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Creeping flow about a spherical particle
Physica 113A (1982) 90-102 North-Holland Publishing Co.
CREEPING FLOW ABOUT A SPHERICAL PARTICLE R. SCHMITZ and B.U. FELDERHOF Institut fiir Theoretische Physik A, RWTH Aachen, SommerfeldstraBe, FR Germany
5100 Aachen,
Received 14 December 1981
We formulate the solution of the linear Navier-Stokes equations for steady, incompressible flow about a spherical particle as an expansion of scattered waves. The amplitudes of the outgoing waves are simply related to a set of spherical force multipoles. These multipoles are found from the amplitudes of the incident waves with the aid of a resistance matrix, which is expressed in terms of the scattering coefficients of the particle.
1. Introduction In previous work we have studied the solution of the Navier-Stokes equations for steady, incompressible flow in the presence of a hard sphere with mixed slip-stick boundary conditions’) and for a permeable spherical particle2) for arbitrary incident flow. In the space outside the particle our solution is closely related to that of Lamb3”). We showed that the perturbed flow involves scattering coefficients which are characteristic for the particle. In this article we analyze the situation more systematically as a problem in scattering theory. We introduce complete sets of incident and outgoing waves in terms of vector spherical harmonics. The expansion coefficients are found with the aid of two adjoint sets which satisfy orthonormality relations on the surface of a sphere. We derive some antenna theorems which show that the amplitudes of the outgoing waves are very simply related to a set of spherical force multipoles. The induced force density can be replaced by a force density which is localized on a spherical shell and gives rise to the same flow outside the particle. This reduced force density can be constructed from the spherical force multipoles. These multipoles are related to the amplitudes of the incident waves by means of a resistance matrix, which is expressed in terms of the scattering coefficients. The present concise formulation should also be useful in dealing with flows involving many particles. In a following article we shall consider the scattering problem for two spherical particles. 0378-437 1/82/ooo(Mooo /$02.75 @ 1982 North-Holland
CREEPING
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91
2. Basic equations We consider steady flow of an incompressible fluid in the presence of a spherically symmetric particle of radius a. The particle may be a hard sphere with mixed slip-stick boundary conditions, or it may be penetrable to the flow with a radially symmetric permeability. In the case of a hard sphere it is convenient to extend the validity of the flow equations to all space and to represent the boundary conditions by a suitably chosen induced force density5.6). The linearized Navier-Stokes equations for the flow velocity u(r) read qV*u - Vp = - F(r),
V. u = 0,
(2.1)
where q is the shear viscosity, p(r) is the pressure which is determined by the condition of incompressibility, and F(r) is the induced force density acting on the fluid. For a hard sphere centered at the origin the last of these can be chosen as’) F(r) = f(0, cp)S(r - a) + 5a.f~(0,p)S’(r - a),
(2.2)
where the surface force density f(0, cp) is determined by the incident Aow, 5 is a slip coefficient taking the value 5 = 0 for stick and 5 = f for pure slip boundary conditions, and ft is the part of f tangential to the sphere. The solution of (2.1) for r < a is given by u(r) = u(r), where u(r) = U + 0 X r
(2.3)
is the instantaneous solid body motion of the sphere. spherical particle the induced force density is given by F(r) = -Ur)[n(r)
-
For a penetrable
u(r)l,
(2.4)
where [h(r)]-’ is the permeability of the particle. In this case the flow velocity u(r) for r < a is more complicated. More in general we could consider a constitutive equation F(r) = -1 A( r, r’) *[u(r’) - u(r’)] dr’ with the conditions that the kernel symmetric and rotationally invariant: n&r,
r’) = A&r’, r),
A(r, r’) vanishes
(2.5) for r, r’ > a and is
A(r, r’) = D - A(D-’ - r, D-’ - r’) . D-‘,
where D is a rotation matrix.
(2.6)
92
R. SCHMITZ AND B.U. FELDERHOF
The formal solution of (2.1) is given by To(r - r’) - F(r’) dr’,
u(r) = uO(r) + -&-I
(2.7) p(r)=po(r)+&jQ(r-r’)*F(r’)dr’ with the Oseen kernels 1 To(r) = ~(1+
*
*1 rr),
Q(r) = j,
(2.8)
where i = r/r, and where (uo(r), PO(r)) is an arbitrary equations.
solution of the free flow
3. Free flow solutions We consider solutions of the homogeneous ?jv2urj-vp,=o,
equations
v*uo=o.
(3.1)
A complete set of solutions appropriate to spherical symmetry has been presented by Lamb3*‘). In previous articles’*‘) we have rederived Lamb’s solutions in a scheme better suited to the solution of the inhomogeneous equations. In this scheme a complete set of solutions of (3.1) is given by u:,,(r)
= r’Al+l,,,,(f),
o&r(r) = r’C,,(+),
pks(r)
= 0,
(I 2 O),
p h,(r)
= 0,
(1 2 l),
pl m p(r)
=
7)
lc21 - ‘X21+ ‘) r’-‘Y,_,,m(i) l-1
,
(I B 2). (3.2)
Each of the velocity fields is homogeneous of degree I in the Cartesian coordinates. The vector spherical harmonics are defined by’) AI, = lYr,e,+$$e,+&$$e,, Br, = -(1+ 1)Y,me,+$$e,9+&$$e,, 1 au,, Cl, =--ee--e sin 8 acp
(3.3)
aYl,
a0 q'
where Y1,(8, cp) are the spherical harmonics
and e,, e& e, are unit vectors in
CREEPING FLOW ABOUT A SPHERICAL PARTICLE
93
spherical coordinates. The above vector harmonics are related to the vector spherical harmonics Y 2” (with c = i) defined by Normands), according to B,, = V/(1 + 1)(21 + 1) Y In+?
A,,,, = d/1(21 + 1) Y ;‘*‘, C,, = -id&l + 1) Yf;f.
(3.4)
The vector spherical harmonics Y,“[‘I form a complete orthonoimal set for vector functions on the unit sphere. For our purposes it is more convenient to use the harmonics A,,,,, I#,,,,, Cl,,, which are not normalized to unity, so as to avoid unpleasant square root factors in the formulae. We shall denote the set of solutions (3.2) by {u&&r)}, where the discrete index u indicates the tensor character of the solution and takes the values S (symmetric), T (tangential), P (pressure). We have previously’) discussed the relation to Lamb’s solutions. In the following the pressure field can be left out of consideration. Any solution q,(r) of (3.1) can be expanded as
00(r)=
z.cik,uik&>= 2
ciL,luhJ.
(3.5)
In order to find the expansion coefficients c& it is convenient to define an adjoint set of functions {wTW(r)}. We require this set of functions to be orthonormal to the set {uk,} on a sphere of arbitrary radius b. It is easily seen that such a set of functions is given by
d&r> =
(l
uhl+
+
3)
1
r
-1-l
[ AI+I,,(+)-
21+5 2
BI+l,, (*)I, r
(12 01,
&&r)
- -l(1
1) r+‘C,,(i),
(I 3 11,
wL(r)
1 = -1(21_ 1) r-‘-‘B,_,, ,,,(i),
(I 2 2).
These functions
+
satisfy the orthonormality
relations
(WE&% 1 ~Ltlw) = &~mm’~m~, where we have introduced cf 1t.9 = j f*(r)
(3.6)
(3.7)
the scalar product
- g(r) dr
(3.8)
and the abbreviation &, =$8(r-b). The expansion c:*=
(3.9) coefficients
(%ku~b I uo).
CT,, in (3.5) are now given by (3.10)
94
R. SCHMITZ AND B.U. FELDERHOF
The reason for the choice of radial dependence in (3.6) will become evident later.
4. Outgoing waves We wish to write the solution of the inhomogeneous equations (2.1) for the flow in the presence of the particle as a spherical expansion of scattered waves u(r) = =E
u0(r)
+
udr)
chaL4r)
+ E
cidiidrh
(r
>
(4.1)
ah
where the outgoing waves uL,(r) tend to zero at infinity so that at large distances the solution u(r) tends to the unperturbed flow uo(r). We require the functions u&(r) to be solutions of the homogeneous equations except at the origin. Such a set of solutions is given by u,s(r)
l-1 = 1(21- 1)2(21+ 1) r+1%,(i),
1 -‘-‘c*,(i), uLT(r) = I(1 + 1)(21+ 1) r nlmr(r) = (21 : 3)2r
-1-l
p&(r) = 0,
(13 21,
p kT(r)
(12 I),
=
0,
1+2 1 AI+L#) -zBl”,m(i’]. (I + 1)(21 + 1) [ 1 -I-2 pi&r)
=
qmr
Y,+,.,V),
(I
2
0).
(4.2) where the subscripts S, T, P indicate the tensor character. Again there is a close relation to Lamb’s solutions. The reason for the choice of normalization factors will become evident later. Like in the previous section we define a set of adjoint functions {wi’A(r)} which satisfy the orthonormality relations (w;msL 1UI’&‘) = 6,~&,d,~
(4.3)
for any radius b. It is easily seen that such a set of functions
is given by
Wlms(r) = (21- I)(21 + 1) #_I21- 3 2 A,-,,,,,(+) + &r,,,,(f) , I-l I [
(I 2 3,
wkT(r)
(I 2 11,
= (21 + l)+&,(f),
w~P(r)=(21+1)(21+3)r1Al+l,m(P) 1+2
,
(I 2 0).
(4.4)
CREEPING
The expansion ck
=
(Wiidb
coefficients 1 VI>
(b
FLOW ABOUT
A SPHERICAL
PARTICLE
95
c;~ are given by >
(4.5)
a).
The solution of the scattering problem will allow us to find the coefficients c;~,, more directly from the coefficients CL,.
5. Antenna theorems It is of interest to consider the flow due to a force density spherical shell of radius b. Since any surface force density expanded in terms of the vector spherical harmonics A,,, sufficient to study the response due to any of these harmonics. article’) by one of us (B.U.F.) the problem was already solved stationary flow. Applying the method outlined in ref. 7, section the zero frequency limit in ref. 7, eq. (6.2), one obtains
localized on a f(0, cp) can be BI,,,, C,,, it is In a previous for the case of 5, or by taking
l-l
To(r - r’) . A&‘)
+ 1) $=z Adi),
dS’ = 4?r(21 _:ii,
r < b,
S(b)
bl+l i-t1 = 49r (21 - 1)(21+ 1) Yr- Alm(i) BI,(+),
I
r>b,
(5.1)
1+1
Tdr - r’) aBdi’)
dS’= 4~ (21 + 1:(21 + 3) 5
Bdf)
S(b) r
+2~$$($-g)A,,,,(9, b (21+ 1)(21 + 3) 7 I
=
I
4fl
r’
1+3
Bdi)9
r> b,
C,,(~),
r < b,
4~ b’+’ = 21+l;;m GmV),
r > b.
To(r - r’) * C,,,,(P) dS’ = &r=r
S(b)
We call these useful identities antenna theorems, since they exhibit the flow caused by a spherical antenna. By linear combinations of the identities (5.1) and by use of (3.6) and (4.2)
96
R. SCHMITZ AND B.U. FELDERHOF
we find To(r - r’) wLs(r’) dS’ = 47rboLp(r),
r > b,
To(r - r’) - wkT(r’) dS’ = 4mbuI,T(r),
r > b,
To(r - r’) wh,(r’) dS’ = 4vbu&s(r),
r > b.
l
S(b) I S(b)
I
l
(5.2)
S(b) Our choice of radial dependence in (3.6) was made to achieve simplicity in the above expressions. The value of the integrals for r < b is more complicated, but will not be needed. Forming a different linear combination of the identities (5.1) and using (3.2) and (4.4) we find To(r - r’) - w&r’)
dS’ = 4rrboLp(r),
r < b,
S(b)
I
To(r - r’) 9 wkT(r’) dS’ = 4mbu&(r),
r < b,
I
To(r - r’) - wLP(r’) dS’ = 4TbuLS(r),
r < b.
(5.3)
S(b)
S(b) The value of the integrals for r > b is more complicated, but will not be needed. Our choice of normalization factors in (4.2) has been made to achieve simplicity in (5.2) and (5.3).
6. Force multipoles The identities (5.2) and (5.3) show which surface force densities are required to excite the elementary incident and outgoing waves. We show in this section that the force density F(r) can be related to a reduced force density Fb(r) which is localized on a spherical shell of radius b and can be decomposed in terms of the set {Wan}. First we define the reduced force multipoles (fimU}by f Imo
=
u;:(r)
l
F(r) dr = (oh, 1F).
(6.1)
CREEPING
FLOW ABOUT A SPHERICAL
91
PARTICLE
It follows from (2.7), (4.1), (4.5) and (5.3) that crms = q-%0,
CinT
=
q-‘f,mT,
CLIP
= 7)-%mS,
(6.2)
so that the amplitudes of the outgoing waves {u&(r)} are simply related to the force multipole strengths cfiW}. It is evident from the definition (6.1) that the force multipoles cfiM} contain only limited information about the force density F(r). The relationship cannot be inverted to reconstruct the force density F(r) from the multipoles cfiW}. This is in contrast to the set of Cartesian multipoles (defined as the moments of the force density with all products of powers of the Cartesian coordinates), which are fully equivalent to the force density. On the other hand it follows from (6.2) that the spherical multipoles Cfrmo}suffice to determine the flow outside the particle. It has been shown earlier by one of us? (RX) in a Cartesian tensor representation how to extract the reduced multipoles, which are relevant for the external flow, from the complete set of Cartesian multipoles. The irreducible parts of the reduced tensors thus obtained are equivalent to the reduced force multipoles CfrmU}. The multipoles cfmur}can be used to construct a force density &(r) which is localized on a spherical shell of radius b and which gives rise to the same flow ul(r) for r > b as the force density F(r). Thus we define
from which f~mcr = (u:m~1Fd,
(6.4)
so that Fb(r) has the same reduced multipoles as F(r). By making b infinitesimally small Fb(r) can be localized at the origin. We call Fb(r) for b C a the reduced force density. The relation between F(r) and Fb(r) can be expressed by means of a reduction operator #-b(t) = j- Rb(r, t’) - F(r’) dr’
(6.5)
with
The reduction operator Rb relocates the force density on the spherical shell of radius b and removes all information which is irrelevant for the external flow. It clearly is a projector: R$ = Rb.
R. SCHMITZ AND B.U. FELDERHOF
98
7. Resistance matrix We have yet to indicate the relation between the force multipoles Cfimtr}and the coefficients {CL,,} for the incident flow. This relation can be found from the explicit solution of the flow problem. In general the induced force density is related to the incident flow velocity by a resistance operator Z(r, r’) F(r) = -j
Z( r, r’)
* [I
-
U - 42 x
r’ dr’. I
(7.1)
For the constitutive equation (2.5) one can express the operator 2 formally in terms of the permeability kernel A according to 2 = A[1 + GA]-‘,
(7.2)
where G is the Green function G = (47~~)-‘1~. The resistance operator is symmetric and rotationally invariant. For simplicity we take the sphere to be fixed and put U = 0, G = 0. Using (3.5) and (6.1) one then finds from (7.1) f&
= -
x Z&no; I’m’o’
l’m’u’)c;&’
(7.3)
with Z@rla;
l’m’a’) = (u:mo]Z]e;t;m’~‘) = -(uL
(7.4)
1Fr,kTh
where Flmo(r) = -
I
Z(r, r’) - uL,,(r’) dr’.
(7.5)
In shorthand
notation (7.3) reads + f = -&c .
(7.6)
We shall call Z, the resistance matrix. The subscript indicates that the matrix & is a reduced version of the resistance operator 2. From the definition it follows that the matrix 2, is symmetric. Writing the relation (6.2) in the form c-=Go.f we find for the relation waves c- = -GO&c + = xc+.
(7.7) between
the amplitudes
of incident
and outgoing (7.8
The elements of the matrix Go are
(7.9)
CREEPING FLOW ABOUT A SPHERICAL PARTICLE
99
and from (5.2) and (5.3) they are explicitly given by (7.10)
G&ma; I’m’s’) = + 6,,&,,~
in agreement with (6.2) and (7.7). In order to find the resistance matrix Z, we must use the solution of the flow problem which is known from previous work’.?. We consider separately the three incident flows ohs, uk T, ok p. For the incident flow o&, s(r) the perturbed flow outside the particle is given by ul(r.
lrn
s)
=
,
_
2(1+ 1)(2f+ ‘It21+ 3, ASo-1 1+2
(r)
ImP
(r > a),
(7.11)
where As and Bs are scattering coefficients characteristic the incident flow u&r(r) the perturbed flow is
for the particle. For
- (2I+ 3)2(2i + 5)B?~1+~,~s(r),
r > a,
ul(r; Im T) = -I(Z + 1)(21+ l)ATo,,(r), and for the incident flow ol’,dr) ur(r; Im P) = -(21-
(7.12)
it is
3)(21- 1)2Aro&,,P(r)
-
‘c21- 1)3(21 + l) BPo-
I
2(1- 1)
(r)
ImS
r > a.
9
(7.13)
Using (6.2) and (7.3) one can now read off the elements of the resistance matrix. Since the matrix must be symmetric one obtains the relation (7.14)
(21 - 3)A: = (21+ 1)B?e2, which was found earlier by other meansr”‘” ). Explicitly the resistance reads l’m’u’) = T$,,,
Z&ma; 2(1 + ‘)ty;;)(21
K
matrix
+ 3, Ass,,,
0 (2f- 3)(21- 1)2A:6L,,+2
0
(21+ 1)(21+ 3)2A:+&,‘-2
I(I + 1)(2Z+ l)A:& 0
i
.
0 1(21- 1)3(21+ l) B:S”, 2(1- 1) i (7.15)
For a hard sphere with mixed slip-stick boundary coefficients are given by’)
conditions
the scattering
R. SCHMITZ
100
AND B.U. FELDERHOF
AT
’ AP,21+1 I
2
=
1 - (I + 2)t l+(l-l)5o
l-36 *I-‘, BT = ?!$ 1+ 2(1- 2)5 a
zl+l
’
(7.16)
1 +:;I 32)5 a21+‘_
The coefficients AZ, A:, BE, A; and By vanish by definition. For a uniform permeable sphere of radius a the coefficients are given in ref. 10, eq. (4.3). Combining eqs. (7.10) and (7.15) one also finds the matrix X defined in (7.8). The structure of the resistance matrix (7.15) is remarkably simple. Each multipole fimD is determined by only one or two coefficients c;‘,,. For the multipoles foms, flmT and f,,,,s this statement is equivalent with Fax&r’s theorems for the force, the torque and the force dipole moment, respectively. In the appendix we relate these spherical multipoles to their Cartesian counterparts.
8. Discussion We have studied the solution of the linear Navier-Stokes equations for steady flow in the region outside a spherical particle. The amplitudes of the outgoing waves are related to the spherical force multipoles by the very simple relations (6.2). The force multipoles can in turn be expressed in terms of the amplitudes of the incident waves via the resistance matrix. That matrix is given explicitly by (7.15) in terms of the scattering coefficients for the particle. The advantage of the present formulation is that it concentrates on scattered fields. In a fully Cartesian framework much of the information contained in the Cartesian multipoles is superfluous for the calculation of the outside flow?. We have shown that the force density induced in the particle can be effectively replaced by a reduced density localized on a spherical shell. This replacement is based on the antenna theorems (5.2) and (5.3). In a following article we shall show that the present formalism can be used with advantage in the problem of hydrodynamic interactions between two spherical particles.
Appendix
We express the force multipoles f o,,,s, fl,,, T and f,,,, s in terms of the Cartesian components of the force, the torque and the force dipole moment. One has
(A. 1)
CREEPING FLOW ABOUT A SPHERICAL PARTICLE
101
where the e, are the spherical unit vectors’) eo=e,,
-&(c,+ie,),
el=
em,=--; l ( e,-ie,). V/2
From (3.2) and (6.1) one therefore fO,,,s= dget;.jF(r)dr=
(A.3
finds dge$,*S,
where % is the total force exerted -i x A,, one finds
on the fluid. Using the relation
(A-3) Cl,,, =
(A.4) where r is the total torque exerted on the fluid. Using the explicit expressions for the spherical harmonics Yz,(i) one finds by straightforward calculation
f12~- f~.-2.~ = -i ~IIS+ fl, -1.~
=
d15 I G
WY + YF,) dr,
idg1WY +
YE)
dr,
(A.3
OHS-fl,-LS=-~~f(zF,+xF,)dr, xF, + yF, - 22F,] dr. In addition one has the relation
I
r
l
F(r) dr = 0,
(A.6)
which follows from a Faxen theorem and the condition of incompressibility?. Hence, all components of the symmetric force dipole moment can be determined from the spherical multipoles flmS. References 1) 2) 3) 4)
R. Schmitz and B.U. Felderhof, Physica !UA (1978) 423. B.U. Felderhof and R.B. Jones, Physica 93A (1978) 457. H. Lamb, Hydrodynamics (Dover, New York, 1945) p. 595,632. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff. Leiden, 1973) p. 62.
102 5) 6) 7) 8)
R. SCHMITZ AND B.U. FELDERHOF
R.G. Cox and H. Brenner, J. Fluid Mech. 28 (1%7) 391; Chem. Eng. Sci. 26 (1971) 65. P. Mazur and D. Bedeaux, Physica 76 (1974) 235. B.U. Felderhof, Physica 84A (1976) 557. J.-M. Normand, A Lie Group: Rotations in Quantum Mechanics (North-Holland, Amsterdam, 1980). 9) R. Schmitz, Physica 102A (1980) 161. 10) P. Reuland, B.U. Felderhof and R.B. Jones, Physica 93A (1978) 465. 11) R. Schmitz, Dissertation R.W.T.H. Aachen 1981.
CREEPING FLOW ABOUT A SPHERICAL PARTICLE R. SCHMITZ and B.U. FELDERHOF Institut fiir Theoretische Physik A, RWTH Aachen, SommerfeldstraBe, FR Germany
5100 Aachen,
Received 14 December 1981
We formulate the solution of the linear Navier-Stokes equations for steady, incompressible flow about a spherical particle as an expansion of scattered waves. The amplitudes of the outgoing waves are simply related to a set of spherical force multipoles. These multipoles are found from the amplitudes of the incident waves with the aid of a resistance matrix, which is expressed in terms of the scattering coefficients of the particle.
1. Introduction In previous work we have studied the solution of the Navier-Stokes equations for steady, incompressible flow in the presence of a hard sphere with mixed slip-stick boundary conditions’) and for a permeable spherical particle2) for arbitrary incident flow. In the space outside the particle our solution is closely related to that of Lamb3”). We showed that the perturbed flow involves scattering coefficients which are characteristic for the particle. In this article we analyze the situation more systematically as a problem in scattering theory. We introduce complete sets of incident and outgoing waves in terms of vector spherical harmonics. The expansion coefficients are found with the aid of two adjoint sets which satisfy orthonormality relations on the surface of a sphere. We derive some antenna theorems which show that the amplitudes of the outgoing waves are very simply related to a set of spherical force multipoles. The induced force density can be replaced by a force density which is localized on a spherical shell and gives rise to the same flow outside the particle. This reduced force density can be constructed from the spherical force multipoles. These multipoles are related to the amplitudes of the incident waves by means of a resistance matrix, which is expressed in terms of the scattering coefficients. The present concise formulation should also be useful in dealing with flows involving many particles. In a following article we shall consider the scattering problem for two spherical particles. 0378-437 1/82/ooo(Mooo /$02.75 @ 1982 North-Holland
CREEPING
FLOW ABOUT A SPHERICAL
PARTICLE
91
2. Basic equations We consider steady flow of an incompressible fluid in the presence of a spherically symmetric particle of radius a. The particle may be a hard sphere with mixed slip-stick boundary conditions, or it may be penetrable to the flow with a radially symmetric permeability. In the case of a hard sphere it is convenient to extend the validity of the flow equations to all space and to represent the boundary conditions by a suitably chosen induced force density5.6). The linearized Navier-Stokes equations for the flow velocity u(r) read qV*u - Vp = - F(r),
V. u = 0,
(2.1)
where q is the shear viscosity, p(r) is the pressure which is determined by the condition of incompressibility, and F(r) is the induced force density acting on the fluid. For a hard sphere centered at the origin the last of these can be chosen as’) F(r) = f(0, cp)S(r - a) + 5a.f~(0,p)S’(r - a),
(2.2)
where the surface force density f(0, cp) is determined by the incident Aow, 5 is a slip coefficient taking the value 5 = 0 for stick and 5 = f for pure slip boundary conditions, and ft is the part of f tangential to the sphere. The solution of (2.1) for r < a is given by u(r) = u(r), where u(r) = U + 0 X r
(2.3)
is the instantaneous solid body motion of the sphere. spherical particle the induced force density is given by F(r) = -Ur)[n(r)
-
For a penetrable
u(r)l,
(2.4)
where [h(r)]-’ is the permeability of the particle. In this case the flow velocity u(r) for r < a is more complicated. More in general we could consider a constitutive equation F(r) = -1 A( r, r’) *[u(r’) - u(r’)] dr’ with the conditions that the kernel symmetric and rotationally invariant: n&r,
r’) = A&r’, r),
A(r, r’) vanishes
(2.5) for r, r’ > a and is
A(r, r’) = D - A(D-’ - r, D-’ - r’) . D-‘,
where D is a rotation matrix.
(2.6)
92
R. SCHMITZ AND B.U. FELDERHOF
The formal solution of (2.1) is given by To(r - r’) - F(r’) dr’,
u(r) = uO(r) + -&-I
(2.7) p(r)=po(r)+&jQ(r-r’)*F(r’)dr’ with the Oseen kernels 1 To(r) = ~(1+
*
*1 rr),
Q(r) = j,
(2.8)
where i = r/r, and where (uo(r), PO(r)) is an arbitrary equations.
solution of the free flow
3. Free flow solutions We consider solutions of the homogeneous ?jv2urj-vp,=o,
equations
v*uo=o.
(3.1)
A complete set of solutions appropriate to spherical symmetry has been presented by Lamb3*‘). In previous articles’*‘) we have rederived Lamb’s solutions in a scheme better suited to the solution of the inhomogeneous equations. In this scheme a complete set of solutions of (3.1) is given by u:,,(r)
= r’Al+l,,,,(f),
o&r(r) = r’C,,(+),
pks(r)
= 0,
(I 2 O),
p h,(r)
= 0,
(1 2 l),
pl m p(r)
=
7)
lc21 - ‘X21+ ‘) r’-‘Y,_,,m(i) l-1
,
(I B 2). (3.2)
Each of the velocity fields is homogeneous of degree I in the Cartesian coordinates. The vector spherical harmonics are defined by’) AI, = lYr,e,+$$e,+&$$e,, Br, = -(1+ 1)Y,me,+$$e,9+&$$e,, 1 au,, Cl, =--ee--e sin 8 acp
(3.3)
aYl,
a0 q'
where Y1,(8, cp) are the spherical harmonics
and e,, e& e, are unit vectors in
CREEPING FLOW ABOUT A SPHERICAL PARTICLE
93
spherical coordinates. The above vector harmonics are related to the vector spherical harmonics Y 2” (with c = i) defined by Normands), according to B,, = V/(1 + 1)(21 + 1) Y In+?
A,,,, = d/1(21 + 1) Y ;‘*‘, C,, = -id&l + 1) Yf;f.
(3.4)
The vector spherical harmonics Y,“[‘I form a complete orthonoimal set for vector functions on the unit sphere. For our purposes it is more convenient to use the harmonics A,,,,, I#,,,,, Cl,,, which are not normalized to unity, so as to avoid unpleasant square root factors in the formulae. We shall denote the set of solutions (3.2) by {u&&r)}, where the discrete index u indicates the tensor character of the solution and takes the values S (symmetric), T (tangential), P (pressure). We have previously’) discussed the relation to Lamb’s solutions. In the following the pressure field can be left out of consideration. Any solution q,(r) of (3.1) can be expanded as
00(r)=
z.cik,uik&>= 2
ciL,luhJ.
(3.5)
In order to find the expansion coefficients c& it is convenient to define an adjoint set of functions {wTW(r)}. We require this set of functions to be orthonormal to the set {uk,} on a sphere of arbitrary radius b. It is easily seen that such a set of functions is given by
d&r> =
(l
uhl+
+
3)
1
r
-1-l
[ AI+I,,(+)-
21+5 2
BI+l,, (*)I, r
(12 01,
&&r)
- -l(1
1) r+‘C,,(i),
(I 3 11,
wL(r)
1 = -1(21_ 1) r-‘-‘B,_,, ,,,(i),
(I 2 2).
These functions
+
satisfy the orthonormality
relations
(WE&% 1 ~Ltlw) = &~mm’~m~, where we have introduced cf 1t.9 = j f*(r)
(3.6)
(3.7)
the scalar product
- g(r) dr
(3.8)
and the abbreviation &, =$8(r-b). The expansion c:*=
(3.9) coefficients
(%ku~b I uo).
CT,, in (3.5) are now given by (3.10)
94
R. SCHMITZ AND B.U. FELDERHOF
The reason for the choice of radial dependence in (3.6) will become evident later.
4. Outgoing waves We wish to write the solution of the inhomogeneous equations (2.1) for the flow in the presence of the particle as a spherical expansion of scattered waves u(r) = =E
u0(r)
+
udr)
chaL4r)
+ E
cidiidrh
(r
>
(4.1)
ah
where the outgoing waves uL,(r) tend to zero at infinity so that at large distances the solution u(r) tends to the unperturbed flow uo(r). We require the functions u&(r) to be solutions of the homogeneous equations except at the origin. Such a set of solutions is given by u,s(r)
l-1 = 1(21- 1)2(21+ 1) r+1%,(i),
1 -‘-‘c*,(i), uLT(r) = I(1 + 1)(21+ 1) r nlmr(r) = (21 : 3)2r
-1-l
p&(r) = 0,
(13 21,
p kT(r)
(12 I),
=
0,
1+2 1 AI+L#) -zBl”,m(i’]. (I + 1)(21 + 1) [ 1 -I-2 pi&r)
=
qmr
Y,+,.,V),
(I
2
0).
(4.2) where the subscripts S, T, P indicate the tensor character. Again there is a close relation to Lamb’s solutions. The reason for the choice of normalization factors will become evident later. Like in the previous section we define a set of adjoint functions {wi’A(r)} which satisfy the orthonormality relations (w;msL 1UI’&‘) = 6,~&,d,~
(4.3)
for any radius b. It is easily seen that such a set of functions
is given by
Wlms(r) = (21- I)(21 + 1) #_I21- 3 2 A,-,,,,,(+) + &r,,,,(f) , I-l I [
(I 2 3,
wkT(r)
(I 2 11,
= (21 + l)+&,(f),
w~P(r)=(21+1)(21+3)r1Al+l,m(P) 1+2
,
(I 2 0).
(4.4)
CREEPING
The expansion ck
=
(Wiidb
coefficients 1 VI>
(b
FLOW ABOUT
A SPHERICAL
PARTICLE
95
c;~ are given by >
(4.5)
a).
The solution of the scattering problem will allow us to find the coefficients c;~,, more directly from the coefficients CL,.
5. Antenna theorems It is of interest to consider the flow due to a force density spherical shell of radius b. Since any surface force density expanded in terms of the vector spherical harmonics A,,, sufficient to study the response due to any of these harmonics. article’) by one of us (B.U.F.) the problem was already solved stationary flow. Applying the method outlined in ref. 7, section the zero frequency limit in ref. 7, eq. (6.2), one obtains
localized on a f(0, cp) can be BI,,,, C,,, it is In a previous for the case of 5, or by taking
l-l
To(r - r’) . A&‘)
+ 1) $=z Adi),
dS’ = 4?r(21 _:ii,
r < b,
S(b)
bl+l i-t1 = 49r (21 - 1)(21+ 1) Yr- Alm(i) BI,(+),
I
r>b,
(5.1)
1+1
Tdr - r’) aBdi’)
dS’= 4~ (21 + 1:(21 + 3) 5
Bdf)
S(b) r
+2~$$($-g)A,,,,(9, b (21+ 1)(21 + 3) 7 I
=
I
4fl
r’
1+3
Bdi)9
r> b,
C,,(~),
r < b,
4~ b’+’ = 21+l;;m GmV),
r > b.
To(r - r’) * C,,,,(P) dS’ = &r=r
S(b)
We call these useful identities antenna theorems, since they exhibit the flow caused by a spherical antenna. By linear combinations of the identities (5.1) and by use of (3.6) and (4.2)
96
R. SCHMITZ AND B.U. FELDERHOF
we find To(r - r’) wLs(r’) dS’ = 47rboLp(r),
r > b,
To(r - r’) - wkT(r’) dS’ = 4mbuI,T(r),
r > b,
To(r - r’) wh,(r’) dS’ = 4vbu&s(r),
r > b.
l
S(b) I S(b)
I
l
(5.2)
S(b) Our choice of radial dependence in (3.6) was made to achieve simplicity in the above expressions. The value of the integrals for r < b is more complicated, but will not be needed. Forming a different linear combination of the identities (5.1) and using (3.2) and (4.4) we find To(r - r’) - w&r’)
dS’ = 4rrboLp(r),
r < b,
S(b)
I
To(r - r’) 9 wkT(r’) dS’ = 4mbu&(r),
r < b,
I
To(r - r’) - wLP(r’) dS’ = 4TbuLS(r),
r < b.
(5.3)
S(b)
S(b) The value of the integrals for r > b is more complicated, but will not be needed. Our choice of normalization factors in (4.2) has been made to achieve simplicity in (5.2) and (5.3).
6. Force multipoles The identities (5.2) and (5.3) show which surface force densities are required to excite the elementary incident and outgoing waves. We show in this section that the force density F(r) can be related to a reduced force density Fb(r) which is localized on a spherical shell of radius b and can be decomposed in terms of the set {Wan}. First we define the reduced force multipoles (fimU}by f Imo
=
u;:(r)
l
F(r) dr = (oh, 1F).
(6.1)
CREEPING
FLOW ABOUT A SPHERICAL
91
PARTICLE
It follows from (2.7), (4.1), (4.5) and (5.3) that crms = q-%0,
CinT
=
q-‘f,mT,
CLIP
= 7)-%mS,
(6.2)
so that the amplitudes of the outgoing waves {u&(r)} are simply related to the force multipole strengths cfiW}. It is evident from the definition (6.1) that the force multipoles cfiM} contain only limited information about the force density F(r). The relationship cannot be inverted to reconstruct the force density F(r) from the multipoles cfiW}. This is in contrast to the set of Cartesian multipoles (defined as the moments of the force density with all products of powers of the Cartesian coordinates), which are fully equivalent to the force density. On the other hand it follows from (6.2) that the spherical multipoles Cfrmo}suffice to determine the flow outside the particle. It has been shown earlier by one of us? (RX) in a Cartesian tensor representation how to extract the reduced multipoles, which are relevant for the external flow, from the complete set of Cartesian multipoles. The irreducible parts of the reduced tensors thus obtained are equivalent to the reduced force multipoles CfrmU}. The multipoles cfmur}can be used to construct a force density &(r) which is localized on a spherical shell of radius b and which gives rise to the same flow ul(r) for r > b as the force density F(r). Thus we define
from which f~mcr = (u:m~1Fd,
(6.4)
so that Fb(r) has the same reduced multipoles as F(r). By making b infinitesimally small Fb(r) can be localized at the origin. We call Fb(r) for b C a the reduced force density. The relation between F(r) and Fb(r) can be expressed by means of a reduction operator #-b(t) = j- Rb(r, t’) - F(r’) dr’
(6.5)
with
The reduction operator Rb relocates the force density on the spherical shell of radius b and removes all information which is irrelevant for the external flow. It clearly is a projector: R$ = Rb.
R. SCHMITZ AND B.U. FELDERHOF
98
7. Resistance matrix We have yet to indicate the relation between the force multipoles Cfimtr}and the coefficients {CL,,} for the incident flow. This relation can be found from the explicit solution of the flow problem. In general the induced force density is related to the incident flow velocity by a resistance operator Z(r, r’) F(r) = -j
Z( r, r’)
* [I
-
U - 42 x
r’ dr’. I
(7.1)
For the constitutive equation (2.5) one can express the operator 2 formally in terms of the permeability kernel A according to 2 = A[1 + GA]-‘,
(7.2)
where G is the Green function G = (47~~)-‘1~. The resistance operator is symmetric and rotationally invariant. For simplicity we take the sphere to be fixed and put U = 0, G = 0. Using (3.5) and (6.1) one then finds from (7.1) f&
= -
x Z&no; I’m’o’
l’m’u’)c;&’
(7.3)
with Z@rla;
l’m’a’) = (u:mo]Z]e;t;m’~‘) = -(uL
(7.4)
1Fr,kTh
where Flmo(r) = -
I
Z(r, r’) - uL,,(r’) dr’.
(7.5)
In shorthand
notation (7.3) reads + f = -&c .
(7.6)
We shall call Z, the resistance matrix. The subscript indicates that the matrix & is a reduced version of the resistance operator 2. From the definition it follows that the matrix 2, is symmetric. Writing the relation (6.2) in the form c-=Go.f we find for the relation waves c- = -GO&c + = xc+.
(7.7) between
the amplitudes
of incident
and outgoing (7.8
The elements of the matrix Go are
(7.9)
CREEPING FLOW ABOUT A SPHERICAL PARTICLE
99
and from (5.2) and (5.3) they are explicitly given by (7.10)
G&ma; I’m’s’) = + 6,,&,,~
in agreement with (6.2) and (7.7). In order to find the resistance matrix Z, we must use the solution of the flow problem which is known from previous work’.?. We consider separately the three incident flows ohs, uk T, ok p. For the incident flow o&, s(r) the perturbed flow outside the particle is given by ul(r.
lrn
s)
=
,
_
2(1+ 1)(2f+ ‘It21+ 3, ASo-1 1+2
(r)
ImP
(r > a),
(7.11)
where As and Bs are scattering coefficients characteristic the incident flow u&r(r) the perturbed flow is
for the particle. For
- (2I+ 3)2(2i + 5)B?~1+~,~s(r),
r > a,
ul(r; Im T) = -I(Z + 1)(21+ l)ATo,,(r), and for the incident flow ol’,dr) ur(r; Im P) = -(21-
(7.12)
it is
3)(21- 1)2Aro&,,P(r)
-
‘c21- 1)3(21 + l) BPo-
I
2(1- 1)
(r)
ImS
r > a.
9
(7.13)
Using (6.2) and (7.3) one can now read off the elements of the resistance matrix. Since the matrix must be symmetric one obtains the relation (7.14)
(21 - 3)A: = (21+ 1)B?e2, which was found earlier by other meansr”‘” ). Explicitly the resistance reads l’m’u’) = T$,,,
Z&ma; 2(1 + ‘)ty;;)(21
K
matrix
+ 3, Ass,,,
0 (2f- 3)(21- 1)2A:6L,,+2
0
(21+ 1)(21+ 3)2A:+&,‘-2
I(I + 1)(2Z+ l)A:& 0
i
.
0 1(21- 1)3(21+ l) B:S”, 2(1- 1) i (7.15)
For a hard sphere with mixed slip-stick boundary coefficients are given by’)
conditions
the scattering
R. SCHMITZ
100
AND B.U. FELDERHOF
AT
’ AP,21+1 I
2
=
1 - (I + 2)t l+(l-l)5o
l-36 *I-‘, BT = ?!$ 1+ 2(1- 2)5 a
zl+l
’
(7.16)
1 +:;I 32)5 a21+‘_
The coefficients AZ, A:, BE, A; and By vanish by definition. For a uniform permeable sphere of radius a the coefficients are given in ref. 10, eq. (4.3). Combining eqs. (7.10) and (7.15) one also finds the matrix X defined in (7.8). The structure of the resistance matrix (7.15) is remarkably simple. Each multipole fimD is determined by only one or two coefficients c;‘,,. For the multipoles foms, flmT and f,,,,s this statement is equivalent with Fax&r’s theorems for the force, the torque and the force dipole moment, respectively. In the appendix we relate these spherical multipoles to their Cartesian counterparts.
8. Discussion We have studied the solution of the linear Navier-Stokes equations for steady flow in the region outside a spherical particle. The amplitudes of the outgoing waves are related to the spherical force multipoles by the very simple relations (6.2). The force multipoles can in turn be expressed in terms of the amplitudes of the incident waves via the resistance matrix. That matrix is given explicitly by (7.15) in terms of the scattering coefficients for the particle. The advantage of the present formulation is that it concentrates on scattered fields. In a fully Cartesian framework much of the information contained in the Cartesian multipoles is superfluous for the calculation of the outside flow?. We have shown that the force density induced in the particle can be effectively replaced by a reduced density localized on a spherical shell. This replacement is based on the antenna theorems (5.2) and (5.3). In a following article we shall show that the present formalism can be used with advantage in the problem of hydrodynamic interactions between two spherical particles.
Appendix
We express the force multipoles f o,,,s, fl,,, T and f,,,, s in terms of the Cartesian components of the force, the torque and the force dipole moment. One has
(A. 1)
CREEPING FLOW ABOUT A SPHERICAL PARTICLE
101
where the e, are the spherical unit vectors’) eo=e,,
-&(c,+ie,),
el=
em,=--; l ( e,-ie,). V/2
From (3.2) and (6.1) one therefore fO,,,s= dget;.jF(r)dr=
(A.3
finds dge$,*S,
where % is the total force exerted -i x A,, one finds
on the fluid. Using the relation
(A-3) Cl,,, =
(A.4) where r is the total torque exerted on the fluid. Using the explicit expressions for the spherical harmonics Yz,(i) one finds by straightforward calculation
f12~- f~.-2.~ = -i ~IIS+ fl, -1.~
=
d15 I G
WY + YF,) dr,
idg1WY +
YE)
dr,
(A.3
OHS-fl,-LS=-~~f(zF,+xF,)dr, xF, + yF, - 22F,] dr. In addition one has the relation
I
r
l
F(r) dr = 0,
(A.6)
which follows from a Faxen theorem and the condition of incompressibility?. Hence, all components of the symmetric force dipole moment can be determined from the spherical multipoles flmS. References 1) 2) 3) 4)
R. Schmitz and B.U. Felderhof, Physica !UA (1978) 423. B.U. Felderhof and R.B. Jones, Physica 93A (1978) 457. H. Lamb, Hydrodynamics (Dover, New York, 1945) p. 595,632. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff. Leiden, 1973) p. 62.
102 5) 6) 7) 8)
R. SCHMITZ AND B.U. FELDERHOF
R.G. Cox and H. Brenner, J. Fluid Mech. 28 (1%7) 391; Chem. Eng. Sci. 26 (1971) 65. P. Mazur and D. Bedeaux, Physica 76 (1974) 235. B.U. Felderhof, Physica 84A (1976) 557. J.-M. Normand, A Lie Group: Rotations in Quantum Mechanics (North-Holland, Amsterdam, 1980). 9) R. Schmitz, Physica 102A (1980) 161. 10) P. Reuland, B.U. Felderhof and R.B. Jones, Physica 93A (1978) 465. 11) R. Schmitz, Dissertation R.W.T.H. Aachen 1981.