Applicability of quasi-steady assumption during the numerical simulation of the start-up of weakly compressible Herschel-Bulkley fluids in pipelines

In the numerical study of the start-up of viscoplastic fluids in pipelines, the quasi-steady assumption, i.e., the assumption of a linear radial distribution of shear stress, is widely introduced into the governing equations to capture the radial position of the yield surface. However, few studies in the existing literature have examined the condition that the quasi-steady assumption is applicable. In the present work, we define a dimensionless number Ut for the start-up of weakly compressible Herschel-Bulkley fluids in pipelines. Ut increases with increasing pipeline aspect ratio, Reynolds number and flow index, and decreases with increasing fluid compressibility and Bingham number. Scale analysis and numerical studies show that the effect of these five parameters on the applicability of the quasi-steady assumption depends only on the value of Ut. The smaller the value of Ut is, the less likely the quasi-steady assumption would affect the simulation results. When Ut < 0.065, the difference in the time it takes for the outlet fluid starts to flow computed with and without the quasi-steady assumption is below 2%, and the difference in the outlet velocity during its recovery process computed with and without the quasi-steady assumption is below 1%. Moreover, the computed deviations slowly decrease as Ut decreases. Conversely, when Ut > 0.065, the quasi-steady assumption shortens the computed time it takes for the outlet fluid starts to flow and intensifies the computed transient processes of the velocities and pressures.