A review of A new determination of molecular dimensions by Albert Einstein

Transcription

1 A reie f A ne determinatin f mleclar dimensins by Albert Einstein Fang X Jne th 7 Hatsls Micrflids Labratry Deartment f Mechanical Engineering Massachsetts Institte f Technlgy

2 Einstein s Miracls year: 95 the htelectric effect: "On a Heristic ieint Cncerning the Prdctin and Transfrmatin f Light", Annalen der Physi 7: 48,95 receied n March 8 th. A ne determinatin f mleclar dimensins. This PhD thesis as cmleted n Aril th and sbmitted n Jly th, 95. Annalen der Physi 9: 89-6, 96; crrectins, 4: 59-59, 9 Brnian mtin: "On the Mtin Reqired by the Mleclar Kinetic Thery f Heat f Small Particles Ssended in a Statinary Liqid", Annalen der Physi 7: ,95 receied n May th. secial thery f relatiity: "On the Electrdynamics f Ming Bdies", Annalen der Physi 7: 89 9, 95 receied n Jne th. mass-energy eqialence: "Des the Inertia f a Bdy Deend Un Its Energy Cntent?", Annalen der Physi 8: 69 64, 95 receied Setember 7 th. Albert Einstein at his des in the Siss Patent ffice, Bern 95

3 References fr iscsity f dilte ssensins Albert Einstein, A ne determinatin f mleclar dimensins. Annalen der Physi 9: 89-6, 96; crrectins, 4: 59-59, 9 English translatin: Albert Einstein, Inestigatins n the thery f the Brnian mement, Edited ith ntes by R. FÜrth, translated by A.D. Cer, Der, Ne Yr, 956 G. K. Batchelr, An intrdctin t flid dynamics, 46 Cambridge Uniersity ress, Cambridge, 99 Gary Leal, Laminar Fl and Cnectie Transrt Prcesses Scaling Princiles and Asymttic Analysis, 75, Btterrth-Heinemann, Netn, 99 William M. Deen, Analysis f transrt henmena,, Oxfrd niersity ress, 998

4 tline In the resence f ne article elcity rfile the rate f dissiatin f mechanical energy In the resence f mltile articles elcity rfile the rate f dissiatin f mechanical energy iscsity Highly cncentrated ssensins: beynd Einstein s frmla An alicatin: Calclate Aagadr nmber N sing iscsity and diffsin cefficients

5 Mtins f the liqid Mtins f the liqid Rigid bdy mtins Mtins f the liqid Nn rigid bdy mtin Straining mtin Parallel dislacement rtatin C B A T [ ] C B A z z y y x x x, y, z x, y, z Transrt mmentm Stress C B A N bdy trqe Princial dilatatins incmressibility

6 Gerning eqatins and bndary cnditins fr the fl ast a shere x, y, z x, y, z Naier-Stes eqatin: [ ] g t Steady state N inertia Neglect graity : iscsity f the flid Stes eqatins Incmressibility 4 Δ, Δ, Δ, B.C. at the srface f the shere

7 T find sltins T find sltins C B A distrbance de t the article elcity itht the resence f the article x, y, z x, y, z z z y y x x, as

8 T find sltins T find sltins The strctre f :,, ith and Which satisfies the gerning eqatins,, Harmnic fnctins

9 T find sltins T find sltins c Gerning eqatins a b c,, c Harmnic fnctin

10 5 sltins fr fl ast a shere cnst P 5 P P A A B C A B C A P 5 P P B A B C A B C B P 5 P P C A B C A B C C and, as,

11 The rate f dissiatin f mechanical energy n Frce acting n the srface s R R>>P P s W E [ t n ] ds t I E [ T ] n t ds P Neglect higher rder terms Extra mechanical r de t the article Pre flid

12 With the resence f mltile articles x,y,z x ν, y ν, z ν x x y y z z,, d: Distance beteen t articles P: The radis f the article d >> P N interactin beteen t articles ν :The distrbance de t ne article :the elcity f the re flid x,y,z Σ ν ν n: # f articles er lme flid Φ: lme f ne article

13 the rincial dilatatins f the mixtre Ax A x A x A x By A B C Az A n B n C n x y x ν ν ν Diergence therem d d d A n B n C n r r x y r z x x x B y B y B y y y y C z C z C z z z z ds ds ds y x x z z y Princial dilatatins A B C A φ B φ C φ δ A B C δ φ

14 iscsity f the mixtre : mixtre W δ W δ φ lme fractin f articles φ φ 5 φ φ O φ φ< % φ iscsity f the mixtre iscsity f the re flid

15 The rhelgy f cncentrated ssensins [] iscsity des nt bey Einstein s frmla Nrmal stress Particles migrate Stress des nt reach steady state instantanesly [] Andreas Acris, The rhelgy f cncentrated ssensins: latest ariatins n a theme by Albert Einstein, Ind-US Jint Cnference 4, Mmbai, Indian, 4

16 An alicatin: frm iscsity and diffsin cefficients t Agadr nmber ω: elcity; N: Agadr nmber; D: diffsin cefficient; : density m: mleclar eight; K: drag frce Stes s la K ω 6πP D RT 6 π NP NP RT 6π D n N φ n πp m NP m π