Motion of a sphere in an oscillatory boundary layer: an optical tweezer based s

Size: px

Start display at page:

Download "Motion of a sphere in an oscillatory boundary layer: an optical tweezer based s"

Phillip Miller
5 years ago
Views:

1 Motion of a sphere in an oscillatory boundary layer: an optical tweezer based study November 12, 2006 Tata Institute of Fundamental Research Co-workers : S. Bhattacharya and Prerna Sharma

2 American Journal of Physics vol 45, pages 3-11, 1977

3 Motion of a sphere near Liquid-Solid interface Liquid-liquid interface

4 Oscillatory motion in a viscous fluid Optical tweezer :

5 Power spectral density Measurement of k S(f ) = f 0 = k opt /2πβ k B T ρ 2 π 2 β(f f 2 ) β = 6πηF(h)

6 The bottom plate is moved with y p = y p0 sin(ωt)

7 Boundary layer The boundary layer is notionally a thin layer near the surface of a bounding surface where the effects of viscosity is significant irrespective of how high the Reynolds number is. Navier-Stokes Equation [ ] v ρ t + (v. )v = p + η 2 v In the boundary layer the viscous term is comparable to the inertial term. Hence ( ) ρu u ( x ρu 2 ) /l ø = η 2 u ηu/δ 2 1 y 2 δ = lν u ν = η/ρ

8 Naiver Stokes equation ( ) v ρ t + 0 (v. )v = 0 p + η 2 v. Traveling wave solutions ρ v y t = v y η 2 z 2 v y (z, t) = v 0 e i(kz ωt) (1 + ı) ; k = ± δ Stokes boundary layer, δ = 2η ωρ

9 Relevant Length Scales Introduction Relevant length scale perpendicular to the flow Particle diameter, D. Oscillatory Stokes boundary layer, δ = η ρω Relevant length scale parallel to the flow Particle diameter, (D 1µ). Associated with vortex formation, shedding, and potential interaction is u/ω The other important length scale is the particle- plate separation (h 1µ)

10 inertial contact: The motion of the bead is in-phase with the motion of the plate. In the velocity coupled regime : σ = η v/ z F s = 2πηav 0 e h+a ( 2a δ ) (1 e δ )e iωt,

11 Introduction WATER GLYCEROL Strong distortion of motion of the sphere at high frequencies. Motion of the sphere is sinusoidal and monochromatic.

12 Introduction Phase Vs Frequency The equation of motion of a forced damped harmonic oscillator is x + γx + ω20 x = F0 cos ωt m x = A cos ωt F0 1 A= 2 m [(ω ω2 )2 + (ωγ)2 ]1/2 0 φ = arctan γω 2 ω ω20. γ is the only fitting parameter.

13 Comparing it with the equation a forced damped harmonic oscillator with the equation of motion of sphere m eff ẍ + 6πηaẋ + k op x = F h + 0 F plate + 6πηau, Spring Constant, Effective mass γ = 6πηa kop, ω 0 = m eff m eff Liquid k opt = (6πηa/γ)ω 2 0 k opt (PSD) m eff Water 13.4x10 7 N/m 11.4x10 7 N/m 0.085x10 9 Kg Glycerol 5.96x10 4 N/m 2.7x10 4 N/m 95x10 9 Kg Bare mass of a colloidal sphere Kg Note : The effective mass scales with viscosity and not with the density

14 Introduction Effective mass Systematic variation of the fitted data from the experimental data. meff (f ) = meff (f = 0) + Af 4 Liquid Water Glycerol meff (f = 0) 0.085x10 9 Kg 95x10 9 Kg

15 The motion of the sphere comprises of two kinds of degrees of freedom Translation : Drag Force + Lift Force rotational : Generates Vorticity: Angular velocity Ω = (Ω, 0, 0)

16 The Navier Stokes equation which defines the flow of the liquid about the sphere is given by 2 q p = ρ η (q. ) q + ρ q η t Where q = (q 1, q 2, q 3 ) is the velocity of the fluid. The origin of the cartesian coordinate is taken to coincide with the center of the sphere. Boundary Conditions q 0 when z q = Ω r for the flow on the sphere q velocity of the plate for the flow on it

17 Motion of a sphere near a liquid-liquid interface

19 Microdroplets Before After The fluid flow driven by the sphere is dynamically coupled to instantaneous conformation of the interface. Multiple backflows scattered from the interface.

20 Conclusions Effective mass of the sphere much higher than its bare mass The phase near the interface is highly unstable. The effective mass scales with the liquid viscosity and not the density. Shaking of a trapped sphere near the interface causes formation of micro-droplets.

Sound Pressure Generated by a Bubble

Sound Pressure Generated by a Bubble Adrian Secord Dept. of Computer Science University of British Columbia ajsecord@cs.ubc.ca October 22, 2001 This report summarises the analytical expression for the