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Tumbling of small non-spherical
particles in a shear flow
B. Mehlig1)
based on joint work with J. Einarsson1), F. Candélier2), F. Lundell3) & J. R. Angilella4)
1)Department
of Physics, University of Gothenburg, Sweden
3)KTH, Stockholm, Sweden
4)Université de Caen, France
Effect of weak fluid inertia on Jeffery orbits,
J. Einarsson, F. Candélier, F. Lundell, J. R. Angilella & B. Mehlig, submitted to Phys. Rev. E
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Shear flow
Shear flow u(y) = sy x̂ , flow-gradient matrix A =
Degenerate orientational dynamics of small
axisymmetric particles: Jeffery orbits (tumbling).
Jeffery, Proc. Roy. Soc. London Ser. A 102, 161 (1922)
Expect that perturbations have strong effect.
0
0
0
s
0
0
0
0
0
.
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Tumbling in simple shear
Axisymmetric particles in shear flow rotate periodically (Jeffery orbits).
Infinitely many degenerate periodic orbits. Jeffery, Proc. Roy. Soc. London
Ser. A 102, 161 (1922)
ˆ
(and ⌦)
J. Einarsson, B. Mihiretie et al.,
arxiv:1503.03023 (2015)
fast
ẑ
n -dynamics
slow
x̂
Micron-sized glass rods
in a micro-channel flow.
forward,
backwards.
ŷ
nz = 1 : log-rolling,
nz = 0 : tumbling. Symmetry axis n spends a long
time aligned with flow-direction x̂ .
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λ>1
Dimensionless parameters
Axisymmetric particles: shape parameter
(aspect ratio = a/c ).
=(
2
1) (
2
2a
+1)
2c
Fluid inertia: Res = a2 s/⌫
shear Reynolds number
⇢p
Res
Particle inertia: St =
⇢f
Stokes number
⌫ kinematic viscosity
s shear rate
p particle density
f fluid density
Brownian rotation: Pe = s/D where D rotational diffusion constant
Péclet number
Jeffery equation obtained for small particles for Res = 0 , St = 0 , Pe =
Degeneracy: must consider effect of perturbations. Here: Res > 0 .
.
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Inertial effects - particles in flows
Particle- and Navier Stokes equations coupled by boundary conditions.
Maxey-Riley equation,
correction to Stokes law
Tumbling of a neutrally buoyant
fibre in shear flow,
slender-body limit.
Log-rolling unstable.
Saffman lift on small sphere
due to shear.
Tumbling of a nearly spherical
neutrally buoyant particle in shear.
Log-rolling stable.
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Effective equation of motion
Neutrally buoyant spheroid.
Find effective vector field, correction to Jeffery’s
equation (caveat: caustics)
ṅ = F 0 (n) + StF 1 (n) + Res F 2 (n) + . . .
Jeffery
F 1 (n) known.
particle inertia
fluid inertia
Einarsson, Angilella & Mehlig, Physica D 278-279, 79 (2014)
Now F 2 (n) . Difficulty: need to calculate torque on particle.
Requires solving Navier Stokes equations.
2
2
Re
St
St
Perturbation theory in Res , , neglect terms of order St Res ,
, s ,... .
Still very difficult problem. Solve it by exploiting the symmetries of problem.
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Equations of motion
Orientational motion of neutrally buoyant spheroid
ṅi =
ijk
j nk
, St(Iij ˙ j + I˙ij
j)
= Ti
Summation convention.
angular velocity j , particle inertia tensor Iij = AI ( ij Pij ) + B I Pij with
Pij = ij ni nj , AI and B I are moments of inertia along and ? to n .
Hydrodynamic torque Ti determined by integrating fluid stresses over particle
surface S . Requires solving Navier Stokes equations. Dimensionless variables
(scales: time T = s 1, length L = a , velocity U = sa , pressure P = µs ):
Res ( t ui + uj
i ui
= 0 , ui =
ui =
p pressure,
j ui )
(
ui
=
ijk j rk
)
as r
ip
+
j j ui
for r
S,
.
(1)
particle angular velocity, ui
shear in dimensionless variables.
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Perturbation theory
Use `reciprocal theorem´ to compute hydrodynamic torque.
Lovalenti & Brady, J. Fluid Mech. 256, 561 (1993)
Subramanian & Koch, J. Fluid Mech. 535, 383 (2005)
Result (simple shear flow)
Z
(0)
Tk = Tk
Res
Jeffery
dv Ũik ( @t ui + uj @j ui )
|{z} | {z }
V
unsteady
unsteady
fluidinertia
inertia
fluid
convective
Integral over volume
V outside particle.
convective
fluid inertia
fluid
inertia
The coefficients Ũik are obtained by solving auxiliary Stokes problem. Above
relation is exact. But unknown ui , solution of Navier Stokes equations.
Perturbation theory: St = Res = 0 - solutions to evaluate Tk to leading order.
Then expand:
!i =
(0)
!i
+
(St)
St !i
+
(Res )
Res !i
+ ... .
(0)
Insert into angular-momentum equation. Find: !i Jeffery angular velocity,
(St)
(Re )
!i particle-inertia contribution, !i s fluid-inertia contribution, depends
on volume integral. Difficult to calculate. Integrand depends non-linearly on n .
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Symmetries
Make use of the symmetries of
the problem. Simple shear flow
A1 = S1 + O1 .
Symmetries constrain form of equation of motion.
ṅi = Oiq nq + [Sip np (np Spq nq )ni ]
+
1 (np Spq nq )Pij Sjk nk + 2 (np Spq nq )Oij
+
3
Pij Ojk Skl nl +
4
nj
Pij Sjk Skl nl .
First row: Jeffery equation. Remainder: St- and Res -corrections, determined
by only four scalar functions
(St)
↵ = St ↵ (
Can be calculated.
)+Res
(Res )
(
↵
) for ↵ = 1, . . . , 4 .
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Results
Stability of log-rolling and tumbling orbits.
Linear stability analysis at small Res .
Stability exponents
LR ,
T
log rolling
log rolling
tumbling
tumbling
stability exponent
contribution from fluid inertia
contribution from particle inertia
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Conclusions
Orientational dynamics of neutrally buoyant axisymmetric particle in shear.
Stability analysis of Jeffery orbits at infinitesimal Res . Results:
-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.
To do:
-analyse orientational motion for small but finite Res . Rosén, Lundell & Aidun (2014)
-wall effects
-settling p = f (more difficult)
-unsteady flows (more difficult)
Lovalenti & Brady, J. Fluid Mech. 256, 561 (1993)
-turbulence (much more difficult)