Experimental Investigation on the Thermal Conductivity and Viscosity of ZnO Nanofluid and Development of New Correlations

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1 Trans. Phenm. Nan Micr Scales, (): 49-60, Summer Autumn 04 DOI: /tnms ORIGINAL RESEARCH PAPER. Exerimental Investigatin n the Thermal Cnductivity and Viscsity f ZnO Nanfluid and Develment f New Crrelatins S. Abarzadeh, M. Farhadi,*, K. Sedighi, M. Ebrahimi Mechanical Engineering Deartment, University f Babl, Babl, I.R. Iran ARTICLE INFO. Article histry Received 5 Octber 03 Acceted June 04 Keywrds Ethylene Glycl Nanfluid Prylene Glycl Thermal Cnductivity Viscsity Abstract In this aer, the measurement f the viscsity f ZnO in ethylene glycl, rylene glycl, mixture f ethylene glycl and water (60:40 by weight), and a mixture f rylene glycl and water (60:40 by weight) and the thermal cnductivity in ethylene glycl and rylene glycl as base fluids in the range f temerature frm 5 ºC t 60 ºC are investigated. The results indicate that as the temerature increase the viscsity f nanfluid decrease and the thermal cnductivity f bth base fluid and nanfluid increase. Several existing mdels fr thermal cnductivity and viscsity are cmared with the exerimental data, and they d nt demnstrate gd cmarisn agreement. Finally, sme new mdels fr redicting the effective viscsity and thermal cnductivity are rsed. Furthermre, the viscsity f the base fluid affects the thermal cnductivity variatin f the nanfluids. The results indicate that the largest enhancements in thermal cnductivity are 5% and 9% fr EG and PG as base fluids, resectively.. Intrductin Nanfluids are susensins cnsisting f slid nanarticles in sizes less than 00 nm. The cnventinal fluids such as ethylene glycl, il and water have lw thermal cnductivity. Using nanarticles can increase the thermal transrt caacities f base fluids. Research cnducted in the case f thermal cnductivity and viscsity are given. Chandrasear et al. [] investigated Al O 3 /water nanfluid with a nminal diameter f 43 nm at different vlume cncentratins (0.33 5%) at rm temerature. m * Crresnding authr address: mfarhadi@nit.ac.ir A maximum increase in viscsity f Al O 3 /water nanfluids at 5% vlume cncentratin was.36 times that f water. Duangthngsu and Wngwises [] used the TiO nanarticles disersed in water at vlume cncentratin ranging between 0.% and.0% demnstrated 3 7% mre thermal cnductivity than water. Schmidt et al. [3] measured the thermal cnductivity f vl. % Al O 3 /decane nanfluids had a thermal cnductivity enhancement f %. Hrishiesh et al. [4] used Alumina nanarticles in three ranges f size with surface area-averaged size f, 45 and 50 nm. They shwed thermal cnductivity f alumina (50 nm)/ethylene glycl nanfluid with3% article vlume fractin in a temerature range f 0 ºC t 50 ºC varied frm 9.5% t %. 49

2 Abarzadeh et al./ TPNMS (04) Nmenclature μ abslute viscsity, Ns/m c secific heat caacity, J/g K ρ density, g/m 3 d article diameter, m ψ shericity, dimensinless thermal cnductivity, W/mK Subscrits b Bltzmann cnstant, J/K bf Base Fluid n shae factr, dimensinless c Centi Pise T temerature, K Eff Effective Gree Symbls Nanfluid β rati f nanlayer thicness t article radius, dimensinless P Particle vlume cncentratin, % r Relative Vajjha and Das [5] investigated ZnO nanarticles in a mixture f ethylene glycl and water f 60:40 (by mass) as a base fluid. They bserved thermal cnductivity rati enhancement between 98 and 363 K fr a 7% cncentratin was 8%. Yu et al. [6] measured the thermal cnductivity f ZnO in ethylene glycl as a base fluid. The enhanced value f 5.0 vl. % ZnO-EG nanfluid was 6.5%. Palabiyi et al. [7] investigated TiO and Al O 3 nanarticles in rylene glycl as a base fluid. The thermal cnductivity f the samles shwed a nnlinear increase with the article cncentratin. The enhancement f thermal cnductivity was nt temerature deendent and arallels the behavir f ure rylene glycl. Lee et al. [8] investigated the thermal cnductivity f nanfluids with Al O 3 and CuO nanarticles that nanfluids with CuO nanarticles have mre enhancement than Al O 3. In desite f Al O 3 has higher thermal cnductivity than CuO. In this study, the viscsity f ZnO nanarticles in ethylene glycl, rylene glycl, mixture f ethylene glycl and water (60:40 by weight), and a mixture f rylene glycl and water(60:40 by weight) and the thermal cnductivity in ethylene glycl and rylene glycl as base fluids in the range f temerature frm 5 ºC t 60 ºC are investigated. In this wr, viscsity and effective thermal cnductivity mdels are reviewed. The results demnstrate that there are significant differences between exerimental results and theretical mdels. It shws the necessity f exerimental wrings. At the end, sme new mdels fr effective viscsity and thermal cnductivity are resented.. Theretical investigatins.. Viscsity Many researchers have studied the effect f arameters such as the size and vlume fractin f nanarticles, the base fluid, and temerature n viscsity. There were sme existing theretical frmulas due t estimating the article susensin viscsities. The Einstein mdel [9] was the first mdel t calculate the viscsity f the susensin f sherical slids. Einstein [9] rsed a viscsity crrelatin fr nn-interacting article susensin in a base fluid when the vlume cncentratin was lwer than 5%..5 eff l () Brinman [0] cnsidered the effect f the additin f ne slute-mlecule t an existing slutin. The crrelatin was fr article cncentratins less than 4%..5 bf ( ) () The crrelatin was suggested by Wang et al. [5]. (3) bf ( ) Bachelr [] resented a frmula including the effect f Brwnian mtin. f (4) 50

3 Abarzadeh et al./ TPNMS (04) Thermal cnductivity Many researchers have studied the thermal cnductivity f nanfluids. There were several theretical mdels where they culd redict thermal cnductivity f article-fluid mixtures in a secial range f temerature and vlume fractins. Furthermre, many factrs such as size, surface area, and shae f nanarticles can have an effect n the mdels. The Maxwell mdel was the first mdel t determine the effective thermal cnductivity f liquid slid mixtures. Maxwell [] resented the thermal cnductivity mdel fr lw slid cncentratins. The effective thermal cnductivity was: eff / b b ( b) b ( b) (5), b and were the thermal cnductivity f the article and base fluid and the article vlume fractin in the susensin, resectively. Hamiltn and Crsser mdified Maxwell s mdel t determine the effective thermal cnductivity f nnsherical articles by cnsidering a shae factr. The Hamiltn and Crsser mdel [3] was used fr determining the effective thermal cnductivity f a tw-hase mixture: eff / b n b ( n) ( b ) n b ( b ) (6) n was the emirical shae factr given by n = 3/and was the shericity. Fr the sherical shae article, the shericity () is. Bruggeman [4] analyzed the interactins amng randmly distributed articles by using the mean field arach. The Bruggeman mdel [7] was given by 3 ( 3 ) eff 4 b b 4 (7) 3 ( / ) 3 b ( 9 9 )( / ) b (8) eff was the effective thermal cnductivity f liquid with article susensin, was the vlume fractin f articles, and b and were the thermal cnductivities f the base fluid and the article, resectively. Jeffrey [5] develed a theretical mdel which assumed the cnductin f heat thrugh a statinary, randm and statistically hmgeneus susensin f sherical articles in a matrix f unifrm cnductivity. He [5] assumed that the vlume fractin f articles is small. eff b (9) bf (0) bf bf Timfeeva et al. [6] exressed a mathematical mdel. It can be written as: () Vajjha and Das [7] rsed an emirical mdel, which invlved the deendence f thermal cnductivity t temerature and cncentratin. A B T C T 3( bf ) / bf ( ) ( ) ( ) () The cefficients A, B, and C were lynmial functins f cncentratin. K and Kleinstreuer [8] develed a new mdel fr nanfluids. Their mdel cntained the effects f article size, article vlume fractin, temerature and rerties f the base fluid and the articles. Their frmula was bf 5

4 Abarzadeh et al./ TPNMS (04) bf ( bf ) eff bf bf ( bf ) (3) 4 T B (50 ) bf cbf f ( T,, etc.) d It can be seen that there are nt general mdels t exlain the strange behavir f the nanfluids including the viscsity and effective thermal cnductivity. Currently there is n reliable thery t accetably redict the anmalus thermal cnductivity f nanfluids. Many arameters iluence thermal cnductivity and viscsity, including the thermal cnductivities f the base fluid and the nanarticles, the vlume fractin f the nanarticles, the surface area, and the shae f the nanarticle and the temerature. The significant differences between exerimental data and theretical mdels demnstrate the necessity f exerimental wrings. 3. Exerimental rcedure 3.. Prearatin f nanfluids There are tw techniques fr rearing nanfluid, ne ste and tw ste methds. The ne-ste methd entails the synthesis f nanarticles directly in the heat transfer fluid. Tw-ste methd, which is used in the study, first reares the nanarticles and then diserses them in the base fluid. The hysical rerties f nanarticles are summarized in Table. Nanfluid samles are reared by disersing ZnO nanarticles in ethylene glycl/water slutin (EG- W slutin), ure ethylene glycl, rylene glycl/water slutin (PG-W slutin) and ure rylene glycl at vlume fractins 0-3%. Table Physical rerties f nanarticle Purity (%) 99 size 30 nm nanarticle ZnO Furthermre, the ZnO nanarticles have electrstatic stabilizatin. This ccurs by the adsrtin f ins n the electrhilic metal surface. The adsrtin creates an electrical duble/multilayer, which results in a Clumbic reulsin frce between the nanclusters. Fr rearing nanfluid, accrding t the vlume fractin, the amunt f nanarticles is calculated, and then measured with a 0.00 recisin weighing scale (AND-FX300GD, gram) and slightly added t the base fluid. The disersin can be dne with varius hysical treatment techniques, such as the stirrer, the ultrasnic disrutr, and the high-ressure hmgenizer. These techniques deagglmerate the article clusters in rder t btain hmgeneus susensins. Fr disersing ZnO, nanarticles in the base fluid magnetic stirrer and ultrasnic disrutr technique, high intensity ultrasnic rcessr (Snics, 0 Hz and 750 W), are used. Magnetic stirrer is used fr abut 45 minutes and then the susensin ges under ultrasnic vibratin. The time f vibratin differs fr each nanfluid. It is hur and 90 minutes resectively fr ethylene glycl/zno, and rylene glycl/zno susensins. In Figure a, the ZnO nanarticles in rylene glycl as a base fluid befre affected by ultrasnic vibratin are shwn. The hmgenus nanfluids after snicatin are illustrated in Figure b. Because ZnO nanarticles are stable in ethylene glycl and rylene glycl fr mre than 7 and 48 hurs, resectively. After these times, nanarticles settle dwn at the bttm f the cntainer. Fig.. a) Prylene glycl mixed with ZnO (b) rylene glycl nanfluid after snicatin. The Visclite (VL700), which is a vibratinal instrument fr instant measurement f viscsity, is used in this study. This instrument shws the dynamic viscsity f any fluid is surrunding the sensr in terms f centiises (cp). The Viscmeter is examined in air and water fr calibratin. Thermal cnductivity measurements are made using the Decagn devices KD thermal analyzer. This handheld device uses the transient line heat surce technique t evaluate the fluid thermal rerties. Insulatin measurements are best made with the KS- rbe with.3 mm diameter and 60 mm length. The 5

5 Abarzadeh et al./ TPNMS (04) KD meter was calibrated by using the base liquids befre any set f measurements. Minimizing the uncertainty f measurements, three measurements were taen fr each cncentratin f the nanfluids at a given temerature. 4. Results and discussins In this aer, the effect f temerature and vlumetric cncentratin n viscsity and thermal cnductivity f ZnO nanfluids, are investigated. Ethylene glycl, rylene glycl, mixture f ethylene glycl and water (60:40 by weight) and rylene glycl and water (60:40 by weight), were used as the base fluid. The results are resented in tw arts namely, the effective viscsity f nanfluids and effective thermal cnductivity f nanfluids. 4.. Effective viscsity f nanfluids Figures shw the viscsity f nanfluid decrease exnentially by increasing temerature. It exhibits that the temerature deendence f nanfluid viscsity is similar t the base fluid. In the rylene glycl as base fluid, nanfluids have the highest viscsity. As it can be seen ZnO nanarticles in rylene glycl as a base fluid in 3% Vl., 66% reductin in viscsity is bserved. Figure 3 gives the relative viscsity (µr) = (µeff) / (µl), defined as the rati f the effective viscsity f nanfluid and the base fluid, in terms f vlumetric article cncentratin. Exerimental data are cmared with the results f mathematical mdels. It can be bserved that the measured viscsity is much higher than the result f mathematical mdels. It refers t the interactin between the nanarticles. The viscsity increases with an increase in the article vlume fractin. Several viscsity mdels are investigated and cmared with exerimental data in this article. In cnclusin, crrelatins are rsed fr ZnO nanfluids in a mixture f ethylene glycl and water and a mixture f rylene glycl and water as the base fluid. The effect f vlume fractin and temerature tae int accunts in these crrelatins. The temerature and vlume cncentratin ranges 5ºC<T<60ºC and 0-3% vlume fractin, resectively. Figure 4 cmares resent viscsity data fr ZnO/EG nanfluid with similar exerimental data resented by Msavi et al. [9] and Kle et al. [0]. It can be seen that the result f this study is different frm ther investigatrs. Many factrs such as the size and ind f nanarticles, nanfabricatin methds and measurement methds can effect n results. A temerature deendent crrelatin between viscsity and vlume fractin is resented fr ZnO nanarticles susended in EG/W as a base fluid. ex( A T B) (4) µ is the ZnO nanfluid viscsity in terms f (c P ), T is the temerature in ºC and A, B, C and D are functins f article vlume ercentage (). The cefficients A, B, C and D are functins f vlume cncentratin () as: A= (5) B= (6) A temerature deendent crrelatin between viscsity and vlume fractin is resented fr ZnO nan articles susended in PG/W as a base fluid. ex( A T B) (7) A= (8) B= (9) The measured viscsity f ZnO nanfluids cmares with rsed crrelatin in figure 5. It can be seen that there is a gd agreement between them. The exerimental data gives an average deviatin f 7.5% and 4.7% fr EG/W and PG/W as base fluids, resectively. 4.. Effective thermal cnductivity f nanfluids The thermal cnductivity f ZnO (0nm) in ethylene glycl and rylene glycl in the range f ( %) as base fluids, are measured. Figures 5 shw that the thermal cnductivity increases with article vlume fractin. Furthermre, this 53

6 Abarzadeh et al./ TPNMS (04) exerimental data have been cmared t Hamiltn Crsser [3], Bruggeman [4], Jeffrey [5] and Timfeeva [6] mdels. As it can be seen in Figure 6 frm the cmarisn f exerimental results and these theretical mdels, it is bserved that these mdels can't redict the effective thermal cnductivity f nanfluids. This can be due t the fact that these traditinal mdels d nt cnsider article size, article Brwnian mtin, and the effect f nanarticles clustering, which are imrtant in nanfluids. The effect f temerature n the relative thermal cnductivity is shwn in figure 7. The largest enhancements in thermal cnductivity are 5% and 9% fr EG and PG as base fluids, resectively. It can be bserved that by increasing the temerature, thermal cnductivity increases. The variatin f temerature between 5-60 ºC causes an increase in the Brwnian mtin and results in the clustering f nanarticles. Brwnian mtin result in cllisins between nanarticles and it causes energy transfers by nanarticles. The results indicate ZnO in ethylene glycl have higher thermal cnductivity rati. EG and PG are used as base fluids. The viscsity f the base fluid affects the Brwnian mtin f nan articles and it causes the thermal cnductivity variatin f the nanfluid []. Viscsity( cp) /W 3%Vl. %Vl. 0.75%Vl. 0.5%Vl. EG/W Viscsity(cP) % Vl. 0.5%Vl. 0.5% Vl. EG Temerature( C) (a) ZnO PG/W 3%Vl. %Vl. 0.5%Vl. PG/W Temerature( C) (b) ZnO PG 3%Vl. %Vl. 0.5%Vl. PG Viscsity(cP) Viscsity (cp) Temerature( C) Temerature ( C) (c) (d) Fig.. The effect f temerature n the viscsity f ZnO nanfluids (a) EG/water (b) EG (c) PG/water (d) PG. 54

7 Abarzadeh et al./ TPNMS (04) /W Exeriment Batchelr Brinman Einstein Wang 5 5 Batchelr Brinman Einstein Wang µ /µ.4.3 µ /µ ZnO PG/W Batchelr Brinman Einstein Wang (a) v ZnO PG Batchelr Brinman Einstein Wang (b) v µ /µ.6 µ /µ v v (c) (d) Fig.3. The effect f nanarticle vlume fractin n Relative viscsity f ZnO nanfluids (a) EG/water (b) EG (c) PG/water (d) PG Msavi(0.5%Vl.) [9].3 Exeriment Msavi [9] Kle [0] µ.3 µ (cp) v Fig.4. Cmarisn f the measured viscsity with results frm varius researchers. 55

8 Abarzadeh et al./ TPNMS (04) Viscsity(cP) 5 0 ZnO PG/W Crrelatin 3%Vl. Crrelatin %Vl. Crrelatin Crrelatin 0.5%Vl. 3%Vl. %Vl. 0.5% Vl. Viscsity(cP) /W Crrelatin 3%Vl. Crrelatin %Vl. Crrelatin Crrelatin 0.75%Vl. Crrelatin 0.5%Vl. 3%Vl. %Vl. 0.75% Vl. 0.5%Vl Temerature( C) Fig.5. Cmarisn f measured viscsity with rsed crrelatin. / Bruggeman Mdel H-C Mdel Jeffrey Mdel Timfeeva Mdel / ZnO PG Bruggeman Mdel H-C Mdel Jeffrey Mdel Timfeeva Mdel v v Fig.6. The effect f nanarticle vlume fractin n Enhancement f thermal cnductivity f ZnO Nanfluids. In additin, the effect f electric duble layer [] frming arund nanarticles deends n the base fluid. The thermal cnductivity f ZnO nanarticles in ethylene glycl and rylene glycl as base fluids versus the temerature shwed in Figure 8. The redictin f K and Kleinstreuer mdel [8] als demnstrated in this figure. It is clear that the redictin f this mdel can't redict the exerimental results. In figures 9, the results indicated that the measured thermal cnductivity rati f the resent study was larger than that f Msavi et al. (ZnO nm) [9] and lwer than Kle et al. (ZnO-50 nm) [0]. Furthermre, the resent thermal cnductivity results cmare with exerimental resented by Yu et al. [6], Feng et al. [3] and Leng et al. [4]. It is bvius that the result btained by ther researchers is quite different frm thse f the resent data. The main reasn fr this difference is nt clear. Several factrs such as differences in the article size, article rearatin and the measurement technique can affect. Therefre, mre exerimental studies are necessary. Different thermal cnductivity mdels are cmared with exerimental results. These mdels can't redict the variatin f thermal cnductivity f ZnO nanfluids. The new crrelatin is resented fr ZnO nanarticles and have been susended in EG as a base fluid. The effect f vlume fractin and temerature are cnsidered in these crrelatins. The temerature and vlume cncentratin ranges 5ºC<T<60ºC and 0-3% vl, resectively. 56

9 Abarzadeh et al./ TPNMS (04) %Vl. %Vl. 0.5%Vl. ZnO PG 3%Vl. %Vl. 0.5%Vl. / / (a) (b) Fig.7. The effect f temerature n Relative thermal cnductivity f ZnO Nanfluids (a) EG (b) PG. Thermal Cnductivity, (W/m.K) %Vl. %Vl. K and Kleinstreuer Mdel 3% Vl. K and Kleinstreuer Mdel % Vl. K and Kleinstreuer Mdel % Vl. Thermal cnductivity, (W/m.K) ZnO PG 3% Vl. % Vl. % Vl. K and Kleinstreuer Mdel 3% Vl. K and Kleinstreuer Mdel % Vl. K and Kleinstreuer Mdel % Vl Temerature ( C) Fig.8. Cmarisn between exerimental results and thse f btained by K and Kleinstreuer mdel in different temeratures. A temerature deendent crrelatin between thermal cnductivity and vlume fractin is rsed fr ZnO nanarticles susended in EG as a base fluid. bf A B T C T DT E T 3 4 A (0) () bf A B T C T DT E T 3 4 D E (3) (4) (5) B () In figure 0, the thermal cnductivity f ZnO in EG frm the exeriment is cmared with crrelatin. 57

10 Abarzadeh et al./ TPNMS (04) (W/m.C) (3% Vl.) (% Vl.) (% Vl.) (0.5% Vl.) Kle(3.75%Vl.) Kle(% Vl.) Kle(0.5%Vl.) / (% Vl.) Msavi (%Vl.) Msavi Feng Leng Kle Yu / v Fig.9. Cmarisn f the measured thermal cnductivity with results frm varius researchers. It can be bserved that the maximum deviatin is %. There is an accetance agreement between the exerimental data and the existence crrelatin. In figure, the rsed crrelatin cmared with exerimental results resented by Msavi et al. Yu et al. [6], Feng et al. [3] and Leng et al. [4]. It can be bserved that the maximum deviatins are 3.7%, 5.4%, 4.5% and 5.8% resectively. It can be seen that the rsed crrelatin can redict the exerimental results f Msavi et al. [9], Feng et al. [3] and Leng et al. [4]. 5. Cnclusins An exerimental study is dne t investigate the effect f article vlume fractin and temerature n the viscsity and thermal cnductivity f EG, PG, EG/water and PG/water based ZnO nanfluids. Viscsity f nanfluids increases with the increase in article vlume cncentratin. The Einstein mdel culd nt redict it. The thermal cnductivity f the samles dislayed a nnlinear increase with the article vlume fractin, which the Hamiltn and Crsser mdel and Bruggeman mdel culd nt redict. Furthermre, the effect f temerature n thermal cnductivity was investigated. The enhancement f thermal cnductivity with temerature variatin shws that nanfluids have mre alicatin in high temerature. The results indicate ZnO in ethylene glycl has the higher thermal cnductivity rati than rylene glycl. The largest enhancements in thermal cnductivity are 5% and 9% fr EG and PG as base fluids, resectively. 58

11 Abarzadeh et al./ TPNMS (04) / Crrelatin 3%Vl. Crrelatin %Vl. Crrelatin Crrelatin 0.5%Vl. 3%Vl. %Vl. 0.5%Vl Fig.0. Cmarisn f measured thermal cnductivity with rsed crrelatin. / / Crrelatin (% Vl.) Msavi T(K).3 Crrelatin Msavi Feng Leng Yu (a) v (b) Fig.. Cmarisn f rsed crrelatin with the ther researchers exerimental results. References [] M. Chandrasear, S. Suresh and A. Chandra Bse:Exerimental investigatins and theretical determinatin f thermal cnductivity and viscsity f AlO3/water nanfluid, J. f Exerimental Thermal and Fluid Science 34(00) 0 6. [] W. Duangthngsu, S. Wngwises: Measurement f temerature-deendent thermal cnductivity and viscsity f TiO water nanfluids, J. f Ex. Therm. Fluid Sci, 33(009) [3] A. J. Schmidt, M. Chiesa, D. H. Trchinsy, J. A. Jhnsn, K. A. Nelsn and G. Chen: Thermal cnductivity f nanarticle susensins in insulating media measured with a transient tical grating and a htwire, J. f Alied Physics 03(008) [4] E. Hrishiesh, T. Patel, S. Sundararajan, K. Das: An exerimental investigatin int the thermal cnductivity enhancement in xide and metallic nanfluids, J. f Nanart Res (00) [5] S. Ravianth, D. Vajjha, K. Das: Exerimental determinatin f thermal cnductivity f three nanfluids and develment f new crrelatins, J. f Heat and Mass Transfer 5(009) [6] W. Yu, H. Xie, L. Chen, Y. Li: Investigatin f thermal cnductivity and viscsity f ethylene glycl based ZnO nanfluid, J. f Thermchimica Acta 49(009) [7] I. Palabiyi, Z. Musina, S. Witharana, Y. Ding: Disersin stability and thermal cnductivity f rylene glycl-based nanfluids, J. f Nanart Res 3(0) [8] S. Lee, S.U.S., Chi, S. Li, J.A. Eastman: Measuring thermal cnductivity f fluids cntaining xide nanarticles, ASME J. f Heat Transf (999) [9] A. Einstein, N.B. Eine, D. Mleüldimensinen, J. f Ann. Phys 34(906) [0] H. Brinman: The viscsity f cncentrated susensins and slutins, J. f Chem. Phys 0 (95) 57. [] G. Batchelr: The effect f Brwnian mtin n the bul stress in a susensin f sherical articles, J. f Fluid Mech 83(977) [] J. C. Maxwell, A Treatise n Electricity and Magnetism, Clarendn Press, Oxfrd

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