Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 38A / a

Solution

Let

u (r, z)

and

w (r, z)

be the radial and vertical velocity components. For steady incompressible axisymmetric flow without swirl, mass conservation and the relevant cylindrical components of the Navier-Stokes equation are

\frac{1}{r} \frac{\partial ( r u )}{\partial r} + \frac{\partial w}{\partial z} = 0,

ρ (u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z}) = - \frac{\partial p}{\partial r} + μ (\frac{\partial ^{2} u}{\partial r ^{2}} + \frac{1}{r} \frac{\partial u}{\partial r} - \frac{u}{r ^{2}} + \frac{\partial ^{2} u}{\partial z ^{2}}),

ρ (u \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z}) = - \frac{\partial p}{\partial z} + μ (\frac{\partial ^{2} w}{\partial r ^{2}} + \frac{1}{r} \frac{\partial w}{\partial r} + \frac{\partial ^{2} w}{\partial z ^{2}}) .

The vertical scale is

w \sim V

. Incompressibility over radial scale

R

and gap scale

h

gives

u \sim V R / h = V / ε

. In the radial equation, inertia is smaller than transverse viscous stress by

\frac{ρ Vh}{μ} = Re ≪ 1,

while radial viscous derivatives are smaller than

u_{zz}

ε^{2}

. The pressure scale is

μ V R^{2} / h^{3}

. In the vertical equation, both viscous and inertial terms are smaller than the corresponding cross-gap pressure-gradient scale by at least

O (ε^{2})

\frac{1}{r} (r u)_{r} + w_{z} = 0, p_{r} = μ u_{zz}, p_{z} = 0.

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