 Annular viscous extension

ID: annular-viscous-extension

E

in an annular Newtonian fluid gives radial velocity

u_{r} = - E r /2 + C / r

R_{i}, R_{e}

A = R_{e}^{2} - R_{i}^{2}

, and constant gas pressures,

C = \frac{R _{i}^{2} R _{e}^{2}}{2 μ A} [p_{i} - p_{e} - γ (\frac{1}{R _{i}} + \frac{1}{R _{e}})], p = p_{e} - \frac{( p _{i} - p _{e} ) R _{i}^{2}}{A} + \frac{γ ( R _{e} + R _{i} )}{A} - μ E .

The opposite signs of inner and outer curvature in the Young–Laplace equation give these expressions. The kinematic boundary condition implies

\dot{A} = - E A

and

\frac{d}{d t} (R_{e} - R_{i}) = - \frac{E}{2} (R_{e} - R_{i}) + \frac{γ}{2 μ} - \frac{( p _{i} - p _{e} ) R _{i} R _{e}}{2 μ ( R _{i} + R _{e} )} .

 New to topics? Read the docs here!