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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

m0 d2x dt2 + γm0 dx dt + κx = R(t) (1) d2x dt2 + γ dx dt + ω2

0x = R0

m0 cos ωt (2) ω0 =

• κ/m0

I(ω − ω0) = R2 8m0 γ (ω − ω0)2 + γ2

4

The mass m0 = mR + mf where mR is the resonator mass in vacuum and the added mass mf is that of the fluid ‘pushed’ by the resonator. The damping γ = γR + γf where γf is FLUIDIC DISSIPATION

Oscillating particles in fluids: Theory, experiment and numerics

For a spherical slowly moving body of radius a, by the Stokes formula : F = 6πµa˙ x = 4π 3 ρba3γ ˙ x F = 16µa˙ x F = 32 3 µa˙ x For a 2d-disk and ellipsoid, respectively. The mass:. m0 = 4πa3 3 (ρb + 1 2ρ) m0 = πa2(ρb + ρ) for a sphere and disk, respectively.

Oscillating particles in fluids: Theory, experiment and numerics

m0 → m0 + δm; ω1 ≈ ω0(1 − δm 2m0 )

Oscillating particles in fluids: Theory, experiment and numerics

Double-Clamped Beam. Schematic.

diagram.pdf

h w l z(t) Δ

~

h w l z(t) Δ

~

h ≈ 0.1µm, w ≈ 0.5µm, l ≈ 10µm .

Oscillating particles in fluids: Theory, experiment and numerics

Array of Nanobeams. Microscope. Dimensions → ω0

Oscillating particles in fluids: Theory, experiment and numerics

Nanobeams as Sensors.

Frequency Amplitude of Motion Biomolecule NEMS in Fluidic Enviroment

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: m0 ≈ 10−13 − 10−12gr ω0 ≈ 109Hz 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 : ∞

−∞

h2P(h)dh = 4kBT 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 δm: ω0 δm 2m0 ≫ γ 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

p ≈ 800torr; τmol−mol ≈ 0.2 × 10−9sec , τmol−si ≈ 10−9sec . p = 100torr; τmol−mol ≈ 10−9 − 10−8 . 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 ωt; u(∞, t) = 0 w = v = 0 ∂u ∂t + u · ∇u = −∇p/ρ + ν∇2u

Oscillating particles in fluids: Theory, experiment and numerics

∂u ∂t = ν∂2u ∂y 2 u(y, t) = Ue−y

δ cos(ωt + y

δ) δ =

• 2ν

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(ωt + π 4) Mean dissipation per unit time −σu = U2 2 ωµρ 2 Quality factor Q

Oscillating particles in fluids: Theory, experiment and numerics

1 Q = ˙ E 2πEst = γ ω ≈ σ u(0, t)ω = ρµ ω γ = √ρµω = µω RT p

Oscillating particles in fluids: Theory, experiment and numerics

• Dissipation. x = ωτ

20 40 60 80 100 x 0.5 1 5 10 delta 1 2 5 10 20 50 100 x 1 1.5 2 3 5 7 10 gamma

Oscillating particles in fluids: Theory, experiment and numerics

Breakdown of Newtonian Hydrodynamics.

∂ui ∂t + u·∇ui = 1 ρ∇jρσij σij = u′

iu′ j

Kinetic Theory. Boltzmann Equation.

σij = σ(1)

ij

+ σ(2)

ij

+ .... σ(1)

ij

≈ ν 2(∂iuj + ∂jui) ≡ νSij

Oscillating particles in fluids: Theory, experiment and numerics

Different geometries. Different scales.

Oscillating particles in fluids: Theory, experiment and numerics

µ = ρν ≈ ρcsλ σ(1)

ij

U2 ≈ µSij U2 ≈ ρλcsU ρU2L ≈ λ L cs U ≈ Kn Ma σ(2)

ij

≈ λ2 ∂ui ∂xα ∂uj ∂xα Stokes Problems: Si,αSα,j = SiαSj,α = 0 and there is no length scales (Landau-Lifshitz)

Oscillating particles in fluids: Theory, experiment and numerics

σ(2)

ij

≈ λ2Sij∇ · u 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).

∂f ∂t + v · ∇f = −f − f eq τ f eq = ρ (2πθ)

3 2 exp(−(v − U(x, t))2

2θ )

Oscillating particles in fluids: Theory, experiment and numerics

Boltzmann’s collision integral: C =

• vrel(f ′f ′

1 − ff1)dσdp1 ≈

− f τ({f }) +

• vrelf ′f ′

1dσdp1

τ({f }) =

• vreldσf1dp1 ≈ τ = const???

Oscillating particles in fluids: Theory, experiment and numerics

Relaxation time τ ≈ const. τ ≈ λ/v ≈ λ/cs cs-speed of sound. In the air at temperature θ ≈ 300K and pressure p = 1atm ≈ 1000torr τ ≈ 200/p × 10−9sec ≈ 0.2 × 10−9sec At this point this relation does not account for solid walls, strong shear etc.

Oscillating particles in fluids: Theory, experiment and numerics

• C(f )dv =
• C(f )vdv = 0

ρ =

• f (v)dv

∂ρ ∂t + ∇ · ρU(x) = 0 ∂ρUj(x) ∂t + ∂iρUj(x)Ui(x) + ∂iσij = 0

Oscillating particles in fluids: Theory, experiment and numerics

σij = ρ(vi − Ui(x))(vj − Uj(x)) The goal is to derive the expression for the stress σij 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) + ǫf (1) + ǫ2f (2) + · · · ∂t = ǫ∂t0 + ǫ2∂t1 + · · · ∇ = ǫ∇1

Oscillating particles in fluids: Theory, experiment and numerics

In the zeroth order f = f eq σij|eq = ρθδij (ideal gas):

Oscillating particles in fluids: Theory, experiment and numerics

f 1 = −τ θf 0Sij[(vi − ui)(vj − uj) − (u − v)2 d δij f (2) = −2τ 2f (0)[(vi−vj)∂j(Sij−∇·uδij.....)+++

Oscillating particles in fluids: Theory, experiment and numerics

∂Uj(x) ∂t + Ui(x)∂iUj(x) + 1 ρ∇p(x) = 0

Oscillating particles in fluids: Theory, experiment and numerics

Defining θ(x) = 1

d(ci − vi(x))2, we

• btain:

∂θ(x) ∂t + ∇iρUi(x)θ + 1 d ∇iρ(vi − Ui(x))(vj − Uj(x))2 +2 d ρ(vi − Ui(x))(vj − Uj(x))Si,j = 0

Oscillating particles in fluids: Theory, experiment and numerics

In the next order we derive Newtonian approximation and NS Equations: σ1 ≈ −2ρν(Sij − 1 d δij∇ · u) ν = θτ Sij = 1 2(∂ui ∂xj + ∂uj ∂ui )

Oscillating particles in fluids: Theory, experiment and numerics

σ(2)

ij

= 2ρν ˙ (τSij) + 4ρν2 θ (SikSkj − δijSklSkl/d) − −2ρν2 θ (SikΩkj + SjkΩki) ˙ A = (∂t + U · ∇)A Ωij = 1 2(Uj

i − Ui j )

Oscillating particles in fluids: Theory, experiment and numerics

σ(1)

ij

+ σ(2)

ij

= −2ρν(1 − τ∂t)Sij ∂u ∂t + u · ∇u = 2ρν(1 − τ∂t)∂2u ∂y 2

Oscillating particles in fluids: Theory, experiment and numerics

Two dimensionless expansion parameters appear: σij = σ(1)

ij +σ(2) ij

≈ 2ρν(1+τ ∂ ∂t +O(ν|S| θ ))Sij Wi = τ∂t → τω S ≈ U/L; ν = λcs; θ ≈ c2

s

MaKn = νS/θ ≈ U cs λ L

Oscillating particles in fluids: Theory, experiment and numerics

VISCO-ELASTIC TRANSITION. (VY, Colosqui, JFM 2007). For a problem of oscillating plate: SikSkj = 0 ∇ · U = 0 SiαSαβ.....Sθ,j = 0

Oscillating particles in fluids: Theory, experiment and numerics

This is correct to all orders and as ωτ → ∞ (VY, Colosqui) τ ∂2u ∂t2 + ∂u ∂t = ν∂2u ∂y 2 u(0, t) = U cos ωt; u(∞, t) = w = v = 0 As ωτ → 0, we recover Stokes problem When ωτ → ∞, wave equation.

Oscillating particles in fluids: Theory, experiment and numerics

All this can be exactly derived in a non-perturbative way (Chen, Staroselsky, Orszag, Shan, VY...). ∂f ∂t + v · ∇f = −f − f eq τ Interested in a unidirectional flow, which is incompressible, we, without loss of generality, set ρ = const = 1.

Oscillating particles in fluids: Theory, experiment and numerics

f (x, v, t) =

• t

τ

e−sf eq(x − vτs, v, t − τs)ds + f0(x − vt, v)e− t

τ

t ≫ τ f (y, z, v, t) = ∞ e−sf eq(x−vτs, v, t−τs)ds where x = yj + zk, and j and k are unit vectors along the y and z axis.

Oscillating particles in fluids: Theory, experiment and numerics

Spatial shift operator: F(x + a) = ea∂xF(x) f (x, v, t) = ∞ e−se−τsv·∇f eq(x, v, t−τs)ds Unidirectional flow: ρ = const = 1: u(x, t) =

• vdv

∞ e−se−τsv·∇f eq(x, v, t−τs)ds (3)

Oscillating particles in fluids: Theory, experiment and numerics

Simple Gaussian integration gives: U(y, t) = ∞ e−ses2λ2∇2/2U(y, t − τs) λ2 = τ 2θ sec2 ·cm2/sec2 = cm2

Oscillating particles in fluids: Theory, experiment and numerics

Differentiate twice over time; use identity ∂ ∂s

• e−se

s2λ2∇2 2

• = [sλ2∇2 − 1]
• e−se

s2λ2∇2 2

• τ ∂2u

∂t2 + ∂u ∂t = ν∇2u + νλ2 ∞ s2e−se

s2λ2∇2 2

∇4u(y, t − τs)ds

Oscillating particles in fluids: Theory, experiment and numerics

This hydrodynamic equation is exact. τ ∂2u ∂t2 + ∂u ∂t = ν(1 + φ)∇2u (4) φ = λ2 ∞ dss2e−se

s2λ2∇2 2

e−sτ∂t∇2 (5) φ = −Wi 2 ∞ dss2e−s(1−iWi)−s2Wi2

2

Oscillating particles in fluids: Theory, experiment and numerics

Wi = ωτ → 0 ν = θτ; ∇2 ≈ 1/δ2 = ω/2ν = ω 2θτ φ ≈ ωτ 2 ∞ dss2e−se−s2ωτ

4 eisωτ = ωτ → 0

∇2 = 1/δ2 = ω/2ν

Oscillating particles in fluids: Theory, experiment and numerics

Diffusion equation; Stokes’ problem;

Oscillating particles in fluids: Theory, experiment and numerics

Large-Weissenberg Limit Wi = ωτ ≫ 1: φ ∼ −Wi

• 1

√ Wi

s2ds ∼ −Wi−1/2 → 0 ∂2u ∂t2 + 1 τ ∂u ∂t − c2∂2u ∂y 2 ≡ Θu = 0 (6) The TE derived from the CE expansion before (VY, Colosqui. JFM (2007).

Oscillating particles in fluids: Theory, experiment and numerics

Solution:

u = Ue

y δ− cos(ωt − y

δ+ ) δ δ± = (1 + ω2τ 2)

1 4[cos(tan−1 ωτ

2 ) ± sin(tan−1 ωτ 2 )]

Oscillating particles in fluids: Theory, experiment and numerics

We have three scales and three Knudsen numbers:

δ; δ−; δ+ Kn = λ δ Kn− = λ δ− ; Kn+ = λ δ+

Oscillating particles in fluids: Theory, experiment and numerics

λ ≈ τcs = 1 σn = τ √ θ ∝ τ√p Newtonian regime; Stokes’ scale: δ =

• 2ν

ω ≈

• λcs

ω ≈ λ

• 1

ωτ Knδ = √ωτ = √ Wi

Oscillating particles in fluids: Theory, experiment and numerics

as ωτ → ∞,

Kn− → 1 2; Kn+ → ωτ = Wi u(y, t) → U0e−2y

λ cos(ωt − ωτ y

λ) In the Knudsen layer y ≤ λ we have PROPAGATING SHEAR (TRANSVERSE) wave. LIKE IN SOLIDS

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

Velocity distribution vs τω. Colosqui, VY. LBM-BGK equation.

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

σ(0, t) = ρν exp(−t τ ) t

−∞

∂u(0, λ) ∂y exp(λ τ )d λ τ (7) ν∂u(0) ∂y = U(−cos ωt δ− + sin ωt δ+ )

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

Ddissipation rate per unit time per unit area of the plate: W = −u(0, t)σ(0, t) W (τ, ω) = 1 2 µU2 1 + ω2τ 2( 1 δ− + ωτ δ+ ) (8) (instead of √p in Newtonian regime.

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

Dissipation rate per cycle vs τω.

2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

Plane oscillator.

M∂ttx + Sγ∂tx + ω2

0x = MR(t)

∂ttx + γ ρphp ∂tx + ω2

0x = R(t)

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

γxt = γu(0, t) = 2ρσ(0, t)/ms ms = ρph γ = g ρ√ωpν ρph(1 + ω2τ 2

p)

3 4 ×

[(1 + ωτp) cos(1 2tan−1ωτp)− (1 − ωτp) sin(1 2tan−1ωτp)]

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

In general: ˙ E ≡ W = SU2 2 ωρµ 2 f (ωτ)

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

f (ωτ) = 1 (1 + ω2τ 2)

3 4 [(1+ωτ) cos(tan−1 ωτ

2 )− (1 − ωτ) sin(tan−1 ωτ 2 )]

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

We predict:

γ ∝ √ω; 1 Q = γ ω ∝ 1/√ω; γ ∝ √p γ = const; 1 Q ∝ ω−1; γ ∝ p

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

• Dissipation. VY, Colosqui (2007); Ekinci, Karabacak

(2007).

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

EXPERIMENTAL DATA. KARABACAK, VY, EKINCI (PRL, 2007).

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

Normalized fluidic dissipation γn vs pressure. a. Cantilever (53 × 2 × 460µm); b. beams (230nm × 200nm × 9.6µm);

• c. (240nm × 200nm × 3.6µm); d. relaxation time τ vs p.

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

Quality factor Q vs pressure and frequency.

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

• Universality. Large quartz resonator. Ekinci,

Karabacak,VY; PRL (2008)

Base Mounting clips Bonding area Electrodes Quartz blank Seal Pins

D ≈ 1cm

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

1/Q vs pressure for different ω

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

1/Q vs 1 ≤ p ≤ 103 ; ≤ 101S/m ≤ 104; nems, mems, macro).

Oscillating particles in fluids: Theory, experiment and numerics Experimental data. Universality.

1/Q vs 10 ≤ p ≤ 103 ; ≤ 101S/m ≤ 104; NEMS, MEMS, Quartz (macro).

Oscillating particles in fluids: Theory, experiment and numerics

Summary.

• 0. Solving the BGK equation we predicted a transition from

viscous to visco-elastic behavior at ωτ ≈ 1 in a simple fluid. (VY; Colosqui).

• 1. This effect has been demonstrated on experimental data on

resonators and numerically using LBGK simulations (Ekinci; Karabacak;).

Oscillating particles in fluids: Theory, experiment and numerics

• 2. Size of the device in a vast range 10−4 ≤ L ≤ 1cm does not

enter the results represented as a function of x = ωτ. This points to the universality which is a much broader class than the well-known SIMILARITY.

• 3. The transition is due to formation of propagating shear wave in

the Knudsen layer. Is it a new physics emerged from LBM? (Mandelshtam/ Leontovich (chem reactions), Boon (relaxation due to polymers etc)...

I do not know any work showing this phenomenon in simple flows. Are you?

Oscillating particles in fluids: Theory, experiment and numerics

• 3. We have found a UNIVERSALITY in the entire range

0 ≤ ωτ ≤ ∞ u = Uφ( r L, λ L, ωτ, Re) with the scaling function f (x) derived from kinetic equation.

Oscillating particles in fluids: Theory, experiment and numerics

• 4. LBGK simulations agree with experimental results. (non-trivial
• utcome).
• 5. The transition is general U(x, 0) = U sin(ky). (Colosqui et al
• Phys. Pluids. (January))

Oscillating particles in fluids: Theory, experiment and numerics

• 6. Bodies of finite size (Colosqui)
• 7. Boundary conditions; slip
• no-slip.

Oscillating particles in fluids: Theory, experiment and numerics

Different geometries. Different scales.

Oscillating particles in fluids: Theory, experiment and numerics

Bluff Body. Linear Dimension L. Fig.1a

σ(2)

ij

≈ ρλ2U2/L2 σ(1) + σ(2) ≈ ρν U L + ρλ2 U2 L2 ≈ ρν U L (1 + U cs λ L) ≈ ρU L (1 + τ T ) Kn = λ/L; Wi = τ/T T is a shedding period.

Oscillating particles in fluids: Theory, experiment and numerics

Boundary Layer.

σ(1) ≈ ρν ∂u ∂y ≈ ρcsλ U δ(x) σ(2) ∝ ∂ui ∂xα ∂uj ∂xα = 0 σ(2) ≈ ρλ2∂uu∇ · u ≈ ρλ2 U2 xδ σ(1) + σ(2) ≈ ρcsλU δ (1 + λ2 δ2 )

Oscillating particles in fluids: Theory, experiment and numerics

In the Stokes problem

Oscillating particles in fluids: Theory, experiment and numerics

Future: 1. This may be a huge area with applications in chemical engineering, combustion, bio/medical engineering etc. Based on what we know LBM may become the most important tool......

Oscillating particles in fluids: Theory, experiment and numerics

TIME The parameter τ i: s a time scale characterizing return of an initially perturbed system to thermodynamic equilibrium. If, for example, at time t = 0, a small volume fraction of a gas is locally perturbed from thermodynamic equilibrium by an excess energy ∆E (local heat release, for example), due to intermolecular collisions or collisions with the walls of the vessel, this local perturbation (inhomogeneity) must disappear on a time-scale τ. The first-principle derivation of this parameter, requiring full description of all microscopic details of interactions is very difficult and, in general, is impossible. It is especially hard if relaxation process involves interactions with the solid bodies and excitations

• f phonons and other internal degrees of freedom by a gas

molecule impacting the surface. It is clear that in the case of a flow generated by an oscillating solid surface, it is the gas -surface interaction which is responsible for the dissipation process. For ideal gas of the number density n colliding with the wall:

Oscillating particles in fluids: Theory, experiment and numerics

τ ≈ λcs ∝ cs σn ∝ kBθ

3 2

σp ≡ I p where the cross-section σ includes all microscopic information. It is easy to estimate the relaxation time in a bulk of a gas of hard spheres of diameter d where σ ∝ d2. For a nitrogen gas it gives τ ≈ 180 × 10−9sec. The relaxation time can experimentally be established from the relation ωτ ≈ 1 marking the bending in a dissipation curve γ(τω (See Fig. [XX]. The observed dependence τ ∝ 1/p is shown on Fig. [XX]. Our experimental data give I ≈ 1850 − 1500 for nanocantilevers and nanobeams and I ≈ 400 − 500 for the quartz resonators with aluminum - covered

• surface. The fact that all silicon -based nanodevices can be

charcterized by the same magnitude of coefficient I ≈ I = 1850 × 10−9sec points to the microscopic details of gas-surface interaction as an important factor.