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Abstract
The capillary driven flow of non-Newtonian fluids into networks is a complex phenomenon with multiple applications in biological systems, industry and engineering. In this work, we extend the classical Lucas-Washburn scenario, where a Newtonian fluid is driven by capillary forces through a single cylindrical vertical tube, by considering the flow of a non-Newtonian power-law fluid through a self-similar root-like branching network, without neglecting hydrostatic effects. We develop a theoretical model that assumes axisymmetric flow in each network branch, and find a complex generalized form for the classical square-root Lucas-Washburn relation between filled length and time: $L\propto t^{1/2}$. The model is validated by comparing our theoretical results with data from experiments and numerical simulations available in the literature. We identify two flow regimes, equilibrium and overflow. In the equilibrium regime the flow asymptotically approaches an equilibrium filling limit, and in the overflow regime the fluid fills the entire network in a finite time. We show that the regime flow is determined by ratio between capillary and hydrostatic forces. Furthermore, we describe an optimal geometrical configuration that maximizes fluid transport across the network. We show that non-Newtonian rheological features play a significant role in the flow dynamics. Finally, we show that the power-law rheology and the structural geometrical complexity, change the simple Lucas-Washburn power-law relation to a fairly complex signomial one. The capillary flow of shear-thinning fluids is significantly faster than shear-thickening ones, but also much more sensitive perturbations on the physical parameters of interest. These insights provide a framework for designing bio-inspired fluidic systems and optimizing capillary-driven engineering applications, such as smart textiles, biomedical microfluidics devices, and enhanced oil recovery.
