2 - One Sphere in Stokes Flow

7/25/2019 2 - One Sphere in Stokes Flow

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Three single sphere flows Hydrodynamic force moments Faxen laws Sphere in shear flow

1 Three single sphere flows

RotationTranslationStraining

2 Hydrodynamic force moments

ForceTorqueStressletComputing the hydrodynamic force

3 Faxen laws for the sphere

4 A sphere in simple shear flow

Elisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension Dynamics

One sphere in Stokes flow
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Decomposition of the relative motion about a point

Fluid motion near a point (Taylor series to linear order) = uniformtranslation + linearly varying field

u(x) = u(x0) + u(x0) (x x0)

= U + x+ E x

Antisymmetric rate-of-rotation tensor

ij = 1

2

uixj

uj

xi

can be expressed as rotation vector: i

= 1

2ijk

jk

= 1

2( u)i =

1

2

i

Symmetric rate-of-strain tensor

Eij = 1

2

uixj

+uj

xi


http://find/


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Decomposition of shear into rotation and strainSimple shear: contributions of rotation and strain equal in magnitude

u = + E



S
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Rotation









Th i l h fl H d d i f F l S h i h fl


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Rotation

Sphere in a rotational field and sphere rotating



Th i l h fl H d d i f t F l S h i h fl
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Rotation

Disturbance problem for the sphere in a rotational field

Disturbance fields=velocity and pressure differences from thoseexisting in the imposed flow in the absence of the body

u(x) = uactual(x) u(x)

p(x) = pactual(x) p(x)

Homogeneous Stokes equations for the disturbance fields

u = 0

2u = p

Boundary conditions

u= x at r= |x| =a

u and p 0 as r= |x|



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Rotation

Solution for the pressure

p=

decaying harmonics

spherical solid harmonics: 1

r and its gradients,

xi

r3 ,

ijr3

3xixjr5 ,

ijxk+ikxj+kjxir5

5xixjxkr7 , . . .

p= scalar and linear in

pi xi

r3

But pseudo-vector whereas pframe independent

p= 0



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Rotation

Solution for the velocity

u=

decaying harmonics

1

r,xi

r3,ijr3

3xixjr5

,ijxk+ikxj+kjxi

r5

5xixjxkr7

, . . .

u= vector and linear in

u(x) =1

x

r3

Velocity boundary condition 1 = a3

u(x) = xar

3



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Rotation

Sphere in a rotational field and sphere rotating

Sphere in a rotational field

No pressure induced by the presence of the sphere

Disturbance velocity:

u(x) =

xar3

decays as r2 and retains the symmetry of the boundary condition

on the sphere

Sphere rotating at p

No induced pressure

Velocity:

u(x) = p xar

3



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Translation











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g p y y p

Translation

Sphere in a translational field and translating sphere



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Translation

Disturbance problem for the sphere fixed in the uniform

stream u

=U


u = 02u = p

Boundary conditions

u= U

at r= |x| =au and p 0 as r= |x|



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Translation


p=

decaying harmonics

1

r,xi

r3 ,ijr3

3xixjr5 ,

ijxk+ikxj+kjxir5

5xixjxkr7 , . . .

p= scalar and linear in U

p=1U

x

r3



http://find/


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Translation


Particular solution

u(p) =px/2=12

xx

r3 U

Homogeneous solutionu(h) =

decaying harmonics:

1

r,xi

r3,ijr3

3xixjr5

,ijxk+ikxj+kjxi

r5

5xixjxkr7

, . . .

u(h) = vector and linear in U

u(h) =21

rU +3(

I

r3

3xx

r5 ) U



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Translation

Determination of constants

Continuity: u= 0 yields (after some work)

uixi

=

12 2

Ux

r3 = 0

2 =1/(2)

Boundary condition u= U at r=a

12a

niU

j nj+U

i

+

3U

j

a3 [ij 3ninj] = U

i

1 = 3a/2 and 3 = a3/4

Boundary conditions at

Automatically satisfied by decaying harmonics

Elisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic A Physical Introduction to Suspension DynamicsOne sphere in Stokes flow

http://goforward/http://find/http://goback/


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Translation

Disturbance velocity and pressure

Disturbance velocity

ui= 3a

4Uj

ijr

+xixj

r3

3a3

4 Uj

ij3r3

xixj

r5

Disturbance pressure

p p= 3a

2

Uj xj

r3

Disturbance solution

velocity induced by a sphere translating in otherwise quiescentfluid at Up = U


http://find/


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Translation

Streamlines for translationDisturbance streamlines for a translating sphere and full streamlines for a particle fixed in

uniform stream


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Translation

Three features of this Stokes flow field

Slow decay of disturbance fields away from the translatingsphere, as r2 for the pressure and as r1 for the dominantportion of the velocity

Fore-aft symmetry: instantaneous streamlines of thedisturbance velocity field converging above the particle anddiverging below

Disturbance twice as large at a point on the axis of symmetry

3Upa

/2r

as at a point at an equal distance in thetransverse direction 3Upa/4r(due to the pressure fieldmaintaining the flow as divergent-free)


http://find/


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Straining








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Straining

Sphere fixed in a strain field


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Straining

Disturbance problem for the sphere at the origin in the

straining flow E x


u = 02u = p

Boundary conditions

u= E

x at r= |x| =au and p 0 as r= |x|


http://find/


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Straining


p=

decaying harmonics

1

r,xi

r3,ijr3

3xixjr5

,ijxk+ikxj+kjxi

r5

5xixjxkr7

, . . .

p= scalar and linear in second-rank tensor E

p

ijr3 3

xixj

r5

Eij

= 1 xiE

ij xjr5

becauseijE

ij =E

ii = u = 0 (continuity)


http://find/


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Straining


Particular solution

u(p)i =

p

2xi =

12

xixjE

jkxk

r5

Homogeneous solution

u(h) =

decaying harmonics:

1

r,xi

r3,ijr3

3xixjr5

,ijxk+ikxj+kjxi

r5

5xixjxkr7

, . . .

u(h) = vector and linear in E

u(h)i =2E

ij

xj

r3+3E

jk

ijxk+ikxj

r5

5xixjxkr7


http://find/


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Straining

Disturbance velocity and pressure

Determination of constants

Continuity: u= 0 2 = 0

Boundary conditions: 1= 5a3 and 3= a

5/2

Disturbance velocity: dominant portion decays as r

2

ui = 5a3

2

xi(xjE

jkxk)

r5

a5

2Ejk

ijxk+ikxj

r5

5xixjxkr7

Disturbance pressure: decays as r3 and quadrupolar form

p= 5a3xiE

ij xj

r5


http://find/


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Straining

Streamlines for a sphere fixed in a strain field


http://find/


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F
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Force







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Torque


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Torque









Torque
http://find/


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Torque

Hydrodynamic torque

Hydrodynamic torque on the particle

Th =

Sp

x ndS

Torque balance for inertialess motion

Th + Te = 0

Te = external torque

Stokes law (sphere held fixed in rotational flow x)

Th = 8a3

linear in and in a3



Stresslet
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Stresslet









Stresslet
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Stresslet

First moment of the surface traction over the surface

Mij=

Sp

xijknkdS

Symmetric portion = stresslet

Sij=1

2

Sp

[ikxj+jkxi]nkdS

Antisymmetric portion = same information as torque

Aij=1

2

Sp

(xijk xjik)nkdS= 1

2ijkTk



Stresslet
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Stresslet for a solid sphere in a straining flow Ex

S=20

3 a3E

linear in E and in a3

Result of the resistance of the rigid particle to the straining motion



Computing the hydrodynamic force
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Computing the hydrodynamic force


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Simpler computation of the hydrodynamic force

Computation of dominantstress decaying as R2

Divergence theorem applied to the Stokesmomentum equations

Vij

xjdV =

Spijn

outj dS+S

ijnout

j dS= 0

Drag force

Fhi =

Sp

ijnjdS =

Sp

ijnout

j dS

=

S

ijn

out

j dS

=

S

(2)ij

noutj dS


http://find/


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Elisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic

A Physical Introduction to Suspension Dynamics




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Faxen laws

Hydrodynamic force and force moments for a sphere in a generalambient flow field

F = 6a

1 +

a2

6

2u

(x= 0)U

pT = 8a3 [(x= 0) p]

S = 20

3a3

1 +

a2

102

E(x= 0)

Additional pieces owing to the curvature of the flow 2u (evaluated at

the position x= 0 occupied by the center of the particle) for the force

and stresslet but not for the torque!

Elisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic

A Physical Introduction to Suspension Dynamics


http://find/


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Lag of a sphere in a Poiseuille flow

uz =Umax

1 rR

2

Force-free suspended sphere at rs

Us = [1 +a2

62]uz (rs)

= uz (rs)2

3

a2

R2Umax

Torque-free sphere in this situation hasrotation vector ofp =

p

=rUmax/R2

(along the azimuthal coordinate )



http://find/


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1 Three single sphere flowsRotationTranslationStraining







http://find/


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Decomposition of a sphere in a shearsphere in a rotation + sphere in strain



http://find/


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Freely mobile sphere in simple shear flow

Homogeneous Stokes equations

u = 0

2u = p

Boundary conditions

u= p x at r= |x| =a

u u 0 as r= |x|

Faxen lawsForce-free: Up =u(x= 0)

Torque-free: p = (x= 0)



http://find/


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Flow caused by a sphere held fixed in linearly varying

ambient flow field u = (y, 0, 0) = (E + )x

Total velocity field

ui =u

i

5a3

2

xi(xjE

jkxk)

r5

a5

2 E

jkijxk+ikxj

r5

5xixjxk

r7

Disturbance flow generated by the sphere only due to its resistanceto the straining component of the shearing flow because no

disturbance created by a freely-rotating sphere embedded in asolid-body rotation

Rotationally dominated motion in the vicinity of the rotating sphere closed streamlines



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Streamlines around freely mobile sphere in simple shear


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2 - One Sphere in Stokes Flow

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