A fractal roughness model for the transport of fractional non-Newtonian fluid in microtubes

•A novel fractal roughness model is proposed for non-Newtonian fluid transport.•A new Poiseuille number combines the memory of fluid and fractal trait of a surface.•The effective roughness in two dimensions is introduced and investigated.•The effects of key parameters and the underlying mechanism are discussed.•This model may be a potential approach for complex fluid transport in microfluidics.