because the given representation enforces incompressibility. This second proof applies whenever the region between the paths lies in the regular fluid domain. The question already assumes a globally defined streamfunction; without that assumption, incompressibility alone would guarantee a global streamfunction on a simply connected domain, but a hole carrying nonzero volume flux could obstruct it.

b

 0  0

Solution

 0  0

Substitute the given Stokes streamfunction representation into the divergence in spherical coordinates. The mixed derivatives cancel:

\nabla \cdot u = \frac{1}{r ^{2}} \partial_{r} (\frac{Ψ _{θ}}{sin θ}) + \frac{1}{r sin θ} \partial_{θ} (- \frac{Ψ _{r}}{r}) = \frac{Ψ _{r θ} - Ψ _{θ r}}{r ^{2} sin θ} = 0.

(1)

Thus incompressibility is automatic for a smooth Stokes streamfunction. Since this is an axisymmetric flow with no azimuthal velocity, the only possibly nonzero component of its vorticity is

(\nabla \times u)_{φ} = \frac{1}{r} [\partial_{r} (- \frac{Ψ _{r}}{sin θ}) - \partial_{θ} (\frac{Ψ _{θ}}{r ^{2} sin θ})] = - \frac{1}{r ^{3} sin θ} [r^{2} Ψ_{rr} + sin θ \partial_{θ} (\frac{Ψ _{θ}}{sin θ})] .

(2)

Define the differential operator in the Stokes streamfunction equation for irrotational flow

L Ψ = r^{2} Ψ_{rr} + Ψ_{θθ} - cot θ Ψ_{θ} .

(3)

The stated equation is

L Ψ = 0

, and therefore

\nabla \cdot u = 0, \nabla \times u = 0 .

(4)

The irrotational flow has a velocity potential locally, and globally on a regular simply connected domain. Regularity of the physical velocity field supplies the limiting interpretation on the polar axis, where the coordinate expressions contain

sin θ

in the denominator.

c

 0  0

Solution

 0  0

Put

s = a^{2} / r

and define the image term

G (r, θ) = (r / a) Ψ (s, θ)

. The key step is to verify that inversion preserves the Stokes streamfunction equation for irrotational flow. Direct differentiation gives

G_{r} = \frac{1}{a} (Ψ - s Ψ_{s}), G_{rr} = \frac{s ^{2}}{a r} Ψ_{ss}, G_{θθ} - cot θ G_{θ} = \frac{r}{a} (Ψ_{θθ} - cot θ Ψ_{θ}),

(1)

where the quantities on the right are evaluated at

(s, θ)

. Hence

L_{r} G = \frac{r}{a} [s^{2} Ψ_{ss} + Ψ_{θθ} - cot θ Ψ_{θ}] = \frac{r}{a} L_{s} Ψ = 0.

(2)

By linearity,

Ψ = Ψ - G

obeys the same equation and describes an incompressible flow and an irrotational flow wherever the original exterior flow is regular.

On the sphere,

s = a

and the two terms cancel identically:

Ψ (a, θ) = 0, u_{r} (a, θ) = \frac{\partial _{θ} Ψ ( a , θ )}{a ^{2} sin θ} = 0.

(3)

Thus the sphere is a stream surface and satisfies the no-penetration boundary condition. Tangential slip is allowed in this potential flow model. For every finite

r > a

, the image argument satisfies

0 < s < a

, where

Ψ

has no singularities. The prefactor

r / a

is regular there. Therefore the construction creates no new finite exterior singularities; any original exterior singularities remain those of the prescribed incident flow. This is the sphere theorem for a Stokes streamfunction.

For the far-field claim, the stated bound gives

∣ G (r, θ) ∣ \leq \frac{r}{a} c \frac{a ^{2}}{r} = c a (r sufficiently large) .

(4)

One must compare velocities, rather than require the correction to the Stokes streamfunction to tend to zero. Smoothness at the original origin, together with

Ψ (0, θ) = 0

, gives

Ψ_{s} = O (1)

and

Ψ_{θ} = O (s)

there. Thus

G_{r} = O (r^{- 1})

and

G_{θ} = O (1)

, and the correction has

δ u_{r} = - \frac{G _{θ}}{r ^{2} s i n θ} = O (r^{- 2}), δ u_{θ} = \frac{G _{r}}{r s i n θ} = O (r^{- 2}) .

(5)

These estimates hold away from the coordinate poles, with the regular-axis limits supplied by the original smooth physical flow. Consequently

u - u ⟶ 0 (r \to \infty) .

(6)

The far-field velocity field is unchanged. For a fully smooth incident velocity field at the origin, the usual regular expansion is stronger and makes the sphere disturbance decay as