6.642 Continuum Electromechanics

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Φωσφόρος Βλαβιανός
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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, Fll 4 Prof. Mrus Zhn Lecture 8: Electrohdrodnmic nd Ferrohdrodnmic instilities I. Mgnetic Field Norml Instilit Courtes of MIT Press. Used with permission. A. Equilirium ( ξ ) μ H μh ; Poe Pod + H H μ μ ρ gx + Pod x > Po ( x) ρ gx + Poe x <. Perturtions: e j( ω t zz) H H ix+ h, H H ix+ h coth h c xc sinh Ψ h d xd coth Ψ sinh coth h e xe sinh Ψ h f xf coth Ψ sinh 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge of 3

3 C. oundr Conditions v v xc xf Ψ Ψ c f v xd,e + vd,e + vzd,e vxd vxe jωξ t n i i i x z i x i iz z + + e i ij j pn T n γ nni ξ ξ n + i x n n i x P + P' Txx nx + Tx n + Txz ξ nz + γ + perturtion perturtion Hh μ Hh μ x x z Equilirium ( ξ ) second order P Txx P od Poe H H μ μ Perturtions P P H h H h μ ' d ξ ' e ξ μ ' 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 ' ' Pe Poe Pe Pe g Pe x oe ( ξ ) ( ξ ) + ξ + ρ ξ , Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge of 3

4 coth c p v c jωρ sinh x p d d coth v sinh x coth e p v e jωρ sinh x p 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 + ( μ h μ h ) i + ( μ h μ h ) i x z x xd xe x d e h h i + μ zd μ ze z μ h μ h xd xe n h ix i iz ( H H ) ix + ( hxd hxe ) ix + ( hd he ) i + ( hzd hze ) iz i h h i h h + H H i H H i z d e zd ze z d ( e) hd he H H + j Ψ Ψ j H H ξ d ( e) h h H H + j Ψ Ψ j H H zd ze z z ( H H ) Ψ Ψ + ξ d e μ coth Ψ μ cothψ d 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 3 of 3 e ξ

5 μ coth Ψ d Ψ μ coth μ coth Ψe ( H H ) ξ μ coth H H ξ μ coth ( ) Ψ e μ coth + μ ( ) Ψ d + μ coth + μ e coth H H ξ μ coth coth D. Dispersion Eqution ωρ ωρ ρgξ coth ξ + ρ gξ coth ξ μh coth H H μ coth ξμh coth H H ξμ coth γ ξ μ coth + μ coth ω ( ) ( μ μ) ( ρ coth + ρ coth ) ( ρ ρ ) g + γ μ coth+ μ coth H H cothcoth H, H H H ( μ μ ) μ μ μ μ μμ ω μ μ ( ρ coth + ρ coth ) g( ρ ρ ) + γ μμ μ +μ tnh tnh E. Short Wvelength Limit (, ) tnh tnh ω g ( μ μ ) ρ + ρ ρ ρ + γ μμ μ +μ f Incipience of Instilit 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 4 of 3

6 f df d ( μ μ ) γc μμ μ +μ c ( μ μ ) γμμ μ +μ μ μ μ μ g( ρ ρ ) + γ γμμ μ + μ μμ μ + μ γ ( μ μ ) 4g ( ρ ρ) γ μμ μ +μ 4g( ρ ρ ) γ γ c g ( ρ ρ ) γ Courtes of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 5 of 3

7 Courtes of MIT Press. Used with permission. II. Electric Field Norml Instilit A. Polriztion Forces Courtes of MIT Press. Used with permission. μ μ ε ε D 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 6 of 3

8 ω D ε ε ( ρ coth + ρ coth ) g ( ρ ρ ) + γ + εε ε ε tnh tnh. Perfectl conducting lower fluid ( ε ) Courtes of MIT Press. Used with permission. V coth coth g coth ω ( ρ + ρ ) ( ρ ρ ) + γ ε 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 7 of 3

9 Courtes of MIT Press. Used with permission. III. Tngentil Grdient Fields Courtes of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 8 of 3

10 A. Equilirium V V x V E i i + ;r R x;e θ θ r θ r R θ R i E x + R lim Δ P x ρ gx + P Po Po + Txx o o P x ρ gx + P o o T ε xx E εe T ( ) E ε ε P P xx o o. Perturtions α e coth Δ x α sinh Δ Φ β coth β e Δ Φ x sinh Δ lim Δ e e x x Φ Φ coth α α p Δ j sinh v ωρ Δ x β p coth v β Δ sinh x Δ v α v β j ωξ x x P jωρ ω ρ vx ξ 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 9 of 3

11 P jωρ ω ρ vx ξ C. oundr Conditions n i i i x z z n E i i i x z ( ex ex ) i + x ( e e ) i + ( ez ez ) i z z ( ) ( z z ) i e e i e e e e j Φ j Φ Φ Φ Φ e z ez jz Φ jz Φ ni εe i i i x zi E ( ε ε) i + ( εex εex) i + x ( εe εe) i e e j E + ξ ε ε ε ε x x j E ε ε ε ε Φ + + ξ ξ je ( ε ε) Φ ( ε + ε ) + ( εez εez) i z pn T n + γ + n i ij j i i x, nx p T + T n + T n γ ξ xx x xz z Txx ε Ex E Ez ε E + e E + Ee ε T T εe e x x ε e e xz x z 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge of 3

12 T x n second order T xz n third order z T d ( ξ ) ε ( E ) xx x de εe ξε Ee ξ ε E e de T εe xx ξ ε Ee de ε ξ jφε E E T ( ε ε ) E ξ j E Φ ( ε ε ) xx de p ξ + p P P g( ρ ρ ) T dp de P P g ρ ρ ξ ε ε E ξ j ΦE ( ε ε) γ ξ ω ( ρ + ρ) ξ g( ρ ρ) ξ ξ de ( ε ε) E ξ γ ξ j E ( ε ε ) ( j E ) ( ε ε ) ( ε + ε ) ξ ( ε ε ) ( ε ε ) ω de E ( ρ + ρ ) g( ρ ρ ) + γ + ( ε ε) E + + E R Uniform tngentil field lws stilizes. Grdient field stilizes, if higher permittivit fluid is in stronger electric field. Sstem stle even if hevier fluid ove if de ( ε ε ) E > g( ρ ρ ) 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge of 3

13 Courtes of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge of 3

14 Courtes of MIT Press. Used with permission. 6.64, Continuum Electromechnics Lecture 8 Prof. Mrus Zhn Pge 3 of 3

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