Complex laminar flows of incompressible fluids have been investigated numerically. Both Newtonian and generalised non–Newtonian fluids are simulated, whilst for generalised non–Newtonian fluids, two shear thinning models, i.e., Bird–Carreau and Power law, are employed. Both models are. For numerical investigation the flow problem adopted is two–dimensional axisymmetric periodically constricted tube. The frame of reference used in the investigation is cylindrical polar co–ordinates. For the simulation two–dimensional momentum transport equation is solved using a Finite Element Method (FEM). Numerical scheme used is known as Taylor–Galerkin/pressure–correction algorithm. In open literature this problem is simulated only for single aspect ratio (amplitude) of the undulation in the absence of fibres. Whereas, in this research investigation principal aim is introduction of fibre reinforced composite material. Different aspect ratios of undulation are also simulated at various inertial forces. For comparison purpose with other experimental results as well as numerical prediction, initially, flow is considered for constant viscosity Newtonian fluids through periodically constricted tube. Subsequently, from dilute to semi–dilute rigid rod like fibre suspended in all fluid models, i.e., constant viscosity, Bird–Carreau and Power law fluids have been examined. For different undulation levels of periodically constricted tube, the critical values of the Reynolds number have been identified, where beginning of embryo recirculation region starts. As inertial force increases, augmentation of vortex is observed in the undulation region. While, introducing fibres vortices shrinks in length and intensity. For Newtonian, power law and Bird–Carreau models pressure isobars are also demonstrated. In all levels of fibre concentration, the graphical illustration of calculated data for friction factor as well as empirical relationships is illustrated for all undulation aspect ratios.