6.642, Continuum Electromechanics, Fall 2004 Prof. Markus Zahn Lecture 8: Electrohydrodynamic and Ferrohydrodynamic Instabilities

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1 6.64, Continuum Electromechnics, Fll 4 Prof. Mrus Zhn Lecture 8: Electrohydrodynmic nd Ferrohydrodynmic Instilities I. Mgnetic Field Norml Instility Courtesy of MIT Press. Used with permission. A. Equilirium ( ξ= ) = H = H ; 1 Poe Pod + H H = ρ gx + Pod x > Po ( x) = ρ gx + Poe x <. Perturtions: ( t yy zz) j e ω H = H i + h = + x, H H ix h 1 coth h c xc sinh Ψ = h 1 d xd coth Ψ sinh 1 coth h e xe sinh Ψ = h 1 f xf coth Ψ sinh 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 1 of 13

2 C. oundry Conditions v = v = xc xf Ψ = Ψ = c f v xd,e = + vyd,e + vzd,e vxd = vxe = jωξ t y ix iy iz y z n = i i i y y x y z e i = ij j γ i pn T n nn n = + y i = x n = n = 1 i x P + P' = Txx nx + Txy ny + Txz ξ ξ nz + γ + y 1 perturtion perturtion Hh Hh x y x z Equilirium ( ξ= ) second order P = Txx P 1 od Poe = H H Perturtions 1 P P H h H h ' d ξ ' e ξ = ' d ξ ' e ξ γ ξ dp P P P P g P od ( ξ ) = ( ξ ) + = ξ + = ρ ξ + ' ' ' ' d od d d d x= dp P P P P g P oe ( ξ ) = ( ξ ) + = ξ + = ρ ξ + ' ' ' ' e oe e e e x= 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge of 13

3 1 coth c p v c jωρ sinh x = p 1 d d coth v sinh x 1 coth e p v e jωρ sinh x = p 1 f f coth v sinh x jωρ d ω ρ p = coth v = coth ξ d x jωρ e ω ρ Pe = coth v = coth ξ x h h xd xe d = coth Ψ e = coth Ψ ni = = i i i i ( ) i + ( h h ) i + ( h h ) i y x y z x xd xe x yd ye y + ( hzd hze) iz = = h = h xd xe n h = = ix iy iz ( H H ) ix + ( hxd hxe ) ix + ( hyd hye ) iy + ( hzd hze ) iz y = i h h i h h + H H i H H i y z yd ye y zd ze z y d ( e) hyd hye = H H + jy Ψ Ψ = jy H H ξ y d ( e) hzd hze = H H + jz Ψ Ψ = jz H H ξ ( H H ) Ψ Ψ = + ξ d e 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 3 of 13

4 coth Ψ = coth Ψ coth Ψ d = Ψ coth d coth Ψe 1 = ( H H ) ξ coth H H ξ coth Ψ e = coth + Ψ d = + coth + e coth H H ξ coth coth e D. Dispersion Eqution ωρ ωρ ρgξ coth ξ+ρ gξ coth ξ H coth H H coth ξ H coth H H ξ coth = γ ξ coth + coth ω ( ) ( ) ( ρ coth + ρ coth ) = ( ρ ρ ) g + γ coth+ coth H H cothcoth H, H H H 1 1 ( ) = = = = ω ( ρ coth + ρ coth ) = g( ρ ρ ) + γ + tnh tnh E. Short Wvelength Limit ( 1, 1) tnh tnh 1 ( ) g f ω ( ρ + ρ ) = ( ρ ρ ) + γ = , Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 4 of 13

5 Incipience of Instility f = df d ( ) = = γc + c ( ) = γ + 1 g( ρ ρ ) + γ = γ + + γ ( ) = 4g ( ρ ρ) γ + 1 = 4g( ρ ρ ) γ = γ c g ( ρ ρ ) γ Courtesy of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 5 of 13

6 Courtesy of MIT Press. Used with permission. II. Electric Field Norml Instility A. Polriztion Forces Courtesy of MIT Press. Used with permission. ε ε D ω D ε ε ( ρ coth + ρ coth ) = g ( ρ ρ ) + γ + εε ε ε tnh tnh 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 6 of 13

7 . Perfectly conducting lower fluid ( ε ) Courtesy of MIT Press. Used with permission. V coth coth g coth ω ( ρ + ρ ) = ( ρ ρ ) + γ ε Courtesy of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 7 of 13

8 III. Tngentil Grdient Fields Courtesy of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 8 of 13

9 A. Equilirium V V x V E= i i 1 + ;r R x;e = θ θ r θ r R θ R y i E x y 1 + R lim P x =ρ gx + P Po Po + Txx = o o P x =ρ gx + P o o T 1 εe 1 ε E xx = y = T = 1 ( ε ε ) E = P P xx o o. Perturtions 1 α e coth x α sinh Φ = β 1 coth β e Φ x sinh lim e e x x = Φ = Φ 1 coth α α p j sinh v ωρ x = β 1 p coth v β sinh x v α v β = = j ωξ x x P jωρ ω ρ = vx = ξ P jωρ ω ρ = vx = ξ 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 9 of 13

10 C. oundry Conditions n = i i i y x y z n E = i i i x y z ( ex ex ) i + x ( ey ey ) i + y ( ez ez ) i = z y z ( y y ) ( z z ) i e e i e e = y e y = ey jy Φ = j y Φ Φ =Φ Φ e z = ez jz Φ = jz Φ ni εe = i i i i E ( ε ε ) i + ( ε e ε e ) i + ( ε e ε e ) i y + ( εez εez) i = z x y z y x x x y y y ε e ε e + j ξe ε ε = x x y Φ ε + ε + j E ε ε ξ = y ξ jy E ( ε ε) Φ= ( ε + ε ) ξ ξ pn = T n + γ + n y i ij j i i = x, nx = 1 p = T + T n + T n γ ξ xx xy y xz z Txx = 1 ε Ex Ey E 1 1 z Ey ey Ey Eey = ε + = ε + Txy = ε Ey ex T = ε e e xz x z 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 1 of 13

11 T xy n second order y T xz n third order z 1 d ( ξ ) = ε ε T E ξ E e = xx y y x de = E ξ ε ε Eey de T =εe xx ξ ε Ee de =εe ξ j Φ ε E y T =( ε ε ) E ξ j E Φ ( ε ε ) xx de y y dp = ξ + = ρ ρ ξ p p P P g T de P P g ρ ρ ξ = ε ε E ξ jy ΦE ( ε ε) γ ξ ω ( ρ + ρ) ξ g( ρ ρ) ξ = de ( ε ε) E ξ γ ξ j E ( ε ε ) ( j E ) ( ε ε ) ξ y y ( ε + ε ) ω de E ε ( ) g( ) ( ) E + ε ρ + ρ = ρ ρ + γ + ε ε + ε ε E R y Uniform tngentil field lwys stilizes. Grdient field stilizes, if higher permittivity fluid is in stronger electric field. System stle even if hevier fluid ove if de ( ε ε ) E > g( ρ ρ ) 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 11 of 13

12 Courtesy of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 1 of 13

13 Courtesy of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 13 of 13

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