# Continuity in the Slender-Jet Expansion Axisymmetric incompressibility: $ \frac{1}{r}\partial_r(r v_r)+\partial_z v_z=0. $ Use the long-wave expansion for axial velocity: $ v_z(r,z,t)=v_0(z,t)+\varepsilon^2 r^2 v_2(z,t)+O(\varepsilon^4 r^4). $ Differentiate in $z$: $ \partial_z v_z=v_0'(z,t)+\varepsilon^2 r^2 v_2'(z,t)+O(\varepsilon^4 r^4). $ Substitute into continuity: $ \partial_r(r v_r) =-r\left[v_0'(z,t)+\varepsilon^2 r^2 v_2'(z,t)+O(\varepsilon^4 r^4)\right]. $ Integrate in $r$: $ r v_r=-\frac{1}{2}v_0' r^2-\frac{1}{4}\varepsilon^2 v_2' r^4+C(z,t)+O(\varepsilon^4 r^6). $ Axis regularity requires $v_r(0,z,t)=0$, so $C(z,t)=0$. Divide by $r$: $ v_r(r,z,t)=-\frac{1}{2}v_0'(z,t)\,r-\frac{1}{4}\varepsilon^2 v_2'(z,t)\,r^3+O(\varepsilon^4 r^5). $ This is the odd-in-$r$ velocity expansion used in the slender-jet reduction. Now apply the kinematic free-surface condition at $r=h(z,t)$: $ h_t+v_z(h,z,t)\,h_z=v_r(h,z,t). $ At leading order, $v_z(h,z,t)=v_0+O(\varepsilon^2)$ and $ v_r(h,z,t)=-\frac{1}{2}h\,v_0'+O(\varepsilon^2). $ Hence $ h_t+v_0 h_z=-\frac{1}{2}h v_0'+O(\varepsilon^2). $ Multiply by $2h$: $ \partial_t(h^2)+\partial_z(h^2 v_0)=O(\varepsilon^2). $ Leading-order 1D continuity equation: $ \partial_t(h^2)+\partial_z(h^2 v_0)=0. $ This is the continuity relation used in [[slender-jets-VE-order-0| VE order 0 slender jet]].