A Method of Regularized Stokeslets for Periodic Boundary...

A Method ofRegularized

Stokeslets forPeriodicBoundaryConditions

Anita Layton

Introduction

PeriodicRegularizedStokeslets

Derivation

NumericalExamples

Periodic arraysof spheres

Beating cilia

A Method of Regularized Stokeslets forPeriodic Boundary Conditions

Anita Layton

May 26, 2011



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Motivation: A carpet of cilia



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Periodic arrays of point forces

Fluid motion at zero Reynolds number due to a point force:

−∇p + µ∇2u = −gδ(x− x0), ∇ · u = 0 (1)

In 3D, the Stokeslet is

S(x, x0) =I

r+

x̂x̂

r3, (2)



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Numerical challenges

Problem: Direct summation over a periodic array of force doesnot coverge (

∑1/r).

Solution: Hasimoto solved the periodically-forced Stokesequations with appropriate boundary conditions.

Problem: Slow convergence of Fourier series for practicalcomputations.Solution: Ewald’s summation



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Regularized Stokeslets

For forces supported along a surface S,

u(x) =1

8πµ

∫SS(x, x0)gdx0 (3)

Problem: Because S is singular, a numerical quadratureapproximation of the integral at a point near S may beinaccurate.Solution: Regularized Stokeslets by Cortez.

−∇p + µ∇2u = −gφε(x− x0), (4)

u(x) =1

8πµSε(x, x0)g (5)



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Deriving the Periodic Regularized Stokeslets

Consider Stokes fluid past a periodic array of regularized pointforces located at the vertices of a 3D lattice

Xn = ı1a1 + ı2a2 + ı3a3

where (ı1, ı2, ı3 = 0,±1,±2, . . . ), and a1, a2, a3 are basevectors that determine the shape of the lattice.Motion of the fluid is described by

−∇p + µ∇2u = −g∑n

φε(x̂n), ∇ · u = 0 (6)

where x̂ ≡ x− x0 − Xn.



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia


Split Sε into

Sε(x, x0) =(∇2I −∇∇)R (7)

=Θε(x, x0) + Φε(x, x0), (8)

where

Θε(x, x0) = (I∇2 −∇∇)R · erfc(ξR), (9)

Φε(x, x0) = (I∇2 −∇∇)R · erf(ξR). (10)

where R =√

r2 + ε2, and ε is the cutoff function parameter.



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia


Directly applying the operator in

Θε(x, x0) = (I∇2 −∇∇)R · erfc(ξR)

gives

Θε(x, x0) = IC (ξR)

R+ xxD(ξR), (11)

where

C (x) = ercf(x) +2x√π

(2x − 3 + ε2(1− ξ2)

)e−x2

, (12)

and

D(x) = ercf(x) +2x√π

(1− 2x2)e−x2. (13)



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia


To evaluate Φε(x, x0), consider its Fourier transform w.r.t. X:

Φ̂ε(k, x̂0) =

∫R3

exp(ik · X)Φε(x, x0)d3X, (14)

=

∫R3

exp(ik · X)(I∇2 −∇∇)R · erf(ξR)d3X, (15)

= (−I|k |2 + kk)

∫R3

exp(ik · X)R · erf(ξR)d3X,

(16)

Lots of algebra...



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia


More algebra, integration by part...

Even more algebra...

Φ̂ε(k, x̂0) =8π

|k|2

(−I +

kk

|k |2

)×[(

1 +ω2

4+ω4

8

)e− |k|2

4ξ2 − erfc(ξε) + cos (|k |ε)− 2ξε√π

]× e ik·x̂0 . (17)



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia


By Parseval’s identity∑n

Φε(x, xn) =1

τ

∑k

Φ̂ε(k, x̂0) (18)

where τ is the volume of the unit cell in the physical space,τ = a1 · (a2 × a3).And the regularized Green’s function is∑

n

Sε(x, xn) =∑n

Θε(x, xn) +1

τ

∑k

Φ̂ε(k, x̂0), (19)



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Three sphere packings

A

B C



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Drag coefficient

1 Assume velocity on spheres is 1.

2 Computer drag forces.

3 F = 6πµaKV , K is the drag coefficient.

Packing Our K Zick & Homsy Hasimoto

A 4.43 4.292 4.50B 4.50 4.447 4.47C 4.50 4.446 4.47

Relative errors ∼< 3%.

Zick & Homsy, JFM, 1982.

Hasimoto, JFM, 1959.



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

A carpet of beating cilia

Fulford & Blake, J Thero Biol, 1986

Questions:

1 Effective vs. recovery stroke?

2 Single cilium vs. carpet of cilia?

3 Spacing between cilia?



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Cilia model

Effective stroke Recovery stroke

Cilium in a periodic box.

Approximate floor.

Specify velocity at markers → boundary forces → pressureand velocity everywhere.



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Streamlines—Effective stroke

1x1x1 Free space

Two cases: same ciliary motion, different BCs.

Different streamlines, different transport velocity



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Streamlines—Recovery stroke

1x1x1 Free space



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Velocity slice—Effective stroke

t = 3/10 of beat period.y velocity at slice y = −0.2.



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Velocity slice—Recovery stroke

t = 7/10 of beat period.y velocity at slice y = −0.2.



Anita Layton

Introduction


Derivation

NumericalExamples


Beating cilia

Acknowledgements

Karin Leiderman Elizabeth Bouzarth

This research is supported by the National Science Foundation,through grant DMS-0715021 (to Layton) and the EMSW21Research Training Groups grant DMS-0943760 to theDepartment of Mathematics at Duke University.

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