Angular dynamics of small particles in flows
B. Mehlig, Department of Physics, University of Gothenburg, Sweden
Collaboration with J. Einarsson, F. Candelier, J.-R. Angilella, T. Rosén, F. Lundell & D. Hanstorp
Stokes law
Equation of motion for small particle in a flow u(r, t)
v̇ = (1
⇢f /⇢p )g + (u(r, t)
buoyancy
⇢f Du
v) +
⇢p Dt
Stokes drag
`pressure-gradient´
Tchen, PhD thesis TU Delft (1947)
Notation:
@u
Du
=
+ (u · r)u
Dt
@t
time derivative along
path of fluid element.
Gravity g , particle size a , particle density ⇢p , fluid density ⇢f , viscosity ⌫ .
Stokes parameter
9 ⇢f ⌫
g⌧
1
=
. Dimensionless numbers: St =
, F=
, and ⇢p /⇢f .
2
2 ⇢p a
u0
⌧
Consequence: heavy particles move out of vortices for St > 0 , light particles move in.
Maxey, J. Fluid Mech. 174 (1987) 441
Inertial effect.
Same approximation for rotation: Jeffery torque and St > 0 .
Jeffery, Proc. Roy. Soc. London Ser. A 102 (1922) 161
Particle inertia ( St ) considered, fluid inertia ( ? ) neglected.
`Maxey-Riley´ equation
Maxey & Riley, Phys. Fluids 26 (1983) 883
Gatignol, J. Méc. Théor. Appl. 1 (1983) 143
Small sphere in a flow u(r, t) . Correction to Stokes law due to unsteady fluid inertia
v̇ = (1
⇢f Du
v) +
⇢p Dt
⇢f /⇢p )g + (u(r, t)
Rt
1 ⇢f ⇣
v̇
2 ⇢p
added mass
Boussinesq (1885),
Basset (1888),
Oseen (1910)
+ a 0 dt1 pu̇u̇ v̇v̇
⌫⇡(t t1 )
Du ⌘
Dt
history
Convective fluid inertia is neglected.
t ui
+ uj
unsteady
fluid
inertia
j ui
=
convective
fluid
inertia
1
f
ip
+
j j ui ,
i ui
=0
(Einstein summation convention)
viscous
terms
Similar approximation for rotation.
Gatignol, J. Méc. Théor. Appl. 1 (1983) 143
Is this a good approximation? Depends on the problem in question. Not for turbulence.
Inertial effects for particles in flows
Particle- and Navier-Stokes equations coupled by boundary conditions.
on particle surface.
Translation
Rotation
Sphere in time-dependent non-linear
flow (effect of unsteady fluid inertia)
Tumbling of a neutrally buoyant
fibre in steady shear flow, slender-body limit:
log-rolling orbit unstable.
Maxey & Riley, Phys. Fluids 26 (1983) 88
Gatignol, J. Méc. Théor. Appl. 1 (1983) 143
Subramanian & Koch, J. Fluid Mech. 535 (2005) 383
Neutrally buoyant prolate nearly spherical
spheroid in steady shear:
log-rolling orbit stable.
Saffman, J. Fluid Mech. 1 (1956) 540
Bifurcation? Oblate particles?
Role of convective fluid inertia?
Unsteady flow?
Gatignol, J. Méc. Théor. Appl. 1 (1983) 143
Log-rolling and tumbling in simple shear
Jeffery, Proc. Roy. Soc. London Ser. A 102, 161 (1922)
ˆ
(and ⌦)
Axisymmetric particles in shear flow tumble periodically (Jeffery orbits).
tumbling in flow-shear plane
Einarsson, Mihiretie, Laas, Ankardal,
Angilella, Hanstorp & Mehlig,
Phys. Fluids 28 (2016) 013302
forward,
backwards.
Micron-sized glass rods
in a micro-channel flow.
log rolling
tumbling
nz = 1 : log-rolling,
nz = 0 : tumbling in flow-shear plane. Symmetry axis n spends
long time aligned with flow direction ê1 .
There are infinitely many different Jeffery orbits. Problem is degenerate.
Angular dynamics of small crystals
Fries, Einarsson & Mehlig (2016)
Angular dynamics of particles with discrete rotational
symmetry and reflection symmetry in plane containing
symmetry axis obey Jeffery’s equation
! = ⌦p ⇤(Sp n) ^ n
vector n points along axis
of discrete rotational symmetry
⌦p vorticity of undisturbed flow at particle position,
Sp strain-rate matrix, ⇤ Bretherton’s shape parameter.
Known long ago for cuboids.
Bretherton, JFM 14 (1964) 284
Particles that do not have this reflection symmetry
spin differently, new shape parameter :
! = ⌦ ⇤(S n) ^ n+ (nT S n)n ,
p
p
p
so that spin even when n ? ⌦p :
! · n = ⌦p · n +
(nT S1
0 n)
6
Fries, Muthu, Hanstorp & Mehlig (2016)
Shear flow - dimensionless parameters
Axisymmetric particles: shape parameter
(aspect ratio = a/c ).
Fluid inertia: Res = a2 s/⌫
shear Reynolds number
⇢p
Res
Particle inertia: St =
⇢f
Stokes number
=(
2
1) (
2
+1)
2a
⌫ kinematic viscosity
s shear rate (fluid-velocity gradient)
p
f
λ>1
particle density
fluid density
Jeffery equation obtained for small particles for Res = 0 , St = 0 .
Neutrally buoyant particle. Perturbation theory in Res .
2c
Effective equation of motion
Neutrally buoyant axisymmetric spheroid ( St = Res but keep St and Res separate).
Find effective equation of motion, correction to Jeffery
equation (caveat: caustics)
n
ṅ = F 0 (n) + StF 1 (n) + Res F 2 (n) + . . .
Jeffery
particle inertia
fluid inertia
F 0 (n) Jeffery equation. F 1 (n) also known.
Einarsson, Angilella & Mehlig, Physica D 278-279, 79 (2014)
Lundell & Carlsson, Phys. Rev. E 81, 016323 (2010)
Now F 2 (n) . Difficulty: need to calculate Res-correction to torque.
Requires solving Navier-Stokes equations.
Perturbation theory in Res , St , neglect terms of order St Res , St2 , Re2s ,... .
Still very difficult problem. Solve it by exploiting the symmetries of problem.
Results and conclusions
Einarsson, Candelier, Lundell, Angilella & Mehlig, Phys. Fluids 27 (2015), 063301
Log rolling stable for prolate spheroids at small Res . This
corrects earlier attempts to calculate stability of log rolling.
Saffman, J. Fluid Mech. 1 (1956) 540
Particle inertia negligible.
log rolling
stable
unstable
⇡ 1/7.3 .
decreases
Bifurcation of tumbling orbit at
Res
s
Both unsteady and convective fluid inertia matter.
theory
symbols: DNS
log rolling
stability exponent
tumbling
contribution from fluid inertia
contribution from particle inertia
bifurcation
Direct numerical linear stability analysis of coupled nonlinear problem.
Rosén, Einarsson, Nordmark, Aidun,
Lundell & Mehlig, Phys. Rev. E (2015)
Angular velocity of spheroid in simple shear
For a sphere (
= 1 ) the stability exponents vanishes.
What is the angular velocity of a sphere in a simple shear?
The angular velocity vector is || ⌦ , and when inertia is
(0)
!
negligible 3 = .2s . Jeffery, Proc. Roy. Soc. London Ser. A 102, 161 (1922)
Inertial corrections? Regular perturbation theory (previous calculation) shows that
the O(Res ) -correction vanishes.
-what is the next order in Res ?
-does inertia increase or reduce the angular velocity?
-how do boundaries affect the angular velocity?
-inertial corrections for non-spherical particle (spheroid)?
Sphere in simple shear
Fluid inertia reduces the angular velocity:
3
s
=
1
2
+ 0.054 Re3/2
s
 = 0.01
0.025
0.05
Re3/2
s
Meibohm, Candelier, Rosén, Lundell & Mehlig (2016)
Stone, Lovalenti & Brady (2016)
Singular perturbation problem. Requires asymptotic
matching of inner solution and outer solution at
r ⇠ aRes 1/2 . Gives rise to Re3/2
s -correction if
system is larger than aRes 1/2 (but not in small system).
Re2s
Computer simulations ( , , ). System size L .
Angular velocity in Stokes limit not quite 2s :
(0)
!3
s
=
1
2
+ 0.22 3 with  = 2a/L .
The numerical results agree well with
Experimental results ( ).
!3
s
=
(0)
!3
s
+ 0.054 Re3/2
s
Poe & Acrivos, J. Fluid Mech. 72 (1975) 605
(
).
Spheroid in simple shear
Angular velocity of log rolling spheroid in simple shear !3 =
(but log rolling is unstable for prolate spheroids).
s
2,
independent of shape
Surface area
Fluid inertia reduces the angular velocity. Shape-dependence of correction:
Small particle in turbulence
Translational slip: Rep = vs a/⌫
slip velocity vs
particle Reynolds number
shear Reynolds number
2
2
strain rate s = htr S i (strain-rate matrix S ).
Sl = 1/(s⌧c )
time scale ⌧c of change of flow that particle sees
Turbulent strain: Res = a2 s/⌫
Unsteadiness:
Strouhal number
Particle inertia:
St = (⇢p /⇢f ) Sl Res
Stokes number
Small neutrally buoyant particle in turbulence: Rep ⇡ 0 , but ⌧c ⇠ ⌧K and so Sl ⇠ 1 and
Res ⇠ (a/⌘K )2 , not negligible for larger particles.
But Maxey-Riley equation: Res ! 0 and Sl ! 1 so that Sl Res ⇠ 1.
Angular velocity of sphere in turbulence
Small sphere with ⇢p ⇡ ⇢f so that Rep ⇡ 0: perturbation theory in Res and St .
Angular velocity of sphere:
! = ⌦p +
1
15
(Res
D⌦
St)
Dt xp
6
15 Res Sp
· ⌦p .
Vorticity of the undisturbed flow: ⌦ = 12 r ^ u , evaluated at particle position xp .
In a steady shear flow
D
Dt ⌦
xp = 0 and Sp · ⌦p = 0 , so no O(Res ) -correction
Unsteady flow: neutrally buoyant sphere ( St = Res ) rotates more slowly than
the fluid in a stretching vortex tube:
4
5 Res
Jeffery theory
|!|2 /|⌦p |2 ⇡ 1
⌦p · (Sp · ⌦p )/|⌦p |2 .
vortex stretching
Angular velocity and vortex stretching
Inertial particle dynamics sensitive to time-irreversibility of turbulence.
4
5 Res
Jeffery theory
|!|2 /|⌦p |2 ⇡ 1
vortex stretching
DNS of Y ⌘ ⌧K ⌦p · Sp · ⌦p /|⌦p |2 and
2
⌧K
|⌦p |2 show that inertial correction
can be substantial.
Y (t)
Jucha, Xu, Pumir & Bodenschatz, Phys. Rev. Lett. 113 (2014) 054501
Xu, Pumir, Falkovich, Bodenschatz, Shats, Xia, Francois, Boffetta,
PNAS 111 (2014) 7558
Pumir, Xu, Bodenschatz & Grauer, Phys. Rev. Lett. 116 (2016) 124502
⌧K2 |⌦p |2
Advected particles:
⌦p · (Sp · ⌦p )/|⌦p |2 .
JHU isotropic turbulence data set
Yu, Kanov, Perlman, Graham, Frederic, Burns,
Szalay, Eyink & Meneveau, J. Turbulence 13 (2012)
Experiment ( )
Guala, Luthi, Liberzon, Tsinober & Kinzelbach, JFM 533 (2005) 339
Conclusions
Inertial dynamics of small particles in flows - beyond the Stokes approximation
Neutrally buoyant particles in shear flow: inertial correction to Jeffery equation lifts
degeneracy of Jeffery orbits. How? Open problem for 60 years. We find:
-log-rolling unstable for prolate particles, tumbling in shear plane stable.
-for oblate particles (but not too disk-like) stabilities are reversed
-fluid inertia contributes more strongly than particle inertia
-both unsteady and convective fluid inertia matter.
It would be qualitatively wrong to neglect either.
Angular dynamics of small sphere in turbulence. Computed first inertial correction
to angular velocity
-both unsteady and convective fluid inertia matter. Unsteady approximation
Basset, Boussinesq,...
not sufficient
-angular velocity sensitive to vortex stretching, inertial correction of order O(Res ) .
Questions
1. Shear flows
2
3

Re
- wall effects, expect
s -corrections (  = 2a/L )
- Re3/2
s -corrections to Jeffery stability exponents can be substantial
2. Angular dynamics of small particles in turbulence
-non-spherical particles (spheroids): inertial correction depends
on alignment of n , ⌦p , and eigen-system of Sp
Xu, Pumir & Bodenschatz, Nature Physics 7 (2011) 709
-check predictions by particle-resolving DNS
-translational problem, lift forces
3. Crystals (discrete rotational symmetry) in turbulence
-spheroids: spinning vs. tumbling
Parsa, Calzavarini, Tosch & Voth, Phys. Rev. Lett. 109 (2012)
Chevillard & Meneveau, JFM 737 (2013) 571
Byron, Einarsson, Gustavsson, Voth, Mehlig & Variano, Phys. Fluids (2016)
-crystals at Res = 0 ?
4. Particles in visco-elastic fluids
Gustavsson, Einarsson & Mehlig,
Phys. Rev. Lett. 112 (2014) 014501