Abstract In this work, analytical predictions of the rectilinear flow of a non-Newtonian liquid under a combined pulsating, time-dependent pressure gradient and a random longitudinal vibration flow is analyzed. The fluctuating component of the combined pressure gradient and oscillating flow is assumed to be of small amplitude and can be adequately represented by a weakly stochastic process, for which a quasi-static perturbation solution scheme is suggested, in terms of a small parameter. This flow is analyzed with the Tanner constitutive equation model with the viscosity function represented by the Ellis model. According to the coupled Tanner-Ellis model, the flow enhancement can be separated in two contributions (pulsatile and oscillating mechanisms) and the power requirement is always positive and can be interpreted as the sum of a pulsatile, oscillating and the coupled systems respectively. Both expressions depend on the amplitude of the oscillations, the perturbation parameter, the Reynolds, Deborah and Weissenberg numbers and the exponent of the Ellis model associated to the shear thinning or thickening mechanisms respectively. At small wall stress values, the flow enhancement is dominated by the axial wall oscillations whereas at high wall stress values, the system is governed by the pulsating noise perturbation. The flow transition is obtained for a critical shear stress which is a function of the Reynolds number, dimensionless frequency and the ratio of the two amplitudes associated with the pulsating and oscillating perturbations respectively. Under some considerations, the total flow enhancement due to the coupling flows can be considered as the sum of the pulsating and oscillating flow effects. In addition, the flow enhancement is compared with analytical and numerical predictions of the Reiner-Phillipoff and Carreau models respectively. Finally, flow enhancement and power requirement are predicted using biological rheometric data of blood.