Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 38C (Fluid Dynamics II)

Table of contents
- a 38C
  - Solution a
- b 38C
  - Solution b
- c 38C
  - Solution c

a

Solution

With streamwise and normal velocities

u, v

, zero exterior pressure gradient gives the steady boundary-layer equations

u_{x} + v_{y} = 0, u u_{x} + v u_{y} = ν u_{yy} .

At the organism,

v (x, 0) = 0, ρ ν u_{y} (x, 0) = S x^{- 1/2},

and matching to the quiescent exterior requires

u (x, y) \to 0

y \to \infty

.

b

Solution

Let

U

be the streamwise speed and

δ

the layer width. Continuity gives

v \sim U δ / x

, while inertia--viscosity balance gives

\frac{U ^{2}}{x} \sim \frac{νU}{δ ^{2}} .

The imposed stress gives

ρ νU / δ \sim S x^{- 1/2}

. Eliminating

U

yields

δ \sim (\frac{ρ ν ^{2}}{S})^{1/3} x^{1/2},

and

U \sim (\frac{S ^{2}}{ρ ^{2} ν})^{1/3},

which is independent of

x

.

c

Solution

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

and choose stream function

ψ = U_{0} A x, f (η) .

Then

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

Substitution gives

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

with

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

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)

.

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