= 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.
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