Cilia, hair-like, organelles that are found in the respiratory tract (nasal cavity, pharynx, trachea, and bronchi) rhythmically beat to clear mucus from the airways. Here, we formulate a novel model of fluid flow due to the movement of cilia, and in the companion paper, Part II, the model is numerically solved under simplifying assumptions using physical data from lung bronchi. The model is based on a porous media model, modified so that instead of fluid moving through a solid porous structure, the solid moves the fluid. Two macroscale regions are considered: a porous medium and a free-fluid domain. We use hybrid mixture theory to derive the governing equations so that we have a broader understanding of the assumptions used to obtain the model. The resulting model is the classical Brinkman Stokes equations generalized to account for the movement of the cilia. The model can be used as a prototype to determine the movement of fluid due to the given movement of a solid component of a porous material.