= Annular viscous extension
Uniform <axial strain rate> $E$ in an annular <Newtonian fluid> gives radial <velocity> $u_r=-Er/2+C/r$. For radii $R_i,R_e$, <area> parameter $A=R_e^2-R_i^2$, and constant gas <pressures>,
$$
C=\frac{R_i^2R_e^2}{2\mu A}\left[p_i-p_e-\gamma\left(\frac1{R_i}+\frac1{R_e}\right)\right],\qquad
p=p_e-\frac{(p_i-p_e)R_i^2}{A}+\frac{\gamma(R_e+R_i)}A-\mu E.
$$
The opposite signs of inner and outer curvature in the <Young–Laplace equation> give these expressions. The <kinematic boundary condition> implies $\dot A=-EA$ and
$$
\frac d{dt}(R_e-R_i)=-\frac E2(R_e-R_i)+\frac\gamma{2\mu}-\frac{(p_i-p_e)R_iR_e}{2\mu(R_i+R_e)}.
$$
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