small variaion around he erminal speed. This is due o he sochasic force. In Fig. 8 posiion is ploed as a funcion of ime. The sochasic effec is no observed a his scale. The graph shows sraigh lines as if all paricles moved a consan speed. When he paricle ouches he boom, he posiion remains consan. To sudy he effec of he sochasic force in more deail, he simulaion migh be carried ou wihou graviy as shown in Fig. 9. The paricles oscillae abou he origin, his figure shows an insan of ime. Figures 10 and 11 show he speed and posiion of he paricles as a funcion of ime. The paricles have a random moion abou he origin wih an ampliude ha seems o depend on he mass. These resuls are valid in a direcion perpendicular o he fall. For his simulaion he disance inerval beween op and boom was reduced FIGURE 7. Small variaions abou he erminal speed due o he sochasic force. FIGURE 8. Posiion is ploed as a funcion of ime for PM10 paricles. The sochasic effec is no observed a his scale. The graph shows sraigh lines as if all paricles moved a consan speed. When he paricle ouches he boom, he posiion remains consan. FIGURE 6. Velociy as a funcion of ime for each of he PM10 paricles. All paricles sar downwards wih acceleraed moion. All paricles seem o aain erminal velociy, and coninue a consan speed. However, if he proper scale is ploed like in Fig. 7, i can be seen ha here is always a small variaion around he erminal speed. This is due o he sochasic force [-5 x10 5, -5 x10 5 m] bu he same ime incremen (0.04s) was used in he inegraion. To sudy random signals i is ofen convenien o use saisical mehods. Figure 1 shows a hisogram of he posiion of coal wihou graviy for a very long run. The hisogram shows he number of imes a cerain value is aained during he run. I can be observed ha he highes values corresponds o he origin. The paricle goes up or down randomly. I could be expeced ha as we le he ime go o infiniy, he peak a he origin will increase and he hisogram will be more symmeric showing ha he number of imes he paricles goes above he origin is he same as he number of imes i goes below.

7 17 R.M.VALLADARES,P.GOLDSTEIN,C.STERN,ANDA.CALLES FIGURE 9. Simulaion of he moion of PM10 paricles wihou graviy. lead paricles will remain in he air for over 1 minues, bu he ones made ou of he four oher maerials will ake over an hour o ouch he ground. In he case of he smaller PM.5 polluans, he lead ones will remain in he air over 3 hours bu he oher maerials will remain floaing beween 0 o 30 hours if emied by a all chimney. If we consider only he firs 3m above he ground, where we breahe, he less dense paricles will remain in he air over hours. Thus, hese smaller paricles may be considered hazardous o healh and represen one of he main causes of air polluion. FIGURE 10. Posiion of PM-10 paricles wihou g as a funcion of ime. The paricles have an erraic moion whose ampliude seems o depend on he mass. The speeds and posiions for he fall of smaller conaminans, PM-.5 (wih a radius of 1.5 x 10 6 m.) are shown in Figs. 13 and 14. These paricles being smaller fall more slowly, and he effec of he sochasic force can be deeced in he velociy. All posiions sar as a curve as in acceleraed moion bu evenually become sraigh lines. If we consider ha polluans are ejeced ino he amosphere by chimneys abou 5m high, using he daa obained in he simulaion we can conclude he following. The 5µ FIGURE 11. Velociy of PM-10 paricles wihou g as a funcion of ime.

8 SIMULATIONOFTHEMOTIONOFASPHERE THROUGHAVISCOUSFLUID 173 FIGURE 1A. Posiion of PM-.5 (radius of 1.5 x10-6 m) paricles as a funcion of ime. The paricles have an erraic moion whose ampliude seems o depend on he mass. FIGURE 14. Velociy of PM-.5 paricles as a funcion of ime. The paricles have an erraic moion superposed o he fall. Noe ha he speed diminishes before i becomes approximaely consan. I is ineresing o noe ha even his simple model gives informaion on phenomena sudied by meeorologiss. 6. Animaion as a eaching ool FIGURE 1B. Hisogram corresponding o Fig. 1a. The highes frequency corresponds o he origin. FIGURE 13. Posiion of PM-.5 (radius of 1.5 x10-6 m) paricles as a funcion of ime. The eacher can use he sofware presened in his paper in various ways. 1. In early undergraduae Physics courses, sudens may easily compare experimenal measuremens of differen falling spheres in differen fluids wih he numerical predicions. A his level we recommend eliminaing he sochasic erm. This program is friendly enough o allow sudens o change he parameers according o he real experimens ha may be achieved in he lab. This experience will enable sudens o undersand he uiliy of numerical experimens as useful ools o plan acual experimens, deermine he bes parameers o be used in he laboraory, and predic resuls.. In an advanced course, he sochasic erm may be included and he suden may visualize he numerical resuls of a sochasic phenomenon described by he Langevin equaion. In order o enhance he sochasic behavior, he graviy erm migh be omied. Saisical mehods can be used o deermine he mean speed and he rms values. 3. In a numerical analysis course, his program may be useful as an example of he applicaion of he Verle algorihm as well as he use of C code in visualizaion. 4. This program can be used in differen research problems. I acually may be useful in dealing wih problems such as he precipiaion of raindrops and he dispersion of polluans. We have specifically presened in his work he cases of he PM-10 and PM-.5 paricles ha represen he main cause of polluion by paricles in Mexico Ciy.

9 174 R.M.VALLADARES,P.GOLDSTEIN,C.STERN,ANDA.CALLES Appendix How o use he program This program is easier o use in an MS-DOS environmen even hough i can also be accessed from Windows. The main execuable file is called Spheres. This program uses a file called fall.da where he informaion for he experimen is kep. The srucure of fall.da is described below. Informaion for differen experimens should be kep in files called fall*.da where * denoes a number. Once Spheres is execued he names and work places of he auhors will appear. The leer (C) is used o coninue. The program firs les he user choose among four languages. Once he language is chosen an (C) is yped, i asks if i should run wih he daa kep in file fall.da. If his is he case, he leer (C) should be yped. If no, he leer (o) is yped and he program asks for he number in fall*.da. Once he daa file is chosen and (C) is used o coninue, a friendly screen like he one in figure (5) appears. The screen can be easily modified following he insrucions on he op. Wih he leer N (n) he number of ubes can be increased (decreased) from 1 o 5. Wih he leer (G) he program keeps he daa of he new run in daa*.res. To change he characerisics of one of he ubes, he user has o ype he number of he ube. Then he characerisics are highlighed in anoher color. Wih (E) one can modify he maerial of he sphere (densiy), wih R he radius, wih (F) he fluid (densiy and viscosiy), wih (T) he emperaure, wih (G) he acceleraion of graviy and wih (D) he posiions of he op and boom of he ube. Wih (Q) one eners he non-highlighed screen. All daa is given in SI unis. Once he parameers of each experimen have been chosen, he program runs while showing he simulaion. The program ends when he slowes paricle ouches he boom or he op of he ube bu i can be sopped by he user a any insan wih [S] and coninued wih [C]. Wih (E) i can be reiniiaed. Wih [Esc] he user can exi he program a any ime. The oal ime appears on he op righ corner of he screen. The ime a which he sphere ouches he boom and is insananeous posiion and velociy appear a he op of each ube. The posiions of he op and boom of he ube are marked. The Reynolds number, he size of he radius of he sphere if i is no consan andhe characerisics of he sphere and of he fluid appear a he boom of each ube. For he sphere he characerisics are he maerial wih is densiy and radius and for he fluid he densiy, viscosiy and emperaure. The value of he acceleraion due o graviy can also be changed. Some daa files wih various characerisics have been creaed bu he user can creae new files depending on he naure of he sudy. For example, he daa file used o sudy he moion of waer drops of differen sizes as hey fall in air is kep as fall1.da, see Table AI. TABLE AI E E E E E In he firs line, he firs number indicaes he number of possible ubes. The second number indicaes wheher i should sop when he firs paricle ouches he ground (-1), when he las one ouches he ground (1) or coninue indefiniely (0). The hird number indicaes he ime incremen in he inegraion. The following lines have he same forma. The firs number corresponds o he solid as found in solids.da (see Table I). The second number corresponds o he fluid as i appears in fluids.da (see Table II). The following numbers correspond respecively o he iniial posiion and velociy, he radius, he posiions of he op and boom of he ubes measured from he saring poins, he acceleraion of graviy and he opion of having consan (0) or variable (1) radius. The only subrouine for variable radius implemened in he program considers linear growh. However oher models can be implemened in he sofware. If he leer [G] is chosen on he screen, he daa of velociy and posiion as a funcion of ime is saved in file res*.da. They can be rerieved easily and ploed. The execuable file can be obained wih any of he auhors. A deailed manual in Spanish is in process. 1. M.P. Allen and D.J. Tildesley, Compuer Simulaions of Liquids (Clarendon Press, Oxford, 1997).. R.W. Fox and A.T. Mc Donald, Inroducion o Fluid Dynamics (John Wiley and Sons, 4h ediion, 1998). 3. R.L. Panon, Incompressible Flow (John Wiley and Sons, 1984). 4. M.C. Poer and J.F. Foss, Fluid Dynamics (John Wiley and Sons, 1975). 5. F. Reif, Fundamenals of Saisical and Thermal Physics (Mc Graw-Hill Inernaional Ediions, 1965).