The “Stokeslet”: A Fundamental Solution for the Creeping-Flow Equations

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Creeping Flows – Three-Dimensional Problems



This method is particularly simple to apply in the case of a point force f. The (disturbance)

pressure must be a decaying harmonic function that is linear in f and a true scalar. Hence,

p = c1



f·x

.

r3



(8–89)



The (disturbance) velocity is thus

u=



c1 f · x

r3



x

2



+ uH ,



(8–90)



where uH is harmonic, decaying, linear in f and a true vector, that is,

f

·f+β ,

r



xx

I

− 3

5

r

3r



uH = α



(8–91)



where α and β are coefficients that must still be determined.

If we apply the continuity equation, (8–88a), we find that

c1

=β

2

so that

u=



c1 f

x(f · x)

+α

+

2 r

r3



xx

I

− 3

5

r

3r



· f.



(8–92)



To this point, the solution is essentially identical to that obtained for uniform flow past a

sphere in Section A. Beyond this, however, the two differ. To satisfy boundary conditions

at the surface of a sphere, a nonzero value of α is necessary. However, for a point force,

α = 0. The simplest way to see this is to note that c1 is dimensionless, whereas α must

have dimensions of (length) 2 . But a point force has no characteristic length scale, so α = 0.

Thus,

u=



c1 f

x(f · x)

.

+

2 r

r3



(8–93)



To determine the scalar constant c1 , we make use of an integral relationship between the

force f and the resultant force exerted on the fluid outside a control surface S.

First, consider the fluid region V between a body of arbitrary geometry ∂ D and control surface S of arbitrary shape that completely encloses it. Because the creeping-motion

approximation is being used,

∇ ·T=0



(8–94)



everywhere in the fluid region V, and thus

∇ · Td V = 0.



(8–95)



V



Thus, by applying the divergence theorem to (8–95), we see that

∂D



T · nd A =



T · ns d A.



(8–96)



S



Here the integral on the left-hand side is taken over the surface of the body ∂D and n is

the outer normal from the body into the fluid. The integral on the right-hand side is taken

over the control surface S with outer normal nS . However, the integral on the left-hand side

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D. Fundamental Solutions of the Creeping-Flow Equations



is just the hydrodynamic force acting on the body (that is, the negative of the force acting

from the body onto the fluid). It thus follows that

F=



T · ns d A.



(8–97)



S



In the present case, we have a point force at the origin, and the same expression holds but

with −f replacing F.

To obtain cl , we apply (8–97) to the solution (8–89) and (8–93), with S chosen for

convenience as a sphere centered at the origin. The surface stress, evaluated with (8–89)

and (8–93), is

T · n = − pI + ∇u + ∇uT

xx · f

= −3c1 4 .

r



·n

(8–98)



Hence, integrating over the spherical surface S (arbitrary radius), we obtain from (8–97)

f = 4π c1 f.

Hence,

c1 =



1

4π



(8–99)



(8–100)



so that

u=



x(f · x)

f

,

+

r

r3

1 f·x

p=

.

4π

r3



1

8π



(8–101)

(8–102)



The associated expression for the stress tensor T is

T=−



3 xx(x · f)

.

4π

r5



(8–103)



The solutions (8–100) and (8–102) are called the stokeslet solution, apparently a name

coined by Hancock.10 It is customary to choose

f = 8π αδ(x)



(8–104)



so that the point-force singularity is characterized in magnitude and direction by α.

We shall see that the stokeslet solution plays a fundamental role in creeping flow theory.

We have already seen in Section E of Chap. 7 that it describes the disturbance velocity

far away from a body of any shape that exerts a nonzero force on an unbounded fluid.

Indeed, when nondimensionalized and expressed in spherical coordinates, it is identical

to the velocity field, (7–151). In the next section we use the stokeslet solution to derive a

general integral representation for solutions of the creeping-flow equations.

2. An Integral Representation for Solutions of the Creeping-Flow Equations

that is due to Ladyzhenskaya

To obtain a general integral representation for solutions of the creeping-flow equations, it

is necessary first to derive a general integral theorem reminiscent of the Green’s theorem

from vector calculus.

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Creeping Flows – Three-Dimensional Problems



Let us then consider the space outside a closed surface ∂D, and let u and u be any smooth

ˆ

vector fields (such as velocity fields) in this exterior domain that satisfy the conditions

∇ · u = 0, ∇ · u = 0

ˆ



(8–105)



(such vector fields are called solenoidal) and vanish at infinity as r−1 or faster. In addition

to u and u, we also define the two tensor (stress) functions

ˆ

T ≡ − pI + (∇u + ∇uT ),



(8–106)



ˆ

ˆ

T ≡ − pI + (∇ u + ∇ uT ),

ˆ

ˆ



(8–107)



ˆ

where p and p are any smooth scalar fields that vanish as r−2 or faster. Then a general vector

identity between these functions is

ˆ

ˆ

ˆ

u · ∇ · T − u · ∇ · T = ∇ · (u · T − u · T) + (∇u : T − ∇ u : T).

ˆ

ˆ

ˆ



(8–108)



However, if we utilize (8–106) and (8–107), then it can be shown that the last term in (8–108)

is identically equal to zero. Hence, integrating over V, we obtain

ˆ

[u · (∇ · T) − u · (∇ · T)]d V =

ˆ

V



ˆ

∇ · [u · T − u · T]d V,

ˆ



(8–109)



V



and applying the divergence theorem, we obtain

ˆ

[u · (∇ · T) − u · (∇ · T)]d V =

ˆ

V



ˆ

[u · T − u · T] · nd A −

ˆ

S



∂D



ˆ

[u · T − u · T] · nd A,

ˆ

(8–110)



where S is any surface enclosing ∂ D. Finally, if we let the surface S → ∞, we see that the

integral over S is O (r−l ) and thus vanishes. It follows that

ˆ

[u · (∇ · T) − u · (∇ · T)]d V = −

ˆ

V



∂D



ˆ

n · [u · T − u · T]d A,

ˆ



(8–111)



where V is now the whole space exterior to ∂ D.

The integral theorem (8–111), leads directly to a general integral representation for

solutions of the creeping-flow equations. To see this, let u,T represent a solution of the

creeping-flow equations – arbitrary except that they must be O(r−l ) and O(r−2 ), respectively,

for r → ∞, as we assumed in deriving (8–111). Further, let u, T be the fundamental

ˆ ˆ

solution of the creeping-flow equations for a point force at a point ξ, that is, referring to

(8–100),

u=

ˆ



1

8π



I

(x − ξ)(x − ξ)

+

· e,

3

R

R



(8–112)



where the point force is assumed to be of unit magnitude f = e, and R ≡ |x − ξ| is the

distance between a point x and the point of application of the force ξ. The corresponding

ˆ

stress T is

3 (x − ξ)(x − ξ)(x − ξ)

ˆ

T=−

· e.

4π

R5

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(8–113)



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D. Fundamental Solutions of the Creeping-Flow Equations



Now, substituting (8–112) and (8–113) into (8–111) and noting that

∇ · T ≡ 0,

ˆ

∇ · T = δ(x − ξ)e,



(8–114a)

(8–114b)



we obtain



V



[−δ(x − ξ)u · e]d Vξ = −



∂D



ˆ

n · [(U · e) · T − u · ( ˆ · e)]d Aξ ,



(8–115)



where we have introduced the shorthand notation

ˆ

u = U · e,

ˆ



ˆ

T= ˆ ·e



(8–116)



in place of (8–112) and (8–113). The symbols d Vξ and d Aξ indicate that the variable of

ˆ

integration is ξ , and the fixed point is thus x. The tensor component of U (i.e., U ik ) is the i

component of the velocity field generated by a point force of unit magnitude in the k direction.

Similarly, the component i jk is the ij component of the stress tensor corresponding to the

flow produced by a point force in the k direction.

To simplify (8–115), we can factor out the constant unit vector e from all terms and use

the integral property of the delta function to evaluate the term on the left-hand side. The

ˆ

result, after substituting for U and ˆ from (8–112), (8–113), and (8–116), is

u(x) = −



3

4π



+



1

8π



∂D



∂D



(x − ξ)(x − ξ)(x − ξ)

· u(ξ) · nd Aξ

R5

I

(x − ξ)(x − ξ)

· T(ξ) · nd Aξ .

+

R

R3



(8–117)



The corresponding form for the pressure is

p(x) =



1

4π



(x − ξ)

· T(ξ) · nd Aξ

R3



∂D



1

+

2π



∂D



I

3(x − ξ)(x − ξ)

· u(ξ) · nd Aξ .

−

R

R3



(8–118)



This is the famous integral representation for the solution of the creeping-flow equations

that is usually attributed to Ladyzhenskaya.11 Because the derivation requires u ∼ r−1 for

large r, we recognize that u must be interpreted as the disturbance velocity field if we wish

to apply (8–117) to a problem that involves an undisturbed velocity field u∞ (x) (which is

a solution of the creeping-flow equations) at large distances from the body (or boundary)

that is denoted as ∂ D. To apply (8–117) directly to the actual velocity field, we let u =

u in (8–117), where u = u – u∞ ; u is now the true velocity, and u∞ is the undisturbed

velocity. This gives

u(x) = u∞ (x) −

1

+

8π



3

4π



∂D



∂D



(x − ξ)(x − ξ)(x − ξ)

· u · nd Aξ

R5



I

(x − ξ)(x − ξ)

· T · nd Aξ .

+

R

R3



(8–119)



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Creeping Flows – Three-Dimensional Problems



The term

∂D



n · [u · T∞ − u∞ · ˆ ]d Aξ

ˆ



is identically equal to zero, as the reader may wish to verify.

The formula (8–117) [or (8–119)] provides a formal solution of the creeping-motion

equations in a compact form. The first integral on the right-hand side is denoted as the

double-layer potential and has a density function that is just the velocity u on the boundaries

∂ D of the flow domain. The second integral on the right-hand side is termed the single-layer

potential, and its density function is the surface-stress vector t = T · n. Of course, (8–117)

and (8–119) do not provide a solution for any specific problem until the density functions

u and T · n are specified on ∂ D. In fact, all that we really have done is to obtain an integral

formula for u that is equivalent to the differential form of the creeping-flow equation,

(7–6). To obtain a solution for any particular problem, we must determine the density

functions so that the velocity field u satisfies the boundary conditions on ∂ D. In general,

this requires numerical solution of the integral equations that result from applying boundary

conditions to (8–117) or (8–119). In fact, this is the essence of the so-called boundaryintegral method for solution of creeping-flow equations; this technique has been used widely

in research and is especially suitable for free-surface and other Stokes’ flow problems with

complicated boundary geometries.12 At the end of this section, we discuss some principles

of the boundary-integral technique. First, however, we discuss some alternative techniques

for the analytic solution of problems involving flow exterior to a solid body.



E. SOLUTIONS FOR SOLID BODIES BY MEANS OF INTERNAL

DISTRIBUTIONS OF SINGULARITIES



One important class of Stokes’ flow problems involves motion past a stationary solid surface.

In this case, the no-slip boundary condition is

u=0



for all x on ∂ D,



(8–120)



and the integral formula (8–119) can be applied for x on ∂ D to obtain

u∞ (x) = −



1

8π



∂D



(x − ξ)(x − ξ)

I

+

· t(ξ)d Aξ .

R

R3



(8–121)



The solution of this integral equation gives us the unknown surface-stress vector t(ξ) =

T(ξ) · n. Then the general solution of the creeping-flow equations is

u(x) = u∞ (x) +



1

8π



∂D



(x − ξ)(x − ξ)

I

+

· t(ξ)d Aξ ,

R

R3



(8–122)



where x is now a fixed point in the flow field and t(x) is the distribution of surface stress

on the boundary ∂ D that we obtain by solving (8–121). We do not discuss the solution of

the integral equation (8–121) here. The main objective in deriving (8–122) is to show that a

general solution of the creeping-flow equations for flow past stationary solid surfaces can be

expressed completely as a superposition of surface forces (stokeslets) at the boundaries ∂ D.

In fact, a solution of the creeping-flow equations can always be written solely as a distribution

of stokeslets over the bounding surfaces, even if these are not solid and stationary, but the

simple identity of the stokeslet density function with the actual surface stress is valid only

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E. Solutions for Solid Bodies by Means of Internal Distributions of Singularities



for this special case. To retain the simple physical interpretation of the density functions

for other kinds of boundaries, we must use the more general form (8–119).

1. Fundamental Solutions for a Force Dipole and Other

Higher-Order Singularities

In this section, we pursue the basic idea that the solution for creeping flow past a body,

in terms of a surface distribution of point forces (stokeslets) on the surface of that body,

can sometimes be replaced with an internal distribution of point forces and higher-order

singularities. This is based on two key points. First, if we begin with a solution of the

creeping-flow equations, then a derivative of any order of that solution is still a solution of

the creeping-flow equations. In particular, if we start with the stokeslet solutions, (8–101),

(8–102), and (8–104), which we denote here as (us , ps ),

α (α + x)x

,

+

r

r3

α·x

ps (x; α) = 2 3 ,

r



us (x; α) =



(8–123)



then a derivative of any order of us and ps is also a solution of the creeping-flow equations,

corresponding to a point singularity that is a derivative of the same order of a point force f.

Second, a stokeslet solution for a point force at x = ξ can be expressed in terms of a formal

multipole expansion in the form of a generalized Taylor series about x,

1

us (x − ξ) = us (x) − (ξ · ∇) us (x) + (ξ · ∇)2 us (x) + · · · +,

(8–124)

2

with a similar series expression for ps . However (ξ · ∇)us and ( ξ · ∇)2 us are the velocity

fields generated by a force dipole, (ξ · ∇) fs , and a force quadrapole, ( ξ · ∇)2 fs , respectively.

Thus, for flow past a solid body, we always can use the generalized Taylor series to replace

the surface distribution of stokeslets in the solution (8–122) with an equivalent internal

distribution of stokeslets and higher-order singularities (inside the body).

The obvious question is whether there is any advantage to be gained, especially in view of

the fact that we must replace a surface distribution of stokeslets only with a whole hierarchy

of higher-order singularities inside. One possibility is that we may be able to replace the surface distribution with an internal distribution of lower spatial order. Thus, for example, for

axisymmetric bodies, it may be possible to express the solution in terms of a line distribution

of singularities along the axis of symmetry of the body. In this case, the 2D integral equation

that arises from application of boundary conditions [Eq. (8–121)] would be replaced with

a 1D, though more complicated, integral equation. Another possibility is that we may be

able to replace a stokeslet distribution on a body that has a very complicated surface (for

example, lots of “bumps”) with an internal distribution of singularities on a nearby surface

that has a much simpler geometry. In any case, however, the use of internal distributions of

singularities will be an advantage only if the number of terms in the multipole expansion

can be limited to a relatively small set. In other words, it will be an advantage only if the

Taylor series, (8–124), can be truncated after a finite number of terms. Intuitively, for an

exact solution, this will require bodies of simple geometry in relatively simple flows. Alternatively, if the internal surface (or line) is close enough to the surface of the body, it should

be possible to approximate the multipole expansion by a small number of terms (or even one

term), as the higher-order terms will decrease rapidly in magnitude. Beyond these generalities, we cannot offer more definitive criteria for recognizing problems for which internal

distributions of singularities offer an advantage over the solutions in terms of a surface

distribution of stokeslets [Eq. (8–122)]. This is, in fact, a subject of current research.

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