D. TWO-DIMENSIONAL CREEPING FLOWS: SOLUTIONS BY MEANS OF EIGENFUNCTION EXPANSIONS (SEPARATION OF VARIABLES)

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Creeping Flows – Two-Dimensional and Axisymmetric Problems



where γm is an arbitrary constant. The solution of (7–57) is the sum of a homogeneous

solution of the form (7–56) plus a particular solution to reproduce the right-hand side. After

some manipulation, we find

ψm = αm emx eimy + βm xemx eimy + δm yemx eimy .



(7–58)



Hence, the most general solution for ψ expressed in Cartesian coordinates is

ψ=



ψm



(7–59)



m



with arbitrary complex values of m.

Starting with (7–51) and (7–52), we can also obtain a general solution for ψ in terms of

a circular cylindrical coordinate system, and this solution is more immediately applicable

to real problems. The governing equations in polar cylindrical coordinates are

1 ∂

r ∂r



r



∂ψ

∂r



+



1 ∂ 2ψ

= −ω,

r 2 ∂θ 2



(7–60)



1 ∂

r ∂r



r



∂ω

∂r



+



1 ∂ 2ω

= 0.

r 2 ∂θ 2



(7–61)



We seek a solution of (7–60) and (7–61) in the separable form

ω = R(r )F(θ )



and



ψ = S(r )H (θ ).



(7–62)



Substitution for ω in (7–61) yields

r ∂

R ∂r



r



∂R

∂r



=−



F

= λ2 ,

n

F



where λn is an arbitrary constant, either real or complex. Hence for λn = 0 there are two

independent solutions for each of the functions R and F, namely,

F = sin λn θ,



cos λn θ,



and R = r λn , r −λn .



(7–63)



For λn = 0, on the other hand,

F = a0 , θ and R = c0 , ln r.



(7–64)



Hence, the most general solution for ω in the separable form of Eq. (7–62) is

ω = (a0 + b0 θ )(c0 + d0 ln r )

+



∞



(an cos λn θ + bn sin λn θ) cn r



(7–65)

λn



+ dn r



−λn



.



n=1



To determine ψ, we must solve (7-60) with the general form (7–65) substituted for ω. The

general solution for ψ consists of a homogeneous solution of the same form as that of

(7–65), plus a particular solution. To obtain terms of the particular solution corresponding

to the summation in (7–65), we try

ψ p = r s sin λn θ, r s cos λn θ.

450



(7–66)



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D. Two-Dimensional Creeping Flows

θ=α



1



0.8



0.6



0.4



0.2



0.2



0.4



0.6

U



0.8



1



θ=0



Figure 7–5. 2D flow in a sharp corner, caused by motion of the bottom surface (at θ = 0) with the velocity

U. The plot shows streamlines, ψ = ψ/U , calculated from Eq. (7–79) for α = π/3. Contour values range

from 0 at the walls in increments of 0.02.



Substitution into the left-hand side of (7–60) yields

s 2 − λ2 r s−2 sin λn θ, s 2 − λ2 r s−2 cos λn θ.

n

n



(7–67)



Thus, comparing (7–67) and (7–65), we see

s = λn + 2, −λn + 2,



(7–68)



except for λn = 1. In this case, the second of the values for s is s = 1, and the particular

solution form (7–66) reduces to a solution of the homogeneous equation. Thus, in this case,

the corresponding particular solutions take the form

r 3 sin θ, r 3 cos θ, r θ sin θ, and r θ cos θ.



(7–69)



Finally, for λn = 0, the particular solution is

{(a0 θ + b0 )[c0r 2 + d0r 2 (ln r − 1)]}.



(7–70)



Thus, the general solution for ψis

ψ = [c0 + d0 ln r + c0r 2 + d0r 2 (ln r − 1)](a0 θ + b0 )

+ (c1r + d1r −1 + c1r 3 )(a1 sin θ + b1 cos θ) + d1r ( a1 θ sin θ + b1 θ cos θ)

+



∞



(7–71)



cλn r λn + dλn r −λn + cλn r λn +2 + dλn r 2−λn (aλn sin λn θ + bλn cos λn θ).



n=2



2. Application to Two-Dimensional Flow near Corners

The general solution (7–71) can be applied to examine 2D flows in the region between two

plane boundaries that intersect at a sharp corner. This class of creeping motion problems

was considered in a classic paper by Moffatt,11 and our discussion is similar to that given

by Moffatt. A typical configuration is shown in Fig. 7–5 for the case in which one boundary

at θ = 0 is moving with constant velocity U in its own plane and the other at θ = α is

stationary.

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A noteworthy feature of this configuration is that it lacks a definite physical length scale.

One rationalization of this fact is that the corner is generally a localized part of some more

complicated global geometry. In any case, an obvious question is the range of validity of

the creeping-motion approximation for this situation in which a fixed characteristic length

scale (and thus a fixed Reynolds number) does not exist. To answer this question, we need

to obtain an estimate for the magnitudes of the inertial and viscous terms in the equations of

motion. A starting point is the magnitude of the velocity, corresponding to the form (7–71)

for the streamfunction, namely,

|u| = O Ar ξ −1 ,



(7–72)

ξ



where ξ is the real part of λ and A is a constant with dimension of velocity/(length) . Then,

the magnitudes of the inertia and viscous terms in the equations of motion can be estimated

as

|u · ∇u| = O A2r 2ξ −3 ,

|ν∇ 2 u| = O ν Ar ξ −3 .

Hence, the ratio of inertia to viscous terms is

|u · ∇u|

=O

|∇ 2 u|



Ar ξ

ν



,



(7–73)



and we see that the creeping-motion approximation is valid provided that

Re ≡



Ar ξ

ν



1,



(7–74)



where Re is effectively a Reynolds number based on the distance from the corner. Hence,

for ξ > 0, inertia is negligible for sufficiently small values of r, whereas for ξ < 0, neglect

of inertia requires r to be sufficiently large. We focus here on problems in which ξ > 0.

Because the resulting solution in this case is a local approximation, certain features of the

flow will generally remain indeterminate. In reality, they are determined by the features of

the flow at a large distance from the comer where the creeping-flow approximation breaks

down.

The simplest problem of the type considered here is the one sketched in Fig. 7–5,

which was originally solved by Taylor.12 The problem may be considered to be a local

approximation for the action of a wiper blade on a solid surface that is completely covered

by liquid. The boundary conditions in this case are

u r = U,



uθ = 0



ur = u θ = 0



at



at



θ = 0,



θ = α,



(7–75)



where the r and θ components of velocity are related to the streamfunction by means of the

definitions

ur =



1 ∂ψ

,

r ∂θ



uθ = −



∂ψ

.

∂r



(7–76)



In view of (7–76), it is clear that the requirement ur = U (constant) at θ = 0 can be satisfied

only by the solution form

ψ = r F(θ ).

Referring to the general solution (7–71), we find that the terms that are linear in r are

ψ = r (A1 sin θ + B1 cos θ + C1 θ sin θ + D1 θ cos θ),

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(7–77)



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D. Two-Dimensional Creeping Flows



where A1 , B1 , C1 , and Dl are constants. Applying the boundary conditions (7–75) to this

solution, we find that

U = A 1 + D1 ,

0 = (A1 + D1 ) cos α + (C1 − B1 ) sin α + C1 α cos α − D1 α sin α,

0 = B1 ,

0 = A1 sin α + B1 cos α + C1 α sin α + D1 α cos α.

Thus, solving for A1 , B1 , C1 , and Dl , we have

B1 = 0,

C1 =



U (α − sin α cos α)

,

sin2 α − α 2



D1 =



U sin2 α

,

sin2 α − α 2



A1 =



U α2

,

sin2 α − α 2



(7–78)



and

ψ =−



Ur

[α 2 sin θ + (sin α cos α − α)θ sin θ − (sin2 α)θ cos θ ]. (7–79)

(sin α − α 2 )

2



The streamlines corresponding to (7–79) are shown in Fig. 7–5. Although the velocity

components ur and u θ are perfectly well behaved, the shear stress τr θ is singular in the limit

r → 0. Indeed, if we calculate τr θ |θ =0 , we find that

τr θ |θ=0 = −



1

2U μ

(sin α cos α − α).

(sin2 α − α 2 ) r



(7–80)



Clearly, the solution breaks down in the limit r → 0. In fact, according to (7–80), an

infinite force is necessary to maintain the plane θ = 0 in motion at a finite velocity U,

and this prediction is clearly unrealistic. Presumably, one of the assumptions of the theory

breaks down, although a definitive resolution of the difficulty does not exist at the present

time. The most plausible explanation is that the no-slip boundary condition is inadequate

in regions of extremely high shear stress. However, as discussed in Chapter 2, this issue is

still subject to debate.

Closely related to the Taylor problem is the situation sketched in Fig. 7–6, when a flat

plate is drawn into a viscous fluid through a free surface (that is, an interface). In reality,

of course, the interface will tend to deform as a result of the motion of the plate, but we

assume here that the interface remains flat. Then the problem is identical to the previous

Taylor problem except for the boundary conditions, which now become



τr θ



u r = U, u θ = 0 at θ = −α,

1 ∂u r

= 0, u θ = 0 at θ = 0.

≡

r ∂θ



(7–81)



The solution in this case is

ψ = Ur(sin α cos α − α)−1 [sin α(θ cos θ ) − (α cos α) sin θ ].



(7–82)

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Creeping Flows – Two-Dimensional and Axisymmetric Problems

Moving plate



0.2



Interface



0.4



0.6



0.8



θ=0



−0.2

−0.4

−0.6



θ = −α



U

−0.8



Figure 7–6. 2D flow in a sharp corner created when a flat plate is drawn into a fluid through a flat fluid

interface. The plot shows streamlines, ψ = ψ/U , calculated from Eq. (7–82) for α = π/6. Contour values

range from 0 at the walls in increments of 0.0105.



In this case, it is of interest to calculate the tangential velocity on the free surface,

u r |θ=0 = −U 1 −



(α − sin α)(cos α + 1)

.

− sin α cos α + α



(7–83)



The term in brackets is positive and independent of r. Hence the velocity on the free

surface is constant but smaller than the velocity of the solid plate. The speed of a fluid

particle that travels along the interface must therefore increase discontinuously as it reaches

the plate and turns the corner – that is, it must undergo an infinite acceleration. This infinite

acceleration is produced by an infinite stress and pressure, O(r−1 ), on the plate in the limit

r → 0. Again, we conclude that the solution breaks down in the limit r → 0.

A third interesting example of a flow in the vicinity of a corner is the motion of two hinged

plates. As sketched in Fig. 7–7, we assume that the plates rotate with angular velocities −ω

and +ω, respectively. Thus, the boundary conditions are

u r = 0,

u r = 0,



u θ = −ωr

u θ = ωr



at θ = +α,

at θ = −α.



(7–84)



Because u θ = −∂ψ/∂r , it is evident from the general solution (7–71) and the boundary

conditions (7–84) that λ = 2, so that

ψ = r 2 (A2 + B2 θ + C2 sin 2θ + D2 cos 2θ ).



(7–85)



It is convenient to use the conditions (7–84) at θ = α and the symmetry conditions

u θ = ∂u r /∂θ = 0 at θ = 0 to determine the constants A2 , B2 , C2 , and D2 . After some

manipulation, we find that the solution

1

ψ = ωr 2 (sin 2α − 2α cos 2α)−1 (sin 2θ − 2θ cos 2α).

2



(7–86)



In this case, both the velocity components and the stress are bounded in the limit r → 0,

but the pressure exhibits a O(log r) singularity.

Finally, it is of interest to consider the nature of the flow near a sharp comer that

is induced by an arbitrary “stirring” flow at large distances from the corner. In general,

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D. Two-Dimensional Creeping Flows

θ=α



2

ωr



Figure 7–7. 2D flow in the vicinity of the sharp corner between two hinged,

plane walls that are rotating toward one another with angular velocity ω(−ω).

The plot shows streamlines, ψ = ψ/ω, calculated from Eq. (7–86) for α =

π/44. Contour values range from 0 at θ = 0 in increments of 0.2105.



2



−2



− ωr

θ = −α



there are two fundamental types of flow patterns that can be induced near the corner: an

antisymmetric flow, as sketched in Fig. 7–8(a), and a symmetrical flow, as sketched in

Fig. 7–8(b). The actual flow near a corner will generally be a mixture of antisymmetrical

and symmetrical flow types, but it is permissible in the linear Stokes approximation to

consider them separately (the more general flow can then be constructed as a superposition

of the simpler fundamental flows). Here we consider only the antisymmetric case, which is

the more interesting of the two. Thus we consider the general antisymmetric form for ψ,

namely,

ψ=



∞



r λn f λn (θ ),



(7–87)



n=1



where

f λn (θ ) = An cos λn θ + Cn cos(λn − 2)θ.



(7–88)



The boundary conditions at the walls require

f (±α) = f (±α) = 0.

θ=α



θ=α



θ = −α



θ = −α

(a)



(7–89)



(b)



Figure 7–8. A sketch of the 2D flow near a sharp corner that is induced by an arbitrary “stirring” flow at

large distances from the corner: (a) antisymmetric, (b) symmetric.



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We focus only on the dominant term in the expansion (7–87) for small r, that is, the term

with the largest real part of λn ,

ψ ∼ r λ1 f λ1 (θ).



(7–90)



Thus, applying the boundary conditions (7–89) to (7–90), we find that

A1 cos λ1 α + C1 cos(λ1 − 2)α = 0,

A1 λ1 sin λ1 α + C1 (λ1 − 2) sin(λ1 − 2)α = 0.



(7–91)



Hence, to obtain a nontrivial solution, λ1 must satisfy the condition

det



cos λ1 α

λ1 sin λ1 α



cos(λ1 − 2)α

= 0.

(λ1 − 2) sin(λ1 − 2)α



(7–92)



The resulting value of λ1 is known as the eigenvalue for this problem, and the corresponding

function f λ1 (θ ) is the eigenfunction. The coefficients A1 and C1 corresponding to (7–91)

are

A1 = K cos (λ1 − 2) α,

C1 = −K cos λ1 α,

so that

ψ = Kr λ1 [cos(λ1 − 2)α cos λ1 θ − cos λ1 α cos(λ1 − 2)θ ].



(7–93)



We cannot determine the coefficient K from the local analysis alone but only by matching

the local solution to the stirring flow far from the corner. To obtain λ1 , we express (7–92)

in the form

(λ1 − 2) sin(λ1 − 2)α cos λ1 α = λ1 sin λ1 α cos(λ1 − 2)α,

or, on rearrangement,

−(λ1 − 1) sin 2α = sin[2(λ1 − 1)α].



(7–94)



This equation has a real solution for λ1 when 2α > 146◦ but no real solutions for 0 ≤ 2α <

146◦ . In this range, λ1 is complex.

Let us consider this latter case. If we denote (λ − 1) as p + iq, then we can express

(7–94) in the form

sin ξ cos hη = −kξ,



(7–95)



cos ξ sin hη = −kη,



where ξ = 2αp, η = 2αq, and k is the positive parameter k = sin 2α/2α. Any solution of

these equations must satisfy the condition

(2n − 1)π < ξn < 2n −



1

π,

2



where sin ξn and cos ξn are both negative. The corresponding eigenvalue is

λn = 1 + (2α)−1 (ξn + iηn ).



(7–96)



The eigenvalue with the least positive real part, which dominates near the corner (r < 1),

obviously occurs for n = l. Numerical values of ξ1 and η1 for 2α < 146◦ were tabulated

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D. Two-Dimensional Creeping Flows

θ=α



2



Figure 7–9. The infinite sequence of closed eddies in a

sharp corner with an acute angle 2α < 146◦ . This sequence

of closed streamline flows is commonly known as Moffatt

eddies, after the mathematician H. K. Moffatt,9 who first

discovered their existence. Contours shown here are ψ =

0, −0.005, and 10, and we see the first two eddies in the

sequence. The next eddy, closer to the corner, is too small

to see with this resolution.



0



θ = −α

2



3



by Moffatt. The specific values are not significant for present purposes. What is significant

is the fact that λ1 is complex. This feature of the solution implies the existence, for small r,

of an infinite sequence of closed streamline eddies in the corner, the first two of which are

sketched in Fig. 7–9.

To demonstrate that such a sequence of eddies does exist, we can show that there is an

infinite sequence of dividing streamlines, ψ = 0, for successively smaller values of r. If we

introduce (7–96) into (7–93), the dominant streamfunction can be written in the symbolic

form

ψ1 = r (1+ p) [cos(q · ln r )g1 (θ) − sin(q · ln r )g2 (θ)].



(7–97)



As demonstrated by Moffatt, this streamfunction has infinitely many zeroes as r approaches

zero, namely

q · ln r = tan−1



g1 (θ )

g2 (θ )



−



π

− nπ (n = 0, 1, 2, . . . , ).

2



(7–98)



Hence antisymmetric flow induced in a corner between two solid boundaries with an acute

angle less than 146◦ may be expected to show an infinite sequence of increasingly small

(and weak) eddies.

In this section we have considered the formation of a sequence of eddies near a corner

between two plane boundaries with the suggestion that this type of motion could be driven

by some external “stirring” motion at large distances from the corner. There have, in fact,

been a number of investigations of problems in which the Moffatt eddies occur locally

as part of an overall flow structure. Among these are the studies of Wakiya, O’Neill, and

others13 for simple shear flow over a circular cylindrical ridge on a solid, plane surface,

as depicted in Fig. 7–10. In this case a sequence of eddies is found in the groove formed

at the intersection between the cylinder and the plane wall when the angle of intersection,

φ, is less than 146.3◦ . A closely related problem is the 2D motion of two equal, parallel

cylinders that are touching, either along the line connecting the centers of the cylinders

or perpendicular to it, where a sequence of eddies appears in the neighborhood of the

contact point.14 A summary of these and related problems is available for the interested

reader.15

Later, in Chap. 9, we shall briefly consider the additional 2D problem of creeping flow

past a circular cylinder (that is, a circle) in an unbounded fluid that undergoes a uniform

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Creeping Flows – Two-Dimensional and Axisymmetric Problems



φ



Figure 7–10. A qualitative sketch of the flow over a cylindrical ridge on a plane boundary [adapted from

Fig. 1.3.9(a) in Ref. 13].



motion at large distances from the cylinder. For the moment, however, we turn to general

solution procedures for other classes of creeping-flow problems.



E. AXISYMMETRIC CREEPING FLOWS: SOLUTION BY MEANS OF

EIGENFUNCTION EXPANSIONS IN SPHERICAL COORDINATES

(SEPARATION OF VARIABLES)



We saw in Section C that the creeping-motion and continuity equations for axisymmetric,

incompressible flow can be reduced to the single fourth-order PDE for the streamfunction,

E 4 ψ = 0.



(7–99)



An analytic solution for such a problem can thus be sought as a superposition of separable solutions of this equation in any orthogonal curvilinear coordinate system. The most

convenient coordinate system for a particular problem is dictated by the geometry of the

boundaries. As a general rule, at least one of the flow boundaries should coincide with

a coordinate surface. Thus, if we consider an axisymmetric coordinate system (ξ, η, φ),

then either ξ = const or η = const should correspond to one of the boundaries of the flow

domain.

In this section we consider the general solution of (7–99) in the spherical coordinate

system (r, θ, ϕ). This coordinate system is particularly useful for flows in the vicinity of a

spherical boundary, but we begin by simply deducing the most general solution of (7–99)

that is consistent with the constraint of axisymmetry, namely,

u θ = 0 at θ = 0, π.



(7–100)



Rather than using the polar angle θ as an independent variable, it is more convenient to

introduce

η ≡ cos θ

so that the coordinate variables are (r, η, φ) with −1 ≤ η ≤ 1, and the E2 operator takes the

simplified form

E2 ≡

458



∂2

(1 − η2 ) ∂ 2

+

.

2

∂r

r2

∂η2



(7–101)



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E. Axisymmetric Creeping Flows



The nonzero-velocity components, expressed in terms of the streamfunction, are

ur = −



1 ∂ψ

,

r 2 ∂η



u θ 1 − η2 = −



1 ∂ψ

.

r ∂r



(7–102)



We shall be interested here in flows that involve an axisymmetric body with the origin of

coordinates r = 0 inside the body (and at its centre if the body is spherical).

The symmetry condition (7–100) requires that

∂ψ

= 0 at η = ±1

(7–103a)

∂r

If we assume that the surface of the body is impermeable, than ψ = const on this surface,

and it follows from (7–103a) that

ψ = const



at η = ±1.



(7–103b)



For convenience, we may take const = 0 with no loss of generality.

1. General Eigenfunction Expansion

We seek a solution of (7–99) by means of the method of separation of variables. For this

purpose, it is more convenient to note that E 4 ψ = 0 can be split into two second-order

equations,

E 2 ω = 0,



(7–104)



E 2 ψ = −ω.



(7–105)



Obviously, substituting (7–105) into (7–104), we recover (7–99). In the present analysis,

we first determine the most general solution for ω by solving (7–104), and then we solve

(7–105) for ψ with ω given by this general solution.

To solve (7–104), we assume that

ω = R(r )H (η).



(7–106)



Hence, substituting into (7–104), we obtain

r 2 d2 R

(1 − η2 ) d 2 H

+

= 0.

R dr 2

H

dη2



(7–107)



The first term is a function of r only, whereas the second is a function of η. Hence it follows

that each must equal a constant. We denote this constant as n(n + 1), and thus (7–107)

separates into two equations,

r2



d2 R

− n(n + 1)R = 0,

dr 2



(7–108)



and

(1 − η2 )



d2 H

+ n(n + 1)H = 0.

dη2



(7–109)



Equation (7–108) is a particular case of Euler’s equation, for which a general solution is

R = rs.



(7–110)



Substituting into (7–108), we obtain the “characteristic” equation for s,

s(s − 1) − n(n + 1) = 0.



(7–111)



The two roots of this quadratic equation are

s = n + 1,



s = −n,



(7–112)

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