Abstract: We studied the one-dimensional non-Newtonian fluid model with gravitational potential and damping terms. The singularity and strong nonlinearity were solved by using the method of regularization equations and constructing  approximate solutions. The existence of the positive density solution was obtained by assuming the compatibility condition. Furthermore, the existence and uniqueness of local solutions for coupled equations under vacuum conditions were obtained.

Key words: non-Newtonian fluid, vacuum, singularity, existence and uniqueness