This paper investigates the capillary flows in variable interior corners along the axis when a microgravity environment and the Concus-Finn condition are satisfied. The governing equation for capillary-driven flows in variable interior corners is established, and an approximately analytic solution is obtained. Then we compare the approximately analytic solution with the numerical simulation results obtained using the software FLOW-3D. Results show that the relative error between the approximately analytical and numerical solutions is getting smaller and smaller with increasing time, and it will be less than 5% after 6 s. The influence of different parameters on the interior corner flow is studied using a set of typical parameters. Numerical results show that the liquid-front position and meniscus height decrease with increasing interior angles, contact angle, slope, and power exponent. The liquid meniscus height increases with the duration at different times. But it remains constant at the initial time. The conclusion of this paper can be applied when designing containers and choosing the suitable solution in the space fluid management.