the inverse droplet coagulation problem

The inverse droplet coagulation problem

Colm Connaughton

Warwick Mathematics Institute and Warwick Centre for Complexity Science,University of Warwick, UK

Joint work with Robin Ball and Peter Jones.

Multiphase Turbulent Flows in the Atmosphere and OceanNCAR Geophysical Turbulence Programme

13-17 August 2012

http://www.slideshare.net/connaughtonc ( ) NCAR August 2012 1 / 17

Coagulation of clusters and droplets

Many particles of onematerial dispersed inanother.Transport: diffusion,turbulence advection,ballistic...Particles stick together oncontact.

Applications: surface and colloid physics, atmospheric science,biology, cloud physics, astrophysics...


Mean field theory: Smoluchowski’s equation

Cluster size distribution, Nm(t), satisfies the kinetic equation :

Smoluchowski equation :

∂Nm(t)∂t

=12

∫ m

0dm1K (m1,m −m1)Nm1Nm−m1 (1)

− Nm

∫ M−m

0dm1K (m,m1)Nm1 +

Jm0

δ(m −m0)

− Nm

∫ M

M−mdm1K (m,m1)Nm1

Some additional features:Source of monomers, m0, at rate J.Removal of clusters larger than cut-off, M.


Model kernels

At mean-field level all micro-physics is encoded into the kernel,K (m1,m2). Kernel is often (approximately) homogeneous:

K (am1,am2) = aβ K (m1,m2)

K (m1,m2) ∼ mµ1 mν

2 m1�m2.

A popular model kernel is the Van-Dongen kernel:

K (m1,m2) =12(mµ

1 mν2 + mν

1mµ2 )

K (m1,m2) for some physical models:(m1

m2

)ν+

(m2

m1

)ν+ 2 Brownian coagulation, ν = 1

3 , µ = −13(

m131 + m

132

)2 ∣∣∣∣m 231 −m

232

∣∣∣∣ Gravitational settling, ν = 43 , µ = 0(

m131 + m

132

)2√

1m1

+1

m2Differential rotation, ν = 2

3 , µ = −12


Self-similar solutions of Smoluchowski equation

For homogeneous kernels, clus-ter size distribution often self-similar. Without source:

Nm(t) ∼ s(t)−2 F (z) z =m

s(t)

s(t) is

the typical cluster size. The scaling function, F (z), determining theshape of the cluster size distribution, satisfies:

−2F (z) + zdF (z)

dz=

12

∫ z

0dz1K (z1, z − z1)F (z1)F (z − z1)

− F (z)∫ ∞

0dz1K (z, z1)F (z1).


Stationary solutions of Smoluchowski eq. with source

K (m1,m2) = 1.

Kernel is often homogeneous:

K (am1,am2) = aβ K (m1,m2)

K (m1,m2) ∼ mµ1 mν

2 m1�m2.

Clearly β = µ+ ν. Model kernel:

K (m1,m2) =12(mµ

1 mν2 + mν

1mµ2 )

Stationary state for t →∞, m0 � m� M (Hayakawa 1987):

Nm =

√J (1− (ν − µ)2) cos((ν − µ)π/2)

2πm−

β+32 . (1)

Describes a cascade of mass from source at m0 to sink at M.


Inverse problems

Forward problem: given kernel, K (m1,m2), compute the sizedistribution, Nm(t).Inverse problem: given observations of the size distribution, Nm(t),compute the kernel, K (m1,m2) (Wright and Ramakrishna, 1992).

Inverse problem is useful because:Kernel may not be known.May help in building models and guiding micro-physics theory.Quantifies the sensitivity of the size distribution to variations in thekernel.

butInverse problems are typically ill-posed.


The stationary inverse Smoluchowski problem

The stationary size distribution satisfies:

−J δ(m − 1) = S(Nm)[K (m1,m2)].

where S(Nm) is the integral operator defined by RHS of (1).Linear in K (z1, z2).If we measure Nm, at M discrete masses, then this equationdiscretises to a set of M linear equations for the M2 values of theK (m1,m2).

b = S k.

This system is enormously under-determined⇒ one can findmany solutions but they are mostly determined by the noise in themeasurements.


Dealing with ill-posedness

One strategy is to restrict the allowed K (m1,m2) to a class of functionsdescribed by a much smaller set of parameters and find the “best fit"within this restricted class of functions.

Wright and Ramakrishna (1992)

Suggested the following representation of kernel:

K (m1,m2) =N∑

i=1

N∑j=1

ai,j Li(m1)Lj(m2)

where Li(x) are Laguerre polynomials.

Inverse problem becomes a linear least squares problem for theN2 coefficients ai,j .One can hope to use cross-validation techniques to find optimal N.Not too good for kernels with fractional exponents.


An alternative parameterisation

Inverse problem is very sensitive to the form of the kernel for very largeand very small masses. Can we find a parameterisation which is betterat capturing this?Any homogeneous kernel with asymptotic exponents µ and ν can berepresented as:

K (m1,m2) =12(mµ

1 mν2 + mν

1mµ2 ) f

(m1

m2

)where the “shape function" is by definition

f(

m1

m2

)=

K (m1,m2)12 (mµ

1 mν2 + mν

1mµ2 ).

It is a function of the single variable m1m2

, because it is homogeneous ofdegree zero and has the symmetry f (x) = f (1/x).


Computational strategy

Step 1:Assume f (x) = 1 and find the "best" values of µ and ν.This is a two-dimensional (nonlinear) minimization problem. Weused the Nelder-Mead algorithm.

Step 2:After estimating µ and ν, try to improve our estimate of kernel byfitting the one-dimensional function f (x).Representation of f (x) which enforces required symmetry?

f (x) = g(log(x))

where

g(y) =N∑

n=1

an cos(

n π ylog M

).

This is a linear least squares problem for the an.http://www.slideshare.net/connaughtonc ( ) NCAR August 2012 11 / 17

Results: model kernel (stationary)

Test of step 1 for Van-Dongen kernel:

K (m1,m2) =12

(m

141 m− 1

22 + m

− 12

1 m142

)

Stationary Nm for ν = 1/4, µ = −1/2, M = 250

compared to exact solution fo M =∞.

Slices through the reconstructed kernel. Exponents

obtained are effectively exact.


Results: Stationary Brownian coagulation µ = −ν = 13

Full test for Brownian coagulation of spherical droplets:

K (m1,m2) =

(m1

m2

) 13

+

(m2

m1

) 13

+ 2

Stationary Nm for Brownian coagulation, M = 250

compared to exact solution fo M =∞.

Slices through the reconstructed kernel compared to

exact values. νest ≈ 0.486, µest ≈ −0.484.


Results: shear-like kernel (stationary) ν = 1, µ = −14

Full test for a toy model variation on the shear kernel:

K (m1,m2) =(√

m1 +√

m2)2

√1√m1

+1√m2

Stationary Nm for a toy model with ν = 1, µ = − 14 .


exact values. νest ≈ 0.97, µest ≈ 0.24.


What about the nonstationary case?Method applies to the nonstation-ary case in the scaling limit:

Nm(t) ∼ s(t)−2 F (z) z =m

s(t)

s(t) =M∑

i=1

m2Nm(t)/M∑

i=1

mNm(t)

is typical size used to collapse data.

The

scaling function, F (z) satisfies:

−2F (z) + zdF (z)

dz=

12

∫ z

0dz1K (z1, z − z1)F (z1)F (z − z1)

− F (z)∫ ∞

0dz1K (z, z1)F (z1).

which is similar to stationary Smoluchowski equation.http://www.slideshare.net/connaughtonc ( ) NCAR August 2012 15 / 17

Results: Sum kernel (nonstationary) ν = 14, µ = 0

Nonstationary test case for a toy model variation on the sum kernel:

K (m1,m2) = m141 + m

142

Nonstationary Nm and data collapse for a toy model

with ν = 14 , µ = 0.


exact values. νest ≈ 0.29, µest ≈ −0.03.


Concluding remarks

SummaryDespite clear difficulties, some features of the collision kernel canbe reconstructed from measurements of the size distribution.We have presented an approach to the stationary inverse problemwhich works reasonably well for homogeneous kernels and can beextended to time-dependent problems in the scaling limit.

Issues and future workNeed better ways of selecting the optimal number of Fouriercoefficients to represent the shape function.Sensitivity to noise in the data should be explored.What can be done for non-homogeneous kernels?


the inverse droplet coagulation problem

Technology

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denition m1

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values of thek m1