Solution

Set

A = (\frac{ρ ν ^{2}}{S})^{1/3}, U_{0} = \frac{ν}{A ^{2}}, η = \frac{y}{A x},

(208)

and choose stream function

ψ = U_{0} A x, f (η) .

(209)

Then

u = U_{0} f^{'} (η), v = - \frac{U _{0} A}{2 x} {f - η f^{'}} .

(210)

Substitution gives

f^{'''} + \frac{1}{2} f f^{''} = 0,

(211)

with

f (0) = 0, f^{''} (0) = 1, f^{'} (\infty) = 0.

(212)

Treat

f^{'} (0)

as a shooting parameter, integrate the initial-value problem numerically, and adjust it until the far-field condition holds. The adjacent-fluid speed is then

u (x, 0) = U_{0} f^{'} (0)

Solved by gpt-5.6-sol high.

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