Predicting the distribution of yield-stress fluids in branched pipe manifolds

TL;DR

This work tackles predicting the steady-state distribution of yield-stress fluids in branched pipe manifolds under wall slip. It introduces a reduced-order, one-dimensional network model that incorporates Herschel–Bulkley rheology and a power-law wall-slip boundary condition, with slip parameters measured independently via capillary rheometry. The model is validated against bench experiments and fully resolved CFD simulations, showing excellent agreement and revealing that wall slip significantly enhances distribution uniformity, especially at higher Bingham numbers where a slip-dominated regime emerges. Additionally, the framework supports inversion, enabling slip-parameter estimation from outlet distributions, offering a practical, computationally inexpensive tool for the design and analysis of industrial manifolds handling viscoplastic fluids.

Abstract

We develop a one-dimensional network model to predict the steady-state distribution of yield-stress fluids in branched pipe manifolds under wall-slip conditions. The model accounts for major friction losses between junctions and incorporates wall slip through a power-law relation calibrated independently via capillary rheometry. Predictions from the model are validated against both bench-scale experiments and fully resolved computational fluid dynamics simulations, showing excellent agreement across a range of flow conditions. Our results demonstrate that wall slip strongly influences the uniformity of fluid distribution by modifying the relative resistance between outlet branches. Furthermore, we show that the problem can be inverted: measured distribution profiles can be used to estimate slip parameters, offering a practical method for slip characterisation without pressure measurement. This modelling framework is computationally inexpensive, robust, and adaptable to various network configurations, making it a valuable tool for the design and analysis of industrial manifold systems involving viscoplastic fluids.

Figures (9)

• Figure 1: A schematic diagram of the capillary rheometer used to measure the slip properties of the yield-stress materials and calibrate the pressure measurement system. The setup features a syringe pump which controls the flux of material flowing through a capillary and a pressure sensor mounted upstream.
• Figure 2: Ramp-down steady-shear flow curves of Carbopol (blue circles) and the emulsion (red circles) using profiled parallel plates with a gap size of 1.55 mm. The black dashed curves represent the best fitting to the Herschel-Bulkley constitutive model [cf. Eq. \ref{['eqn:HB']} for the expression and Table \ref{['tab:material rheology']} for the model parameters].
• Figure 3: A schematic diagram of the experimental rig. A syringe pump delivers a constant flux of material to the manifold which divides the flow from one inlet to six outlets. Screws seal off the ends of the main manifold pipe.
• Figure 4: Fluid distribution profiles for Carbopol (a) and the emulsion (b) at $B_\mathrm{in}=0.44$. The blue bars indicate the experimental result, red squares indicate the distribution predicted by the network model without slip ($S_1=S_2=0$), and green circles indicate the distribution predicted by the network model when slip is included in the model.
• Figure 5: Fluid distribution profiles for Carbopol at $B=0.61$ without (a) and with (c, e, g) an outlet blockage and at $B=1.61$ without (b) and with (d, f, h) an outlet blockage. The blue bars indicate the experimental results, the open dark blue crosses indicate the CFD results, the red squares indicate the distribution predicted by the network model without slip ($S=0$), and the green circles indicate the distribution predicted by the network model when slip is included in the model.