The problem of low-gravity isothermal capillary flow along interior corners that are rounded is revisited analytically in this work. By careful selection of geometric length scales and through the introduction of a new geometric scaling parameter $\overline T_c$, the Navier–Stokes equation is reduced to a convenient$\,{\sim}\, O(1)$ form for both analytic and numeric solutions for all values of corner half-angle $\alpha$ and corner roundedness ratio $\lambda$ for perfectly wetting fluids. The scaling and analysis of the problem captures much of the intricate geometric dependence of the viscous resistance and significantly reduces the reliance on numerical data compared with several previous solution methods and the numerous subsequent studies that cite them. In general, three asymptotic regimes may be identified from the large second-order nonlinear evolution equation: (I) the ‘sharp-corner’ regime, (II) the narrow-corner ‘rectangular section’ regime, and (III) the ‘thin film’ regime. Flows are observed to undergo transition between regimes, or they may exist essentially in a single regime depending on the system. Perhaps surprisingly, for the case of imbibition in tubes or pores with rounded interior corners similarity solutions are possible to the full equation, which is readily solved numerically. Approximate analytical solutions are also possible under the constraints of the three regimes, which are clearly identified. The general analysis enables analytic solutions to many rounded-corner flows, and example solutions for steady flows, perturbed infinite columns, and imbibing flows over initially dry and prewetted surfaces are provided.