Characteristics of velocity gradient jumping discontinuity in steady Poiseuille flow of Johnson–Segalman fluid

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The characteristics of velocity gradient jumping discontinuity are studied in pipe flow of a Johnson–Segalman fluid model, which allows for the non-monotonic relationship between the shear stress and velocity gradient in a simple shear flow for a certain domain range of the material parameters, including slip parameter, Weissenberg number and ratio of viscosities. This non-monotonic constitutive equation admitting multiple steady-state solutions is characterized by shear rate discontinuities. Firstly, the analytical solutions of critical maximum pressure gradient and minimum pressure gradient are calculated with material parameters. Then the properties of “top jumping” and “bottom jumping” of velocity gradient, which are introduced by increasing and decreasing of pressure gradient respectively, and their relation with spurt phenomena, are investigated in this paper. We find that the non-overlapping processes of velocity gradient “top jumping” and “bottom jumping” are in analogy to the variation process for amplitude/frequency relationship in non-linear vibration mechanics. It means two jumping processes cannot occur in the same way of pressure gradient increasing or decreasing. This exploring point is distinctly different from the present literatures. And we have made a confirmation that the flow would be subdivided into two layers when velocity gradient jumping happens in the processes of pressure gradient increasing and pressure gradient decreasing in steady flow. Especially the effects of the velocity gradient jumping on the flow rate are considered quantitatively. Besides, there is a linear relationship between pressure gradient and shear stress. However the varying process of velocity gradient is characterized by jumping discontinuities between two critical pressure gradients. Additionally, the effects of material parameters on the top jumping and bottom jumping locations have been considered to find out where the spurt phenomena occur.Finally, the flow rates in different processes of pressure gradient increasing and pressure gradient decreasing have been discussed to provide potential opportunity for practical application in industrial transportation.