Oscillating particles in fluids: Theory, experiment and numerics - PowerPoint PPT Presentation

Oscillating particles in fluids: Theory, experiment and numerics Oscillating particles in fluids: Theory, experiment and numerics Victor Yakhot Boston University, Boston, MA May, 2012, Vienna

Oscillating particles in fluids: Theory, experiment and numerics Boltzmann-BGK Equation in the range 0 ≤ ωτ < ∞ Experimental data. Universality.

Oscillating particles in fluids: Theory, experiment and numerics My collaborators: Kamil Ekinci (BU); Carlos Colosqui (BU → Princeton); Devrez Karabacak (BU → Delft). Hudong Chen, Xiaowen Shan, Ilya Staroselsky (EXA Corp.)

Oscillating particles in fluids: Theory, experiment and numerics Landau and Lifshitz, ”Physical kinetics”. After presenting derivation of Boltzmann equation they mention in passing: C ≈ − f − f eq τ with τ = const ”... is a rough estimate of collision integral” This expression, which is at the foundation of BGK-LBM, has never been derived as a consistent approximation

Oscillating particles in fluids: Theory, experiment and numerics Linear oscillator in liquid/gas.

Oscillating particles in fluids: Theory, experiment and numerics d 2 x dx m 0 dt 2 + γ m 0 dt + κ x = R ( t ) (1) d 2 x dt 2 + γ dx 0 x = R 0 dt + ω 2 cos ω t (2) m 0 � ω 0 = κ/ m 0 I ( ω − ω 0 ) = R 2 γ 0 ( ω − ω 0 ) 2 + γ 2 8 m 0 4 The mass m 0 = m R + m f where m R is the resonator mass in vacuum and the added mass m f is that of the fluid ‘pushed’ by the resonator. The damping γ = γ R + γ f where γ f is FLUIDIC DISSIPATION

Oscillating particles in fluids: Theory, experiment and numerics For a spherical slowly moving body of radius a , by the Stokes formula : x = 4 π 3 ρ b a 3 γ ˙ F = 6 πµ a ˙ x F = 16 µ a ˙ x F = 32 3 µ a ˙ x For a 2d-disk and ellipsoid, respectively. The mass:. m 0 = 4 π a 3 ( ρ b + 1 2 ρ ) 3 m 0 = π a 2 ( ρ b + ρ ) for a sphere and disk, respectively.

Oscillating particles in fluids: Theory, experiment and numerics ω 1 ≈ ω 0 (1 − δ m m 0 → m 0 + δ m ; ) 2 m 0

Oscillating particles in fluids: Theory, experiment and numerics Double-Clamped Beam. Schematic. diagram.pdf l l z(t) z(t) h h Δ Δ w w ~ ~ h ≈ 0 . 1 µ m , w ≈ 0 . 5 µ m , l ≈ 10 µ m .

Oscillating particles in fluids: Theory, experiment and numerics Array of Nanobeams. Microscope. Dimensions → ω 0

Oscillating particles in fluids: Theory, experiment and numerics Nanobeams as Sensors. Biomolecule Amplitude of Motion NEMS in Fluidic Enviroment Frequency

Oscillating particles in fluids: Theory, experiment and numerics Nanobeams. Biodetection

Oscillating particles in fluids: Theory, experiment and numerics Nanocantilevers. Biodetection

Oscillating particles in fluids: Theory, experiment and numerics The modern devices: m 0 ≈ 10 − 13 − 10 − 12 gr ω 0 ≈ 10 9 Hz One can work with higher harmonics and achieve higher frequencies. Now, detection of the mass of a proton is a reality.

Oscillating particles in fluids: Theory, experiment and numerics Pressure dependence of resonance curves of NEMS. (Ekinci & Karabacak (2007)

Oscillating particles in fluids: Theory, experiment and numerics FDT. (a) (b) (c) .

Oscillating particles in fluids: Theory, experiment and numerics FDT .

Oscillating particles in fluids: Theory, experiment and numerics In equilibrium if resonator is thermally excited, variance the displacement : � ∞ h 2 P ( h ) dh = 4 k B T κ −∞ In a confined system this relation is not clear. Experimental test of FDT

Oscillating particles in fluids: Theory, experiment and numerics TO DETECT A MASS δ m : δ m ω 0 ≫ γ 2 m 0 THE MAIN STUMBLING BLOCK IN BIODETECTION IS FLUIDIC DISSIPATION IN WATER (FLUIDS). FIRST WE HAVE TO UNDERSTAND IT.

Oscillating particles in fluids: Theory, experiment and numerics This is not so simple. In air τ mol − mol ≈ 0 . 2 × 10 − 9 sec p ≈ 800 torr ; , τ mol − si ≈ 10 − 9 sec . τ mol − mol ≈ 10 − 9 − 10 − 8 p = 100 torr ; . 0 ≤ ωτ ≤ 10 − 100

Oscillating particles in fluids: Theory, experiment and numerics Stokes’ second problem (1851). Infinite plate oscillating in its plane in the x -direction. Find velocity distribution u ( y , t ) u (0 , t ) = U cos ω t ; u ( ∞ , t ) = 0 w = v = 0 ∂ u ∂ t + u · ∇ u = −∇ p /ρ + ν ∇ 2 u

Oscillating particles in fluids: Theory, experiment and numerics ∂ t = ν∂ 2 u ∂ u ∂ y 2 δ cos( ω t + y u ( y , t ) = Ue − y δ ) � 2 ν δ = ω Overdamped surface wave. No transverse propagating shear waves. (Definition of fluid)

Oscillating particles in fluids: Theory, experiment and numerics Force per unit area (viscous stress) σ = ρν∂ u ∂ y | y =0 = − U √ ωρµ cos( ω t + π 4) Mean dissipation per unit time − σ u = U 2 � ωµρ 2 2 Quality factor Q

Oscillating particles in fluids: Theory, experiment and numerics ˙ � ρµ 1 E = γ σ Q = ω ≈ u (0 , t ) ω = 2 π E st ω � µω γ = √ ρµω = RT p

Oscillating particles in fluids: Theory, experiment and numerics Dissipation. x = ωτ delta gamma 10 7 10 5 5 3 1 2 0.5 1.5 1 x 100 x 0 20 40 60 80 1 2 5 10 20 50 100

Oscillating particles in fluids: Theory, experiment and numerics Breakdown of Newtonian Hydrodynamics. ∂ u i ∂ t + u ·∇ u i = 1 ρ ∇ j ρσ ij σ ij = u ′ i u ′ j Kinetic Theory. Boltzmann Equation. σ ij = σ (1) + σ (2) + .... ij ij ≈ ν σ (1) 2( ∂ i u j + ∂ j u i ) ≡ ν S ij ij

Oscillating particles in fluids: Theory, experiment and numerics Different geometries. Different scales.

Oscillating particles in fluids: Theory, experiment and numerics µ = ρν ≈ ρ c s λ σ (1) U 2 ≈ µ S ij U 2 ≈ ρλ c s U ρ U 2 L ≈ λ c s U ≈ Kn ij L Ma ≈ λ 2 ∂ u i ∂ u j σ (2) ij ∂ x α ∂ x α Stokes Problems: S i ,α S α, j = S i α S j ,α = 0 and there is no length scales (Landau-Lifshitz)

Oscillating particles in fluids: Theory, experiment and numerics σ (2) ≈ λ 2 S ij ∇ · u ij One has to solve dynamic kinetic equation.

Oscillating particles in fluids: Theory, experiment and numerics Stokes’ Second Problem, Revisited. Kinetic Boltzmann-BGK equation. VY, Colosqui (2007). ∂ t + v · ∇ f = − f − f eq ∂ f τ 2 exp( − ( v − U ( x , t )) 2 ρ f eq = ) 3 2 θ (2 πθ )

Oscillating particles in fluids: Theory, experiment and numerics Boltzmann’s collision integral: � v rel ( f ′ f ′ C = 1 − ff 1 ) d σ dp 1 ≈ f � v rel f ′ f ′ − τ ( { f } ) + 1 d σ dp 1 � τ ( { f } ) = v rel d σ f 1 dp 1 ≈ τ = const ???

Oscillating particles in fluids: Theory, experiment and numerics Relaxation time τ ≈ const . τ ≈ λ/ v ≈ λ/ c s c s -speed of sound. In the air at temperature θ ≈ 300 K and pressure p = 1 atm ≈ 1000 torr τ ≈ 200 / p × 10 − 9 sec ≈ 0 . 2 × 10 − 9 sec At this point this relation does not account for solid walls, strong shear etc.

Oscillating particles in fluids: Theory, experiment and numerics � � C ( f ) d v = C ( f ) v d v = 0 � ρ = f ( v ) d v ∂ρ ∂ t + ∇ · ρ U ( x ) = 0 ∂ρ U j ( x ) + ∂ i ρ U j ( x ) U i ( x ) + ∂ i σ ij = 0 ∂ t

Oscillating particles in fluids: Theory, experiment and numerics σ ij = ρ ( v i − U i ( x ))( v j − U j ( x )) The goal is to derive the expression for the stress σ ij in terms of observables. This can be done using the Chapman-Enskog expansion. Hudong Chen et al (2004).

Oscillating particles in fluids: Theory, experiment and numerics f = f (0) + ǫ f (1) + ǫ 2 f (2) + · · · ∂ t = ǫ∂ t 0 + ǫ 2 ∂ t 1 + · · · ∇ = ǫ ∇ 1

Oscillating particles in fluids: Theory, experiment and numerics In the zeroth order f = f eq σ ij | eq = ρθδ ij (ideal gas):

Oscillating particles in fluids: Theory, experiment and numerics θ f 0 S ij [( v i − u i )( v j − u j ) − ( u − v ) 2 f 1 = − τ δ ij d f ( 2) = − 2 τ 2 f ( 0)[( v i − v j ) ∂ j ( S ij −∇· u δ ij ..... )+++

Oscillating particles in fluids: Theory, experiment and numerics ∂ U j ( x ) + U i ( x ) ∂ i U j ( x ) + 1 ρ ∇ p ( x ) = 0 ∂ t

Oscillating particles in fluids: Theory, experiment and numerics Defining θ ( x ) = 1 d ( c i − v i ( x )) 2 , we obtain: ∂θ ( x ) + ∇ i ρ U i ( x ) θ + ∂ t 1 d ∇ i ρ ( v i − U i ( x ))( v j − U j ( x )) 2 +2 d ρ ( v i − U i ( x ))( v j − U j ( x )) S i , j = 0

Oscillating particles in fluids: Theory, experiment and numerics In the next order we derive Newtonian approximation and NS Equations: σ 1 ≈ − 2 ρν ( S ij − 1 d δ ij ∇ · u ) ν = θτ S ij = 1 2( ∂ u i + ∂ u j ) ∂ x j ∂ u i

Oscillating particles in fluids: Theory, experiment and numerics σ (2) ˙ = 2 ρν ( τ S ij ) + ij 4 ρν 2 θ ( S ik S kj − δ ij S kl S kl / d ) − − 2 ρν 2 θ ( S ik Ω kj + S jk Ω ki ) ˙ A = ( ∂ t + U · ∇ ) A Ω ij = 1 2( U j i − U i j )

Oscillating particles in fluids: Theory, experiment and numerics σ (1) + σ (2) = − 2 ρν (1 − τ∂ t ) S ij ij ij ∂ t + u · ∇ u = 2 ρν (1 − τ∂ t ) ∂ 2 u ∂ u ∂ y 2