The study of fluid flow through solid matter by computed tomography (CT) imaging has many applications, ranging from petroleum and aquifer engineering to biomedical, manufacturing, and environmental research. To avoid motion artifacts, current experiments are often limited to slow fluid flow dynamics. This severely limits the applicability of the technique. In this paper, a new iterative CT reconstruction algorithm for improved a temporal/spatial resolution in the imaging of fluid flow through solid matter is introduced. The proposed algorithm exploits prior knowledge in two ways. First, the time-varying object is assumed to consist of stationary (the solid matter) and dynamic regions (the fluid flow). Second, the attenuation curve of a particular voxel in the dynamic region is modeled by a piecewise constant function over time, which is in accordance with the actual advancing fluid/air boundary. Quantitative and qualitative results on different simulation experiments and a real neutron tomography data set show that, in comparison with the state-of-the-art algorithms, the proposed algorithm allows reconstruction from substantially fewer projections per rotation without image quality loss. Therefore, the temporal resolution can be substantially increased, and thus fluid flow experiments with faster dynamics can be performed.