6.642 Continuum Electromechanics

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1 MIT OpenCourseWre Continuum Electromechnics Fll 8 For informtion out citing these mterils or our Terms of Use, visit:

2 6.64, Continuum Electromechnics rof. Mrkus Zhn Lecture 5: Lws, Approximtions, n Reltions of Flui Mechnics Continuum Electromechnics (Melcher) Sections I. Useful Vector Opertions n Ientities Grient χi l = χ - χ Guss s Lw (Divergence Theorem) V i A V = S A in Stokes Theorem S A in = A i l C Some useful Vector Ientities ( f ) = i ( A ) = ( A B ) C=A ( i i B C) (Dot n Cross cn e interchnge in the sclr triple prouct) 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

3 II. Time Derivtive of Flui Volume Integrl ζ = ny sclr quntity such s ensity ρ ( t ) V = lim t ζ V( t) Δt ζ t + Δt V - ζ t V V( t+ Δt) V( t) Δt Linerize ll terms to first orer in ζ ζ( t + Δt ) = ζ( t ) + Δt +... Δ t t ζ V( t) t V = lim Δt ζ ζ ( t ) V - ζ ( t ) V + Δt V V( t+ t) V( t) V( t+ t) Δ Δ Δt s =lim Δ t ΔV ζ t V+ V t Δt ζ Δt V ( s ) ΔV = v Δt i v = flui surfce velocity t V() t ζ () t V = lim Δt S ζ ζ () t v i Δt + Δt V s Δt V() t t ζ() ζ i s V() t ζ t V = V + v V S 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

4 S s V ( ζ ) ζ v i = i v V Divergence Theorem ζ ( t ) V= + v V t ζ i ζ V( t) V III. Conservtion of Mss ( ρ = ζ ) t ρ V = = V + v n ρ ρ s i V V S V i ρ v V ( ρ ) ρ + i v = Volume is ritrry ρ + v i ρ+ ρ i v = D ρ + ρ i v = Dt Incompressile D ρ = i v= Dt IV. Conservtion of Momentum ( ρ v = ) t V ρ v V = F V = ( v ) V + v v n ρ ρ i i i i i s V V V S v + v v V = F V ( ρ i) ( ρ i ) i i i V ρ v + ρ v v = F i i i ρ ρ ρ ( i ) i ( ρ ) v i + v v i + v i i ( i ) v i + v v i = F i ( i ) v ρ + v v = F + v = F i ζ, i th component where i=x, y, or z (Conservtion of mss) 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 3 of 4

5 V. Equtions of Motion for n Invisci Flui ( i ) ex v ρ + v v = - p+f VI. Eulerin Description of the Flui Interfce ξ F x, y, z, t = = y, z, t - x n= F F Courtesy of MIT ress. Use with permission. ( i ) ξ DF F = + v F = on F = = y, z, t - x Dt ξ F F F +v x +v y +v z = x y z - ξ y ξ z ξ ξ ξ v x = + v y + vz y z 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 4 of 4

6 VII. Surfce Tension A. Surfce Force Density δw = γ A s δw s + Τ sa δξ = γ A + Τ sa δξ = Increse of surfce energy Work one on interfce Courtesy of MIT ress. Use with permission. Α + δa = x + δx y + δy xy + yδx + xδ y x+ δx x x = δx = δξ R + δξ R R y+ δy y y = δy = R + δξ R R δξ δa=yδx+xδ y 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 5 of 4

7 xy xy = + δξ R R =xy + δξ R R =A δξ + R R γ A δξ + + Τ sa δξ = R R Τ =- γ + n s [Young n Lplce surfce force ensity] R R B. Interfcil Deformtion V i C V = s C i n (Divergent Theorem) S C=n Courtesy of MIT ress. Use with permission. n=n on top s n = -n on ottom s n n on sie V s i i i ( i ) C V n δξα = n n s A n Α δξ = δα S = A - A = δα top ottom 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 6 of 4

8 - γδα - γ Αδξ i n T s = = Αδξ Αδξ s = - γ in in = + R R Τ =- γ i n n VIII. Bounry Conitions A. Rigi Wll ni v = (norml velocity component is zero) n v = (Viscous flow) (tngentil velocity component is zero) B. Interfce n i v = ; v = v - v ove elow Force Equilirium i ( s) i V F V + T = A m F=F +F e mechnicl electricl T e F e = ij i x j m m T p p F =- p= it =- =-δ m ij ij xj xi xj T = -p δ m ij ij 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 7 of 4

9 T sr Courtesy of MIT ress. Use with permission. m e V F V = T + T i i ij ij n S j T + T n + T = m e ij ij j s i p n = T n - γ i n n e i ij j i 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 8 of 4

10 IX. Bernoulli s Lw g i, r = Xi + yi = zi F = ρ g = ρ g r if ρ = constnt x y z e F = [Specil cse when electricl force is written s grient of sclr] ex ( i ) ( i ) v ρ + v v + p = F = ρ g r - ( v i ) v= ( v ) v+ ( v i v ) v ρ + ω v + p + ρ v v - ρg r + = t i i ω = v vorticity l v v ρ i l+ p + ρ v v - ρg r + = i i Irrottionl Flows ( v= ω = ) v=- θ θ - ρ +p + ρ v i v - ρg i r+ = 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 9 of 4

11 X. Bernoulli Lw rolems A. Cpillry Rise Courtesy of MIT ress. Use with permission. θ =, v=, g =-g i, = z = + ρgξ c = = c - + γ + = R R= αr 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

12 ρ ξ ρ ξ γ = + g - = g = α r ξ = γ ρgαr B. Electriclly Driven Rocket e pld + f x = LD e C V LD C=, f x = V =- x x x e -fx V LD V p= = = LD x LD x Another Wy: p - T e = ; T e = V xx xx x e V p=t xx = x p + ρ v = p + ρ v vld=vd V p = = ρ v - x L 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

13 V v = V p = ρx - L M Thrust=V p =Vp ρ D t C. Mgneticlly Driven Rocket e +p + T yy = μ μ T yy = - H = - z I w μ p= I w p + ρ v = p + ρ v I μ p =, vdw = v w w ρ w D μ I = v - μ I ρ ( w ) v = V = - D 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

14 M t Thrust = V = V Vw = V w ρ ρ D. Dielectric Liqui Rise. Kelvin olriztion Force Density ( i ) ij i j δij k k F= E, T =ED - EE ( ) = - E ( i ) ( )( i ) E = - E E ( ) ( E ) E= ( E i ) E- ( E i E ) ( E i ) E = ( E i E ) ( ) - i E = E i E = - E i E if constnt F=- = ( - ) E i E =- - E E ( ) i + ρgz - ( - ) E i E = constnt 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 3 of 4

15 Courtesy of MIT ress. Use with permission. V E= i αr θ = ( - ) V + ρgξ- = α r c - + T = γ = (negligile surfce tension) - = c T = - E θ = E θ continuous cross interfce = = c = ( - ) V ξ = α r ρ g 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 4 of 4

16 . Korteweg-Helmholtz Force Density F=- E i E, T = EE - δ EE ij i j ij K K =constnt, F=- = = In flui + ρ gz=constnt T = - E = - - ( ) θ V αr = - + T = - = c + ρg ξ = c V V - = ρg ξ = T = - ( - ) = + ( - ) αr αr ( - ) V ξ = α r ρ g E. Mgnetic Flui Rise in Tngentil Mgnetic Fiel Courtesy of MIT ress. Use with permission. 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 5 of 4

17 I H= iθ πr F = μ ( M i ) H, T = HB - μ δ H H ij i j ij K K. Linerly Mgnetizle ( μ ) B= H+M = H M= - H μ μ μ μ F= ( μ -μ)( H i ) H= ( μ - μ) ( H) H+ ( H i H) μ - μ μ - μ i i μ =- F = H H = H H if = constnt =- - H H μ μ i + ρgz - ( μ - μ ) H i H = constnt = = c - + T = T = - H θ = μ = = c = + g - ( - ) I ρ ξ μ μ = πr ( - ) μ μ I ξ = ρ g π r c 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 6 of 4

18 . Nonliner Mgnetiztion Chrcteristics H ( ) M = M H M H H M ( i ) ( i ) F= μ M H= μ H H H H i H = H H + H i H μ M μ μ H F= H = M H= M( H) H=- H H H μ μ = - M H H = - M H; M = M( H) H H Specil cse: Liner Mteril: μ M= μ -μ H =- ( μ -μ) H =- μmh Sturte Mteril: μ M = constnt =-μ MH + ρgz- =constnt 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 7 of 4

19 = = c - + T = + ρg ξ+ = + c c T = HB - H H δμ ij i j ij K K T = - Hθ T = μ = = c = H μm H H - μm H ξ = = = ρg ρg ρg Liner Mteril: =- ( - ) I μ μ π r ( μ μ ) ( π ) - I ξ = r ρg Sturte Mteril: =-μ MH μmh μmi ξ = = ρg πρgr 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 8 of 4

20 F. Mgnetic Flui Rise in Norml Fiel = + ρgδh+ = + ρgδh- μm H = - + T = T = H B - H μ x x x μ T = H - μ H ( H +M ) + μ H μ μ H = H +M μ T = ( H +M) - H ( H +M ) + H μ μ 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 9 of 4

21 μ = M μm = - = μm - = = ρgδh-μ M H μ M h= +M Δ H ρg Liner Mteril: M μ M MB ρg ρg M= Δh= M+ H = Sturte Mteril: μ M M=M Δh= M+ H ρg G. Mgnetic Nozzle 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

22 + ρv + = + ρv - μ M H = = μ v - v = M H ρ π π v = v = v 4 4 v μ μ v = v + M H = v + M H ρ ρv μ = + M H ρv H. Mgnetic Flui Rotry Shft Sel μm H 3 = - μm H T = 3, T =, Assume H tngentil to interfce T = - Ht T = μ, 3,4 = 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

23 4 = = 4 - = Δ = μ ( M H 3 3 -M H ) H3 = μ MH H In well esigne sel H Δ μ M H Typicl numers: μ M = 7 G =.7T μ H = 8, G =.8T ( μ M)( μ H ).7 (.8 ) H 3 Δ= = = Nm 5-7 μ 4 π = kscls = Atmosphere XI. Force on Boy in Mgnetic Flui p+ ρgz-μ M H=constnt=C p= μ M H-ρ gz+c f = - p p n = - μm H n + ρgz n + C n S S S S uoyncy effect Mgneticlly Sturte: M= M=constnt S C n = f = - μ M H n = - μ M H n = - μ M HV M S S Mgneticlly Liner: ( μ - μ ) M= M= H μ ( μ - μ ) V C V f M =- μm H n =- μ H n =- μ -μ H V μ S S V 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge of 4

24 A. Non-mgnetic Boy f oy =uoynt weight + f M f M opposite to irection of incresing H Non-mgnetic oy moves towrs wek fiel region (Sink-Flot Seprtion) 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 3 of 4

25 6.64, Continuum Electromechnics Lecture 5 rof. Mrkus Zhn ge 4 of 4

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6.642 Continuum Electromechanics

6.642 Continuum Electromechanics MIT OpenCourseWre http://ocw.mit.edu 6.64 Continuum Electromechnics Fll 8 For informtion out citing these mterils or our Terms of Use, visit: http://ocw.mit.edu/terms. 6.64, Continuum Electromechnics,