38A (Fluid Dynamics II)

The sink speed scales as

U \sim A / R^{2}

R e \sim U R / ν = A / (ν R)

. The approximation applies in the high-Reynolds-number region

R ≪ A / ν

. Radial inertia scales as

U d U / d R \sim A^{2} / R^{5}

, so the radial pressure gradient has the same

R^{- 5}

dependence.

b

The definitions through the Stokes stream function give

\frac{1}{r} \partial_{r} (r u) + \partial_{z} w = - \frac{1}{r} \partial_{r} \partial_{z} Ψ + \frac{1}{r} \partial_{z} \partial_{r} Ψ = 0,

Balancing inertia

A^{2} / r^{5}

νU / δ^{2}

gives

δ (r) \sim (ν r^{3} / A)^{1/2} .

The streamfunction scale is

r U δ \sim (A ν r)^{1/2}

, motivating

Ψ = (A ν r)^{1/2} F (η), η = z (A / (ν r^{3}))^{1/2} .

The similarity form gives

u = - \frac{A}{r ^{2}} F^{'}, w = \frac{( A ν ) ^{1/2}}{2 r ^{3/2}} (F - 3 η F^{'}) .

ρ^{- 1} d P / d r = 2 A^{2} / r^{5}

in the correctly written radial boundary-layer equation

u u_{r} + w u_{z} = - ρ^{- 1} P_{r} + ν u_{zz}

gives

F^{(3)} + 2 (1 - F^{'2}) - \frac{1}{2} F F^{''} = 0.

No slip and matching to the sink require

F (0) = F^{'} (0) = 0

and

F^{'} (\infty) = 1

. The boundary-layer flux is of order

r U δ

, relative to the sink flux

A

it is

\frac{r U δ}{A} \sim (ν r / A)^{1/2} .

Mass conservation therefore requires an outer correction of this relative size, which is small in the assumed region.

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