PII. S016117120321231X http://ijmms.hindawi.com © Hindawi Publishing Corp.
AN INDIRECT BOUNDARY INTEGRAL METHOD
FOR AN OSCILLATORY STOKES
FLOW PROBLEM
MIRELA KOHR
Received 31 December 2002
The purpose of this paper is to present an indirect boundary integral method for the oscillatory Stokes flow provided by the translational oscillations of two rigid spheres in an incompressible Newtonian fluid of infinite expanse.
2000 Mathematics Subject Classification: 76D05, 76M10.
1. Introduction. The study of oscillatory Stokes flows has a key role in the understanding of the biomechanics of blood flow, the Brownian motion, the motion of swimming microorganisms, and other biological or chemical phe-nomena. The oscillatory Stokes flows have some important characteristics. For example, at low frequencies, they reduce to steady (more exactly quasi-steady) Stokes flows with diffusion of vorticity. At high frequencies, oscillatory Stokes flows reduce to potential flows, located outside of viscous boundary layers. Such a boundary layer, called the Stokes layer, appears in the vicinity of a solid boundary and has the thickness equal to(ν/Ω)1/2, whereνis the kinematic
viscosity of the fluid andΩis the angular frequency of oscillations.
There are several papers devoted to oscillatory Stokes flows. The first paper which refers to an oscillatory Stokes flow problem is that of Stokes (see [15]). The author treated the flow due to the longitudinal oscillations of a sphere or a cylinder in a viscous incompressible fluid. Lawrence and Weinbaum (see [6]) proved that when nonspherical particles are oscillating in a viscous incom-pressible fluid, then an additional term in the expression of the total force on the particles, which is not given by Basset (see [1]), appears. This term is provided by the oscillations and the form of the particles.
boundary integral method for the oscillatory Stokes flow past solid particles and gave a numerical procedure in order to solve the resulting boundary inte-gral equation in the case of axisymmetric flow (see [10]). In his paper, Pozrikidis described the process of separation from the body together with the develop-ment and decay of eddies in the fluid. The results of Pozrikidis concerning the kinematic effects of unsteady Stokes flow were extended by Smith (see [14]).
The purpose of this paper is to present an indirect boundary integral method for the oscillatory Stokes flow due to the translational oscillations of two solid spheres in a viscous incompressible fluid.
2. Mathematical formulation of the problem. We consider an oscillatory Stokes flow due to the oscillations of two solid spheres with the radiiR1and
R2and the velocityUexp(−iΩt)in a quiescent incompressible Newtonian fluid of infinite expanse. HereΩis the angular frequency of oscillations. The velocity and pressure fieldsvandphave to satisfy the following equations:
ρ∂v
∂t =ρf−∇p+µ∇
2v inD, ∇·v=0 inD, (2.1)
whereρandµare the density and the dynamic viscosity of the fluid,ρfis the body force, andDis the domain of the flow, outside of two spheres. Assuming that the body force is constant, we consider the modified pressurePgiven by
P=p−ρ(f·x). (2.2)
Now, using the fact that the flow is an oscillatory flow, the velocity fieldv
and the modified pressurePcan be written as
v(x, t)=u(x)exp(−iΩt), P (x, t)=q(x)exp(−iΩt). (2.3)
From (2.1) and (2.3), we obtain the following equations which describe the oscillatory Stokes flow:
−iρΩu= −∇q+µ∆u inD, ∇·u=0 inD. (2.4)
We now consider the nondimensional variables
x=Rx, u=Uu, q=µUR q, (2.5)
whereU= |U|is the modulus of the amplitude of oscillations of the spheres, andRis one of the two radiiR1andR2. Then (2.4) become
whereλ2= −iΩR2/νis the frequency parameter andν=µ/ρis the kinematic
viscosity of the fluid.
LetΣ1andΣ2be the boundaries of the solid particles. The nondimensional velocity field u have to satisfy the following boundary conditions on these surfaces:
u=e onΣi, i=1,2, (2.7)
whereeis the unit vector of the direction of the particle displacement. Hence
U=Ue.
Also, because the fluid is in rest at great distances, we add the following far-field conditions:
u(x)→0, q(x)→0 as|x| → ∞. (2.8)
In order to simplify the notations, we will omit the symbol prime from (2.6) and conditions (2.7) and (2.8).
3. Uniqueness result of solution. With the help of the flux divergence for-mula, the following result can be easily obtained.
Theorem3.1. LetᏰ⊂R3be a bounded domain with the boundary∂Ᏸof
classC1. Ifu,u ∗∈C2(Ᏸ)∩C1(Ᏸ)are two vectorial functions such that∇·u=
∇·u∗=0, andq, q ∗∈C1(Ᏸ)∩C0(Ᏸ)are two scalar functions, then
Ᏸu
∗·λ2−∆u+∇qdv
=
∂Ᏸu
∗·T[u] ndσ+λ2 Ᏸu·u
∗dv+2 Ᏸ
D[u] :D[u∗]dv,
(3.1)
whereD[u] denotes the deformation tensor, that is,
Du=12∇u+(∇u) T, (3.2)
((∇u) T is the transposed matrix of∇u=(∂u
i/∂xk)i,k=1,...,3),T[u] is the stress
tensor associated to the fieldsuandq, that is,
Tu= −qI +2Du, (3.3)
Note that for two 3×3 matricesA(Aij)andB(Bij), we set
A:B=
3
i,j=1
AijBij. (3.4)
Using the above result and a straightforward computation, we can prove the uniqueness of the classical solution(u, q)of the oscillatory Stokes problem (2.6), (2.7), and (2.8) (i.e.,u∈C2(D)∩C1(D)andq∈C1(D)∩C0(D)).
Theorem3.2. The oscillatory Stokes system (2.6), together with the bound-ary conditions (2.7) and the following conditions at infinity:
u(x)∇u(x)=o|x|−2, u(x)q(x)=o|x|−2 as|x| → ∞, (3.5)
has at most one solution.
4. The Green function of the oscillatory Stokes flow. Consider the oscilla-tory free-space Green function (or oscillating Stokeslet)Gλ2
=(Gλij2)i,j=1,3and
the pressure vectorΠλ2 =(Πλ2
1 ,Πλ 2 2 ,Πλ
2
3 )of the oscillatory Stokes flow due to
a point force located in the pointyofR3. These functions satisfy the following
equations:
λ2−∆Gλ2 ij(x)+
∂Πλ2 j (x)
∂xi =
δijδ(x),
∂Gijλ2(x) ∂xi =
0, (4.1)
whereδijis the Kronecker’ symbol,δ(x)is the three-dimensional delta func-tion, andx=x−y.
The componentsGλij2of the Green functionGλ 2
are given by [8,10]:
Gλ2 ij(x)=
1 4πe
−λr
δij
r A(λr )− xixj
r3 B(λr )
−λ12∂Π λ2 j (x)
∂xi
, (4.2)
wherer= |x|,λ∈Cmeans the particular square root ofλ2which has a positive
real part (Reλ >0), and
A(l)=1+1
l+
1
l2, B(l)=1+
3
l+
3
l2. (4.3)
The componentsΠλ2
i of the pressure vectorΠ λ2
are given by
Πλ2 i ( x)=
1 4π
xi
Next we consider the stress tensorSλ2
, associated to the oscillating Stokeslet and having the following components [11]:
Sijkλ2 = −Πλ 2 j δik+
∂Gλ2 ij
∂xk +
∂Gλ2 kj
∂xi
. (4.5)
From (4.2), (4.3), and (4.4), we obtain
Sλ2 ijk(x−y)
= − 1 4π
1
r3
δijxk+δkjxi
e−λr(λr+1)+2e−λr
1+ 3
λr+
3
λ2r2
− 6
λ2r2
+ 1
r3δikxj
1+2e−λr
1+ 3
λr+
3
λ2r2
− 6
λ2r2
+xixrj5xk
−10e−λr
1+λr3 +λ23r2
+λ302r2−2e−
λr(λr+1).
(4.6)
On the other hand, using the symmetry property of the Green functionGλ2 (i.e., Gλ2
ij(x−y)=Gλ 2
ji(y−x)) and the formulas (4.4), (4.5), we can write the components of the stress tensorSλ2as follows:
Sλ2
ijk(x−y)= 1 4π
yj−xj
r3 δik+
∂Gjiλ2(y−x) ∂xk +
∂Gλjk2(y−x) ∂xi
. (4.7)
Thus the tensorSλ2is determined by a point force (the first term of the right-hand side of (4.7)) and by two point force dipoles (the second and third terms of the right-hand side of (4.7)). Using again the formula (4.4) and the remark that the pressure fieldᏼλ2, due to a point source located at an arbitrary point x, isᏼλ2(x)=λ2/r, we deduce that the pressure tensorΛλ
2
, associated to the stress tensorSλ2
, has the following components [13]:
Λλ2
ik(x−y)= 1 8π
−ᏼλ2(x−y)δik+
∂Πλ2 i (y−x)
∂xk +
∂Πλ2 k (y−x)
∂xi
=41π
δik
r3
λ2r2−2+6xixk
r5
, i, k=1,2,3.
(4.8)
5. Properties of the unsteady hydrodynamic potentials. LetΓ be a closed surface of classC2andh=(h1, h2, h3)a vectorial continuous function onΓ.
Forx∈R3\Γ, we define the unsteady single-layer potentialᐂs
λ2,Γhas follows:
ᐂs λ2,Γh
j(x)=
ΓG
λ2
and the unsteady double-layer potentialᐂdλ2has
ᐂd λ2,Γh
j(x)=
ΓS
λ2
ijk(y−x)nk(y)hi(y)dσ (y), (5.2)
wheren(y)is the exterior unit normal vector toΓ at the pointy. Additionally, we consider the functionsPλs2,ΓhandPλd2,Γhas follows:
Pλs2,Γh
(x)=
ΓΠ
λ2
i (x−y)hi(y)dσ (y),
Pλd2,Γh
(x)=
ΓΛ
λ2
ik(x−y)nk(y)hi(y)dσ (y),
(5.3)
forx∈R3\Γ.
The functions (ᐂsλ2,Γh, Pλs2,Γh)and(ᐂdλ2,Γh, Pλd2,Γh) are smooth functions in each of the domainsR3\D0andD0, respectively, whereD0is the inner domain
with the boundaryΓ. Also these functions satisfy the following equations of oscillatory Stokes flow:
λ2−∆ᐂs λ2,Γh
(x)+∇Pλs2,Γh
(x)=0, ∇·ᐂs λ2,Γh
(x)=0,
λ2−∆ᐂd λ2,Γh
(x)+∇Pd λ2,Γh
(x)=0, ∇·ᐂd λ2,Γh
(x)=0, (5.4)
forx∈R3\Γ.
For our purpose, we also need the surface forceHλ2,Γh, associated to the unsteady single-layer potentialᐂsλ2,Γhand defined in a neighborhoodᐂ⊂R3 ofΓ as follows [5,16]:
Hλ2,Γhj(x)=nk(x)
ΓS
λ2
jik(x−y)hi(y)dσ (y), (5.5)
wherex∈Γ is the unique determined projection ofx∈ᐂontoΓ.
LetᏳbe the free-space Green function (or the Oseen-Burgers tensor) of the steady Stokes flow and let be the associated stress tensor. These tensors have the following components [5,11]:
Ᏻij( x)= 1 8π
δij
r + xixj
r3
, ijk( x)= − 3 4π
xixjxk
r5 . (5.6)
Using again the formulas (4.2), (4.3), and (4.6), we obtain the following de-compositions:
Gλij2(x)=Ᏻij(x)+Gλ 2
ij(x), S λ2
ijk(x)=ijk(x)+Sλ 2
ijk(x), (5.7)
whereGλ2 ij andSλ
2
ijkare continuous functions. Thus, we have
Gijλ2( x)−Ᏻij( x) →0 asr →0(orλ →0),
Sijkλ2( x)−ijk( x)→0 asr →0(orλ →0),
and thus the potentials (5.1) and (5.2) have the same behaviour as in the case
λ=0. With this argument, we obtain the continuity property of the unsteady single-layer potential in the whole spaceR3and the well-known jump formulas
for the unsteady double-layer potential and respectively the surface force of the unsteady single-layer potential across the boundaryΓ.
We introduce the notations
w+x0= lim
x→x0∈Γ x∈R3\D 0
w(x), w−x0= lim
x→x0∈Γ x∈D0
w(x), (5.9)
for the limiting values of a functionwon the two sides of the surfaceΓ. Addi-tionally, the notation
ᐂd
λ2,Γh ∗
j
x0=
P V
Γ Sijk
y−x0nk(y)hi(y)dσ (y) (5.10)
means the principal value of the unsteady double-layer potential in a point
x0ofΓ (i.e., the value of the improper but convergent unsteady double-layer potential whenx0∈Γ). Similarly,
Hλ2,Γh∗jx0=nkx0 P V
Γ Sjik
x0−yhi(y)dσ (y) (5.11)
means the principal value of the surface force of the unsteady single-layer potential in the pointx0∈Γ.
Now, using the above properties as well as the notations (5.9), (5.10), and (5.11), we obtain the following result.
Theorem5.1. LetD0⊂R3be a bounded domain with the boundaryΓ of
classC2and leth∈C0(Γ)be a given vectorial function. Then on the surfaceΓ
ᐂs λ2,Γh
+ =ᐂs
λ2,Γh −
=ᐂs
λ2,Γh, (5.12)
ᐂd
λ2,Γh +
−ᐂd λ2,Γh
∗ =1
2h=
ᐂd
λ2,Γh ∗
−ᐂd λ2,Γh
−
, (5.13)
Hλ2,Γh+−Hλ2,Γh∗= −
1 2h=
Hλ2,Γh∗−
Hd λ2,Γh
−
. (5.14)
Next, we determine the asymptotic behaviour of the unsteady potentials at infinity. First, from (4.2), (4.3), and (4.4), it follows that (see [17])
ᐂs
λ2,Γh
(x)=ᏻ|x|−3 as|x| → ∞,
Ps λ2,Γh
Additionally, from (4.6), we conclude that the componentsSijkλ2nk ofSλ 2
n
can be written in the following manner [17]:
Sijkλ2(y−x)nk(y)= − 1 4π
xinj(y)
r3 +
xknk(y)
r3 δij
a1(λr )
+xjnri3(y)a2(λr )+
xixjxknk(y)
r5 a3(λr )
,
(5.16)
where
a1(λr )=e−λr3+6(λr )−1+6(λr )−2+λr−6(λr )−2,
a2(λr )=1+2e−λr1+3(λr )−1+3(λr )−2−6(λr )−2,
a3(λr )=2e−λr−6−λr−15(λr )−1−15(λr )−2+30(λr )−2.
(5.17)
Hence, we have
a1(λr )=ᏻr−2 asr → ∞,
a2(λr )=1+ᏻr−2 asr → ∞,
a3(λr )=ᏻr−2 asr → ∞,
(5.18)
and thus the unsteady double-layer potential has the following behaviour at infinity [17]:
ᐂd
λ2,Γh
(x)=ᏻ|x|−2 as|x|→ ∞. (5.19)
Finally, formula (4.8) yields (see [17])
Pλd2h
(x)=ᏻ|x|−1 as|x| → ∞. (5.20)
6. The boundary integral formulation of the problem. In this section, we give a boundary integral formulation in order to solve the oscillatory Stokes problem (2.6), (2.7), and (2.8). More exactly, we have the following theorem.
Theorem6.1. The following boundary integral equation:
1 2h
x0+ᐂd λ2,Σ
1∪Σ2h ∗
x0+ᐂs λ2,Σ
1∪Σ2h
x0=e, x0∈Σ1∪Σ2, (6.1)
has exactly one continuous solution h on Σ1∪Σ2 and the boundary integral representations
u(x)=ᐂd λ2,Σ1∪Σ2h
(x)+ᐂs λ2,Σ1∪Σ2h
(x), x∈D, (6.2)
q(x)=Pλd2,Σ 1∪Σ2h
(x)+Pλs2,Σ 1∪Σ2h
(x), x∈D, (6.3)
Remark6.2. The unknown densityhof (6.1) has the following form:
h=
h1 onΣ1,
h2 onΣ2, (6.4)
and the unsteady double-layer and single-layer potentials, ᐂd
λ2,Σ1∪Σ2 and ᐂs
λ2,Σ1∪Σ2, are written as follows:
ᐂd
λ2,Σ1∪Σ2h=ᐂ d
λ2,Σ1h1+ᐂ d λ2,Σ2h2, ᐂs
λ2,Σ
1∪Σ2h=ᐂ s λ2,Σ
1h1+ᐂ s λ2,Σ
2h2.
(6.5)
Proof ofTheorem6.1. Using the decomposition formula (5.7), we de-duce that the single-layer and double-layer operators,ᐂsλ2,Σ
i:C
0(Σ
i)→C0(Σi) andVλd2,Σ
i:C
0(Σ
i)→C0(Σi), where
ᐂs
λ2,Σ
ig
j
x0=
Σi
Gλji2
x0−ygi(y)dσ (y), x0∈Σi,
Vλd2,Σ
ig
j
x0=ᐂd λ2,Σ
ig ∗ j x0 = P V Σi
Sλ2 ijk
y−x0nk(y)gi(y)dσ (y), x0∈Σi,
(6.6)
g∈C0(Σ
i),have weakly singular kernels and hence they are compact operators on the Banach spaceC0(Σ
i)of continuous functions onΣi,i=1,2. Furthermore, we write (6.1) in the following form:
1 2hj
x0+
Σ1∪Σ2
Mjix0,yhi(y)dσ (y)=ej, x0∈Σk, k=1,2, (6.7)
where
Mjix0,y
=
Sλ2 ijk
y−x0nk(y)+Gλ 2 ji
x0−y, fory∈Σ1,x0∈Σ1∪Σ2 Sijkλ2y−x0nk(y)+Gλ
2 ji
x0−y, fory∈Σ2,x0∈Σ1∪Σ2.
(6.8)
The integral operator generated by the kernel matrix(Mji)j,i=1,2,3, withΣ1∪ Σ2as the domain of integration, is compact on the space of continuous func-tions onΣ1∪Σ2, due to the compactness of the integral operatorsᐂs
λ2,Σi and
Vλd2,Σ
i, i=1,2 (generated separately by the kernels G λ2
and Sλ2
n and with
the following homogeneous adjoint equation admits only the trivial solution in the space of continuous functions onΣ1∪Σ2:
1 2h
0x0+H
λ2,Σ1∪Σ2h
0∗x0+ᐂs λ2,Σ1∪Σ2
h0x0=0, x0∈Σ1∪Σ2. (6.9)
Leth0be a continuous solution of the homogeneous adjoint equation (6.9).
Consider the following functions:
u(0)=ᐂs λ2,Σ1∪Σ2
h0, q(0)=Ps
λ2,Σ1∪Σ2
h0. (6.10)
These functions satisfy the equations
λ2−∆ᐂs λ2,Σ1∪Σ2
h0(x)+∇q(0)(x)=0,
∇·ᐂs λ2,Σ1∪Σ2
h0(x)=0
(6.11)
in each of the domainsD,D1, andD2, whereDiis the inner domain with the boundaryΣi,i=1,2.
Furthermore, from (5.14) and (6.9), we deduce that the surface force
Hλ2 ,Σ1∪Σ2h
0, associated to the fields (6.10), has the following limiting value
on the inner side ofΣi:
Hλ2
,Σ1∪Σ2h
0−x0=1
2h
0x0+H
λ2,Σ1∪Σ2h 0∗x0
= −ᐂs λ2,Σ1∪Σ2
h0x0, x0∈Σ
i
(6.12)
or
H
λ2,Σ1∪Σ2h
0−= −ᐂs λ2,Σ1∪Σ2
h0 onΣ
i, i=1,2. (6.13)
Theorem 3.1withu=u(0),q=q(0),u∗=u(0), andᏰ=Di,i=1,2, yields
Di
λ2u(0)2
+2ejk(0)e(jk0)
dv=
Σi
ᐂs λ2,Σ1∪Σ2
h0·H
λ2,Σ1∪Σ2h
0−dσ , (6.14)
where the unit normal vectornpoints outside ofΣiand
e(jk0)=1 2
∂u
(0) j
∂xk +
∂u(k0)
∂xj
. (6.15)
Substituting (6.13) into (6.14), we obtain the identity
Di
λ2u(0)2+2e(0) jke
(0) jk
dv= −
Σi
ᐂs
λ2,Σ1∪Σ2
which implies that
u(0)=0 onDi, i=1,2. (6.17)
From (6.13) and (6.17), we find that
Hλ2,
Σ1∪Σ2h
0−=0 onΣ
i, i=1,2. (6.18)
Returning to (6.11) and using the properties (5.12), (5.15), and (6.17), we conclude that the functionsu(0)andq(0)determine an oscillatory Stokes flow in the unbounded domainD, with zero velocity field on the surfacesΣ1 and
Σ2and at infinity. Thus from the uniqueness result (Theorem 3.2), we obtain
u(0)=0, q(0)=0 inD (6.19)
and consequently
Hλ2
,Σ1∪Σ2h
0+=0 onΣ
i, i=1,2. (6.20)
Now, using the jump formula (5.14) for the surface force due to an unsteady single-layer potential across its domain of integration, as well as the properties (6.18) and (6.20), we deduce that
h0=Hλ2 ,Σ1∪Σ2h
0−−H
λ2,Σ1∪Σ2h
0+=0 onΣ
i, i=1,2. (6.21)
Hence the adjoint homogeneous equation (6.9) has only the trivial solution in the space of continuous functions onΣ1∪Σ2. From Fredholm alternative, we conclude that the Fredholm integral equation of the second kind (6.1) has a unique continuous solutionh. With this function, we determine the boundary integral representationsuandqas in (6.2) and (6.3). Using the properties (5.4), (5.15), (5.19), and (5.20), we find that the functionsu andqsatisfy (2.6) and the asymptotic conditions (3.5) at infinity. Also from (5.12), (5.13), and (6.1), it follows that conditions (2.7) are also satisfied. This completes the proof.
Finally, using (4.1), we may prove that the nondimensional amplitudeF(m) of the forceR(m), exerted by the oscillatory Stokes flow on the surfaceΣm(i.e.,
R(m)=µU RF(m)exp(−iΩt)), has the following components:
F(m)j= −
Σm
h(m)j(x)dσ
+λ2
Σm
h(m)l(y)
Σm
Gλkl2(x)xjnk(x)dσ (x)dσ (y)
+λ2
Σm
h(m)l(y)nk(y)
Dm
Sλ2
ljk(y−x)dv(x)dσ (y),
(6.22)
7. Asymptotic solution for small-frequency oscillations. As an applica-tion of the above boundary integral method, we consider the case of small values of the frequency parameterλ2. With this aim, we expand the density h, the oscillating StokesletGλ2
, and its associated stress tensorSλ2
in Taylor series with respect toλas follows:
h=h(0)+λh(1)+λ2h(2)+··· asλ →0, Gλ2=G(0)+λG(1)+λ2G(2)+··· asλ →0, Sλ2=S(0)+λS(1)+λ2S(2)+··· asλ → ∞,
(7.1)
where (see [10])
Gij(0)=Ᏻij= 1 8π
δ
ij
r + xixj
r3
,
Gij(1)= − 1 6πδij,
Gij(2)= 1 32πr
23δij
r − xixj
r3
.
(7.2)
Substituting (7.1) into the boundary integral equations (6.1) and collecting terms of zeroth and first order with respect to λ, we deduce the following equations:
1 2h
(0)+
P V
Σ1∪Σ2
S(0)nh(0)dσ+
Σ1∪Σ2
G(0)h(0)dσ=e, (7.3)
1 2h
(1)+
P V
Σ1∪Σ2
S(0)nh(1)dσ+
Σ1∪Σ2
G(0)h(1)dσ
= −
Σ1∪Σ2
S(1)nh(0)dσ−
Σ1∪Σ2
G(1)h(0)dσ ,
(7.4)
onΣi,i=1,2.
BecauseG(1)= −(1/6π )I, it follows thatS(1)=0and thus
Σ1∪Σ2
S(1)nh(0)dσ=0 onΣ1∪Σ2. (7.5)
The boundary integral equation (7.3) corresponds to a steady Stokes flow due to the translational motion of the solid spheres with the constant velocity
e. We may prove that the forceF(Σ0i), acting of the solid sphereΣiin this steady Stokes flow, is determined only by the single-layer potential with the density
h(i0)and has the form (for more details, see [5,11])
F(Σ0)
i = −
Σi
h(i0)dσ , (7.6)
whereh(i0)=h(0)|
Hence from (7.3), (7.4), (7.5), and (7.6), we obtain the following boundary integral equations for the unknown densitiesh(0)andh(1):
1 2h
(0)+
P V
Σ1∪Σ2
S(0)nh(0)dσ+
Σ1∪Σ2
G(0)h(0)dσ=e, (7.7)
1 2h
(1)+
P V
Σ1∪Σ2
S(0)nh(1)dσ+
Σ1∪Σ2
G(0)h(1)dσ= −61πFΣ(01)+F(Σ02), (7.8)
onΣ1∪Σ2. Consequently, the zeroth-order problem (7.7) describes the steady Stokes flow generated by the translational motions of two solid spheres with the velocitye, while the first-order problem (7.8) describes the steady Stokes flow due to the steady motion of the solid spheres with the velocities−F(Σ01)/(6π )
and −F(Σ02)/(6π ), respectively. Hence the solution of the first-order problem (7.8) may be computed using only the forcesFΣ(01)andF(Σ02)acting on the spheres
Σ1andΣ2in the steady Stokes flow.
Next using the relation (6.22) and the Taylor series (7.1), we obtain the fol-lowing asymptotic development for the nondimensional amplitudeF(m)of the forceR(m)exerted on the sphereΣm(m=1,2):
F(m)j=−
Σm
h((m)j0) (x)dσ−λ
Σm
h((m)j1) (x)dσ
+λ2 Σm
h((m)l0) (y)
Σm
G(kl0)(x)xjnk(x)dσ (x)dσ (y)
+
Σm
h((m)l0) (y)nk(y)
Di
Sljk(0)(y−x)dv(x)dσ (y)
+ᏻλ3 asλ →0.
(7.9)
8. Asymptotic solution for high-frequency oscillations
8.1. The velocity potential of the flow. In the case of high-frequency limit |λ|r→ ∞, viscous terms in the singularity solutions (4.2) and (4.5) vanish ex-ponentially and hence we obtain
Gλ2 ji(x)= −
1
λ2 ∂Qi(x)
∂xj
, Sλ2
ijk(x)= −δikQj(x), (8.1)
where
Qj(x)=Πλ 2 j (x)=
1 4π
xj
r3, j=1,2,3. (8.2)
More exactly, if we expand the functionGijλ2in an asymptotic series for large |λ|r, then we obtain
Gijλ2=41πλ12
−δrij3 +xrix5j
+41πe−λr
δij
r − xixj
r3
and hence (8.1). Next using the boundary integral representation (6.2) and the property (8.1), we obtain the velocity fielduin the form
uj(x)= − 1
λ2
Σ1∪Σ2 ∂Qi(x)
∂xj
hi(y)dσ (y)
+
Σ1∪Σ2
Qj( x)ni(y)hi(y)dσ (y).
(8.4)
Thus, in the limit|λ|r→ ∞, we have
u(x)= −∇φ(x), (8.5)
forxin the flow field, whereφis the velocity potential given by
φ(x)= 1
λ2
Σ1∪Σ2
Qi(x)hi(y)dσ (y)
+ 1 4π
Σ1∪Σ2 1
rni(y)hi(y)dσ (y).
(8.6)
From the equations
∇·u=0, ∂Tik ∂xk =
λ2u
i (8.7)
and equality (8.5), we obtain the following properties of the velocity potential
φ:
φ=0, Tik= −λ2φδik, (8.8)
whereTikare the components of the stress tensor associated to the fieldsu andq.
The resulting irrotational velocity field vanishes at infinity and satisfies the impenetrability condition on the surfacesΣ1andΣ2, but it does not satisfy the nonslip boundary condition on the same surfaces. This result appears from the nonuniform limiting process|λ|r→ ∞, used to obtain (8.4) from (6.2). At high frequencies, the flow is composed of an outer irrotational flow and two Stokes boundary layers of thickness|λ|−1that reside on each of the particle
surfaces. To compute the outer irrotational flow, we solve the following Fred-holm integral equation of the second kind for the velocity potentialφ:
φ(x)=2
Σ1∪Σ2
G(x,y)eknk(y)dσ (y)
+2
P V
Σ1∪Σ2
∇G(x,y)·n(y)φ(y)dσ (y), x∈Σ1∪Σ2,
(8.9)
For a pointxaway from the particle surfaces, equation (8.9) is replaced by the following formula:
φ(x)=
Σ1∪Σ2
G(x,y)eknk(y)dσ (y)
+
Σ1∪Σ2
∇G(x,y)·n(y)φ(y)dσ (y).
(8.10)
8.2. The boundary-layer solution. In the viscous boundary layer that re-sides on the surfaceΣi, the characteristic length for tangential flow field vari-ations is the local radius of the mean curvature, denoted bybi(x). The char-acteristic length for the flow field variations normal to the surfaceΣiis|λ|−1. For|λ|bi(x)1, the boundary-layer equations for the unsteady Stokes flow field are [7]
∂2w
i
∂z2 −λ 2w
i=λ2∇Σiφ,
∂2v
i
∂z2 −λ 2v
i=
∂q
∂z, ∇Σi·wi+
∂wi
∂z =0,
(8.11)
wherewiis the tangential velocity field,viis the normal velocity component,z is the local normal coordinate,∇Σiis the surface gradient operator, andφis the velocity potential (evaluated on the surfaceΣi,i=1,2) for the outer irrotational flow. The velocity field that satisfies the nonslip boundary condition on the surfaceΣiand matches the potential flow solution (8.5) is [7]
wi=e−λzwi−∇Σiφ, vi= − 1
λ
1−e−λz∇Σi·w i−
∂φ
∂n, (8.12)
∂qi
∂n = ∇Σi·w
, (8.13)
wherew=e+ ∇φ is the associated tangential slip velocity (relative to the surfaceΣi) andλqi=q−λ2φis the excess pressure of the orderᏻ(|λ|)that appears from the boundary layer flow.
For the next arguments, we denote byλqthe excess pressure of the order
ᏻ(|λ|)that appears from two boundary layer flows.
Now using property (8.13) and the fact thatq=0, we obtain the following Fredholm integral equation of the second kind for the excess pressureq[3]:
q(x)= − 1
4π
Σ1∪Σ2 2
r∇Σ1∪Σ2·w(y)−8π q(y)Qk( x)nk(y)
dσ (y), (8.14)
References
[1] A. B. Basset,A Treatise on Hydrodynamics; with Numerous Examples, vol. 2, Deighton, Bell and Co., Cambridge, 1888.
[2] J. Feng, P. Ganatos, and S. Weinbaum,Motion of a sphere near planar confining boundaries in a Brinkman medium, J. Fluid Mech.375(1998), 265–296. [3] J. D. Jackson,Classical Electrodynamics, John Wiley & Sons, New York, 1962. [4] S. Kim and W. B. Russel,The hydrodynamic interactions between two spheres in
a Brinkman medium, J. Fluid Mech.154(1985), 253–268.
[5] M. Kohr,Probleme Moderne în Mecanica Fluidelor Vâscoase. Vol. II[Modern Prob-lems in Viscous Fluid Mechanics. Vol. II], Presa Universitar˘a Clujean˘a, Cluj-Napoca, 2000 (Romanian).
[6] C. J. Lawrence and S. Weinbaum,The force on an axisymmetric body in linearized, time-dependent motion: a new memory term, J. Fluid Mech.171(1986), 209–218.
[7] M. Loewenberg, Axisymmetric unsteady Stokes flow past an oscillating finite-length cylinder, J. Fluid Mech.265(1994), 265–288.
[8] M. McCracken,The resolvent problem for the Stokes equations on halfspace inLp, SIAM J. Math. Anal.12(1981), no. 2, 201–228.
[9] C. Pozrikidis,A singularity method for unsteady linearized flow, Phys. Fluids A1 (1989), no. 9, 1508–1520.
[10] ,A study of linearized oscillatory flow past particles by the boundary-integral method, J. Fluid Mech.202(1989), 17–41.
[11] ,Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992.
[12] ,The motion of particles in the Hele-Shaw cell, J. Fluid Mech.261(1994), 199–222.
[13] , Introduction to Theoretical and Computational Fluid Dynamics, The Clarendon Press, Oxford University Press, New York, 1997.
[14] S. H. Smith,Slow oscillatory Stokes flow, Quart. Appl. Math.55(1997), no. 1, 1–22. [15] G. G. Stokes,On the effect of the internal friction of the fluids on the motion of
pendulums, Trans. Cambridge Philos. Soc.9(1851), 8–106.
[16] W. Varnhorn,An explicit potential theory for the Stokes resolvent boundary value problems in three dimensions, Manuscripta Math.70 (1991), no. 4, 339– 361.
[17] ,The Stokes Equations, Mathematical Research, vol. 76, Akademie-Verlag, Berlin, 1994.
Mirela Kohr: Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, 1 M. Kog˘alniceanu Street, 3400 Cluj-Napoca, Romania
