1. Viscosities: μ = ρν. 2. Newton s viscosity law: 3. Infinitesimal surface force df. 4. Moment about the point o, dm
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1 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess Motition Gien elocit field o ppoimted elocit field, we wnt to be ble to estimte te esultnt foce nd/o toque due to wll se stess on solid bod in pplictions suc s dulic clutc, iscomete, jounl being, etc Bsic Equtions Viscosities: μ ρν Newton s iscosit lw: du τ μ, d du τ t nt μ dn 3 Infinitesiml sufce foce iˆ + ˆ j + kˆ iˆ + ˆj + kˆ : τ τ τ + τ + τ + τ + τ + τ + τ due to stess on te infinitesiml sufce ˆ j τ If ligns long n one coodinte is, eg, τ τ τ Moment bout te point o, dm o, of te foce : ( se ( se j, ten ( noml ˆ foce) foce) foce) dm o o
2 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess Poblem Two pltes, nd B, e septed b n oil-filled gp of widt 3 mm Te oil is SE-W-3 t o C Te pltes nd B e moing to te igt t te speeds of m/s nd m/s, espectiel ssume line elocit distibution coss te gp U B m/s B U m/s Sketc te elocit pofile on te figue boe lso, find te eqution fo te elocit pofile u (), u in m/s nd in m, using te gien coodintes b Find te se stess on te sufce of te uppe plte B Be sue to get te sign of te stess (not foce) igt c If te plte B s te totl e of m, find te net foce due to te oil on te plte B Be sue to get te sign/diection of te foce (not stess) igt d If te plte is insted moing to te left t te cuent speed (ie, m/s to te left), will te mgnitude of te stess cting on plte B incese o decese nd w is it so? Detemine te lue of tis new stess
3 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess 3 U B m/s B u() U m/s Sketc te elocit pofile on te figue boe lso, find te eqution fo te elocit pofile u (), u in m/s nd in m, using te gien coodintes u( ) U U B U u( ) ( U B U ) + U ( ) m / s B / NS 3m Tus, u( ) ( U U ) + U + m / s [ 6667 ( m) + ] m s b Find te se stess on te sufce of te uppe plte B Be sue to get te sign of te stess (not foce) igt constnt nd > tougout te cnnel d ( ) m / s :, / d 3 m s s N s τ ( ) μ μ P NS d m s c If te plte B s te totl e of m, find te net foce due to te oil on te plte B Be sue to get te sign/diection of te foce (not stess) igt F τ B τ ˆj F τ ( + P) ( m ) 8 N NS d If te plte is insted moing to te left t te cuent speed (ie, m/s to te left), will te mgnitude of te stess cting on plte B incese o decese nd w is it so? Detemine te lue of tis new stess Te mgnitude of te stess will incese since te mgnitude of te elocit gdient du / d will incese Tis cn be seen s follows du U B U constnt nd > tougout te cnnel d ( ( )) m / s : 6, / 3, d 3 m s s N s τ ( ) μ μ 6, + P NS d m s
4 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess Poblem Te gp of eigt between te two disks nd B e filled wit oil Bot disks e dius R Te disk is sttion wile te disk B is otting wit te ngul elocit ω ω, ω > ssume tt ) te flow is ismmetic, ) te elocit pofile coss te gp is line, nd 3) te iscosit μ is constnt nswe te following questions ê ê ê Te oigin of te coodintes sstem is locted t te cente of te sufce of disk ω ω, ω > B In nsweing te following questions, use te gien coodintes sstem Wic component of stess (ie, τ?? ) eets te tngentil foce on te sufce of te solid disk? b Is te stess in Poblem positie o negtie? c In wic diection does te net toque due to te oil ct on te sufce of te solid disk? d Find te wll se stess distibution oe te sufce of te solid disk, ie, τ w ( )? Recognie tt stess is tenso, tus it is signed quntit
5 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess Wic component of stess (ie, τ?? ) eets te tngentil foce on te sufce of te solid disk? 5 τ b Is te stess in Poblem positie o negtie? positie c In wic diection does te net toque due to te oil ct on te sufce of te solid disk? + d Find te wll se stess distibution oe te sufce of te solid disk, ie, τ w ( )? Recognie tt stess is tenso, tus it is signed quntit u ( ) u (, ) u B Tngentil elocit pofile ( u ) t n dius ( ) ω u (, ) ω Te tngentil elocit field cn be witten s u (, ) ω Tus, we e u ω, τ u ω μ μ, nd ω τ w ( ) τ μ NS
6 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess Poblem 3 [dpted fom Çengel nd Cimbl, 6, Poblem -7, p 6] Te clutc sstem sown is used to tnsmit toque toug 3-mm-tick oil film [SE3(W) oil] between two identicl 3-cm-dimete disks Wen te diing sft ottes t speed of 5 pm, te dien sft is obseed to otte t 398 pm ssuming line elocit pofile fo te oil film Detemine te input nd output toques e te equl? Detemine te input nd output powes e te equl? If not, find te efficienc, η : Output Powe/Input Powe, of te clutc 3 Fom ou nswes in nd, biefl discuss te ccteistics of te clutc sstem in tems of (Mecnics/Foce/Toque iewpoint) te tnsmission of toque, nd b (Eneg/Powe iewpoint) te tnsmission of powe Is it possible to e te tnsmission of powe witout te slip (no diffeence in te ngul elocities of te input nd output sfts)? 6 Diing sft 3 cm 3 mm Dien sft SE 3W oil
7 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess 7 Diing Disk Oigin o of te coodinte es ω ωˆ e, ω > V ( ω ) V V ( ω) ˆ (, ) u e ω + ( ω ω ) τ d d ( d d) ˆ e Velocit pofile t te tese t dius ê ê ê Nottion: Diing Disk, Dien Disk, gp between te two disks e d min ω 5 π 58 d / s min e 6 s e d min ω 398 π 6 d / s min e 6 s ssumptions: Newtonin fluid V (, ) u (onl component is pesent) is-smmetic flow: u u (,, t) Sted flow, o eluting t n one instnt Line elocit distibution coss te gp 3 μ constnt (unifom) Diing Disk : Infinitesiml e : Infinitesiml foce on + + ( )ˆ e ( τ )ˆ e + + ( )ˆ e ( τ )ˆ e ( d d) + + ( )ˆ e ( τ Since nd, een if te eist, do not contibute to te moment bout te -is, we cn concen ouself wit onl [ psses toug te is, nd is pllel to te is Tus, bot cnnot contibute to te moment bout te is] ( ), τ () 3 dt o : Infinitesiml moment bout point o due to on (Note tt + ) dto ( dto ) : Infinitesiml moment bout te is toug point o due to on )ˆ e () dt o ( + ) (( ) + ( ) + ( ) ) ( )ˆ e + ( )ˆ e + ( )ˆ e
8 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess dt o [Fo simplicit, we sll dop te subscipt o ] (3) 8 Tus, fom Eqs ()-(3) dt τ ( dd) τ () 5 ssumption : Newtonin fluid Newton s iscosit lw: dut du τ nt μ τ μ (5) dn d 6 ssumption : V (, ) u (onl component is pesent) is-smmetic flow: u u (,, t) Sted flow, o eluting t n one instnt Line elocit distibution coss te gp Tus, u ω + ( ω, (6) du ( ω nd ω ) d fi 7 Wll se stess, foce, nd moment, on te sufce of Disk Fom Eqs (5) nd (6), we e du ( ω τ μ μ d τ, du μ d, ( ω μ, (7) fom () ( ω τ μ dd,, (8) fom (3) ( ω 3 dt μ dd (9) 8 Net toque due to wll se stess on te sufce of Disk Fom (9), we e ssumption 3: μ constnt (unifom) T dt ( ω πμ R π μ( ω R R π 3 ( ω 3 μ dd d μ( ω R { πr { R 3 e Se Lengt Tus, te toque due to fluid iscous stess on te diing disk is gien b ( ) R T π μ ω ω (since ω > ω, negtie diection) Consideing te ngul equilibium of te diing sft, we e te input toque t te sft (not te iscous toque due to fluid t te disk fce) gien b ( ) R T T π μ ω ω (since ω > ω, positie diection) NS () Fo μ 38 N s/m, we e π N s e d min T 38 (5 398) π 5 m 59 N m m min e 6 s 3m NS()
9 3- Fluid Mecnics Clss Emple 3: Newton s Viscosit Lw nd Se Stess Dien Disk Since te elocit pofile is line, te slopes du / d t ec dil position t te fces of te two disks e te sme Tus, te se stesses τ μ( du / d) t ec dil position t te fces of te two disks e lso te sme In ote wods, te two disks e te sme se stess distibution Note tt in tis cse ( ω ω ) τ μ < Since te fce of te diing disk is + ê fce wile tt of te dien disk is ê fce, te foce on te diing disk is in te ê diection wile tt of te dien disk is + ê diection Tus, te net toque due to oil on te dien disk s te sme mgnitude, but opposite in diection, s tt on te diing disk nd ( ) R T T π μ ω ω Consideing te ngul equilibium of te dien disk, we teefoe e te output toque t te sft ( ) R T T π μ ω ω (since ω > ω, negtie diection) NS () Tus, te input nd output toques t te sfts e equl NS () 9 Since te powe t sft is gien b P T ω, we e te input powe t te input sft ( ) π μ ω ω R P T ω ω > NS () P 83 3 W nd te output powe t te output sft ( ) π μ ω ω R P T ω ω < NS () P 8 3 W Te two e not equl nd we find tt te efficienc is gien b η P ω 96 P ω NS () Te clutc cn tnsmit te toque of equl mgnitude fom input to output, owee tee is cetin mount of loss in te tnsmission of powe Cetin mount of slip is equied to estblis te elocit gdient nd, ence, te se stess field suc tt te toque cn be tnsmitted, nd te clutc cnnot tnsmit powe witout te slip since in suc cse tee will be no toque NS (3)
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