In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shearthickening fluid becomes more shear thickening. This approach has many useful applications in industry, for the investigation of blood flow and fractal fluid hydrology