C. REPRESENTATION OF TWO-DIMENSIONAL AND AXISYMMETRIC FLOWS IN TERMS OF THE STREAMFUNCTION

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C. Representation of Two-Dimensional and Axisymmetric Flows



The starting point is a general representation theorem from vector calculus that states

that any continuously differentiable vector field can be represented by three scalar functions

φ, ψ, and χ in the form10

a = ∇φ + ∇ ∧ (ψ∇χ ).



(7–29)



In effect, (7–29) represents a decomposition of the general vector field a into an irrotational

part, associated with ∇φ, and a solenoidal (or divergence-free) part, represented by ∇ ∧

(ψ∇χ). It should be noted that general proofs exist that show not only that (7–29) can

represent any arbitrary vector field a but also that an arbitrary, irrotational vector field can

be represented in terms of the gradient of a single scalar function φ and that an arbitrary,

solenoidal vector field can be represented in the form of the second term of (7–29). Because

∇ · [∇ ∧ (ψ∇χ )] = ∇ · (∇ψ ∧ ∇χ ) ≡ 0,

the representation (7–29) can be given also in terms of a solenoidal vector field A, such

that

a = ∇φ + ∇ ∧ A,



where ∇ · A = 0.



(7–30)



Now, because the velocity field u is a continuously differentiable vector field, it can be

represented for any Reynolds number by either form (7–29) or (7–30), that is,

u = ∇φ + ∇ ∧ (ψ∇χ )



(7–31)



or

u = ∇φ + ∇ ∧ A,



where ∇ · A = 0,



(7–32)



and this is true for an arbitrary 3D motion. The function A that appears in (7–32) is known

as the vector potential for the vorticity because

∇ 2 A = −∇ ∧ u = −␻.



(7–33)



Now the question is whether the representations (7–31) and/or (7–32) lead to any simplification of the mathematical problem for u. To answer this question, let us for the moment

stick to the creeping-flow limit. For an incompressible fluid, the continuity equation requires

that

∇ 2 φ = 0,



(7–34)



and the equation of motion reduces to an equation for the vector potential function A,

∇ ∧ (∇ 2 A) = ∇ p



(7–35)



∇ 4 A = 0.



(7–36)



or



Clearly, A must be nonzero in general to satisfy (7–35). On the other hand, the contribution

∇φ will only be nonzero if

(u − ∇ ∧ A) · n = 0,

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Creeping Flows – Two-Dimensional and Axisymmetric Problems



at the boundaries. Generally, this will not be true, and u can then be represented simply as

u = ∇ ∧ A,



(7–37)



with ∇ · A = 0 and A satisfying (7–35) or (7–36). In this case, introduction of the vector

potential function A allows the problem to be simplified to the extent that the general form

(7–37) satisfies continuity for any choice of A, and the pressure can be eliminated to obtain

a single fourth-order equation for A. Nevertheless, for 3D flows, the problem for A is not

all that much simpler than the original problem because all three components of A are

generally nonzero.

For 2D and axisymmetric flows, however, the general representation results, (7–31) and

(7–32), do lead to a very significant simplification, both for creeping flows and for flows at

finite Reynolds numbers where we must retain the full Navier–Stokes equations. The reason

for this simplification is that the vector potential A can be represented in terms of a single

scalar function,

A = h 3 ψ(q1 , q2 )i3 ,



(7–38)



where i3 is a unit vector that is orthogonal to the plane of motion for 2D flows and in the

azimuthal direction for the axisymmetric case. We shall show shortly that this representation

is sufficiently general for arbitrary 2D and axisymmetric flows. First, however, we need to

define all of the symbols that appear. The factor h3 is the scale factor defined for a general,

orthogonal curvilinear coordinate system

(q1 , q2 , q3 )



(7–39)



by means of the equation for the length of a differential line element:

(ds)2 ≡



1

1

1

(dq1 )2 + 2 (dq2 )2 + 2 (dq3 )2 .

2

h1

h2

h3



(7–40)



In general, for 2D flows, q3 can be identified with a Cartesian variable z, orthogonal to the

plane of motion, and h 3 ≡ 1. However, for axisymmetric flows, q3 represents the azimuthal

angle φ about the axis of symmetry, and h3 will generally be a function of q1 and q2 . If we

express A in terms of spherical coordinates, for example,

(q1 , q2 , q3 ) → (r, θ, φ),

then

h 1 = 1,



h2 =



1

,

r



and



h3 =



1

.

r sin θ



The scalar function ψ that appears in (7–38) is known as the streamfunction. The

physical significance of ψ is best seen through the relationship (7–37). In particular, if we

substitute (7–38) into (7–37), we obtain

u = h2h3



∂ψ

∂ψ

, −h 1 h 3

, 0 .

∂q2

∂q1



(7–41)



We see, from (7–41), that the form (7–38) is consistent with the existence of a 2D or

axisymmetric velocity field and, further, that the magnitudes of the two nonzero-velocity

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C. Representation of Two-Dimensional and Axisymmetric Flows

n



t



Figure 7–4. An arbitrarily chosen curve between

two points, P and Q, with unit normal n and unit

tangent t.



Q



P



components are directly related to the magnitudes of the spatial derivatives of ψ. In addition,

if we calculate the derivative of ψ in the direction of motion, we see that

u · ∇ψ

≡ 0.

|u|



(7–42)



Thus lines of constant ψ are everywhere tangent to the local velocity; that is, curves in

space corresponding to constant values of ψ are coincident with the pathlines followed by

an element of the fluid.

It is also worth noting that the volume flux of fluid across a curve joining any two

arbitrary points, say, P and Q, in the flow domain is directly related to the difference in

magnitude of the streamfunction at these two points. For simplicity, let us show that this is

true for 2D motions. The axisymmetric case follows in a very similar way. Thus we consider

a curve, as shown in Fig. 7–4, that passes through the two points P and Q but is otherwise

arbitrary. The volume flux of fluid across this curve, per unit length in the third direction,

is

J=



(u · n)d .



(7–43)



Note that the integral is independent of the particular path between P and Q for an incompressible fluid where ∇ · u = 0. Now, the unit normal is

n = t ∧ i3 ,

where i3 is the positive unit normal in the third direction. Thus,

J=



u · (t ∧ i3 )d ,



where

td ≡



1

dq1 i1 +

h1



1

dq2 i2 ,

h2



and thus,

(t ∧ i3 )d ≡



1

1

dq2 i1 − dq1 i2 .

h2

h1

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Creeping Flows – Two-Dimensional and Axisymmetric Problems



Hence,

Q



J=



dψ

dψ

dq2 +

dq1 ;

dq2

dq1



P



J=



ψQ

ψP



dψ = ψ Q − ψ P ,



(7–44)



where ψ Q and ψ P denote the values of ψ at the points Q and P, respectively.

Introduction of the streamfunction for axisymmetric and 2D problems simplifies their

solution in two ways. First, in view of the definitions (7–37) and (7–38), it is evident that

the continuity equation for an incompressible fluid,

∇ · u = 0,

will be satisfied automatically for any function ψ that satisfies the other conditions of the

problem. Second, the equations of motion are reduced from a coupled pair of equations

relating u and p to a single higher-order equation for the scalar function ψ. In the creepingflow approximation this equation can be obtained directly from (7–36). Substituting from

(7–38), we obtain

E 4 ψ = 0,



(7–45)



where

E2 ≡



h1h2

h3



∂

∂q1



h1h3 ∂

h 2 ∂q1



+



∂

∂q2



h2h3 ∂

h 1 ∂q2



.



In general, the E2 operator is not the same as the more familiar ∇ 2 (≡ ∇ · ∇) which is defined

as

∂

∂

h1 ∂

h2 ∂

∇ 2 ≡ h1h2h3

+

.

∂q1 h 2 h 3 ∂q1

∂q2 h 1 h 3 ∂q2

However, for 2D flows, the third direction corresponds to a Cartesian coordinate direction

and

h 3 = 1.

In this case, E2 and ∇ 2 are identical, and the governing equation for ψ in two dimensions

is therefore normally expressed as

∇ 4 ψ = 0,



(7–46)



which is the familiar biharmonic equation in two dimensions.

We shall discuss the solution of (7–45) and (7–46) shortly. First, however, it is worth

noting the form of the full Navier–Stokes equations when expressed in terms of ψ. For this

purpose, it is convenient first to take the curl of the equations to eliminate the pressure. In

the 2D and axisymmetric flow cases considered here, this gives

Re



∂␻

− ∇ ∧ (u ∧ ␻) + ∇ ∧ (∇ ∧ ␻) = 0.

∂t



(7–47)



Now,

␻ = ∇ ∧ [∇ ∧ (h 3 ψi3 )] = −h 3 E 2 ψi3 ,

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(7–48)



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D. Two-Dimensional Creeping Flows



according to the definitions of ␻ and u (in terms of ψ). Thus, substituting into (7–47), we

obtain

Re



∂ 2

h1h2 ∂

(E ψ) +

∂t

h 3 ∂q1



h2

3



∂ψ 2

∂

E ψ −

∂q2

∂q2



h2

3



∂ψ 2

E ψ

∂q1



= E 4 ψ.



(7–49)



Again, for the 2D case, this reduces to

Re



∂

∂ 2

(∇ ψ) + h 1 h 2

∂t

∂q1



∂ψ 2

∂

∇ ψ −

∂q2

∂q2



∂ψ 2

∇ ψ

∂q1



= ∇ 4 ψ.



(7–50)



Clearly, in the limit Re → 0, these equations reduce to the limiting forms (7–45) and (7–46).

D. TWO-DIMENSIONAL CREEPING FLOWS: SOLUTIONS BY MEANS

OF EIGENFUNCTION EXPANSIONS (SEPARATION OF VARIABLES)



We saw, in the previous section, that problems of creeping-motion in two dimensions can be

reduced to the solution of the biharmonic equation, (7–46), subject to appropriate boundary

conditions. To actually obtain a solution, it is convenient to express (7–46) as a coupled pair

of second-order PDEs:

∇ 2 ψ = −ω,



(7–51)



∇ ω = 0.



(7–52)



2



The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator

∇ 2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most

common cases for 2D flows of Cartesian and circular cylindrical coordinates.

1. General Eigenfunction Expansions in Cartesian

and Cylindrical Coordinates

For Cartesian coordinates, a solution of (7–52) exists in the separable form

ω = X (x)Y (y).



(7–53)



X

Y

=−

= ±m 2 ,

X

Y



(7–54)



Substituting into (7–52), we obtain



where m is an arbitrary complex number. Hence,

X ± m 2 X = 0,



Y ± m 2 Y = 0,



(7–55)



and from this we deduce that

ω = emx eimy



(7–56)



for arbitrary complex m. Now, to obtain a general solution for ψ, we must solve (7–51) with

the right-hand side evaluated using (7–56). Hence,

∇ 2 ψ = −emx eimy γm ,



(7–57)

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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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