 Viscous buoyant conduit

A slender low-viscosity fluid column inside a much more viscous fluid rises under a density contrast. At negligible inertia, its axial flux is approximately $q=\pi a^4(\Delta \rho g-P_{i,z})/(8{\mu }_i)$. Radial expansion strains the outer fluid and gives the normal-stress pressure difference $P_i-P_o=2{\mu }_o{a}_t/a$ at leading order when inner normal viscous stress and interfacial tension are negligible. Coupling that pressure to volume conservation yields the conduit equation. This two-fluid viscous model differs from a turbulent entraining thermal plume.