Several researchers observed a significant increase in water flow through graphene-based nanocapillaries. As graphene sheets are flexible (Wang and Shi 2015 Energy Environ. Sci. 8 790–823), we represent nanocapillaries with a deformable channel-wall model by using the small displacement structural-mechanics and perturbation theory presented by Gervais et al (2006 Lab Chip 6 500–7), and Christov et al (2018 J. Fluid Mech. 841 267–86), respectively. We assume the lubrication assumption in the shallow nanochannels, and using the microstructure of confined water along with slip at the capillary boundaries and disjoining pressure (Neek-Amal et al 2018 Appl. Phys. Lett. 113 083101), we derive the model for deformable nanochannels. Our derived model also facilitates the flow dynamics of Newtonian fluids under different conditions as its limiting cases, which have been previously reported in literature (Neek-Amal et al 2018 Appl. Phys. Lett. 113 083101; Gervais et al 2006 Lab Chip 6 500–7; Christov et al 2018 J. Fluid Mech. 841 267–86 ; White 1990 Fluid Mechanics; Keith Batchelor 1967 An Introduction to Fluid Dynamics ; Kirby 2010 Micro-and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices). We compare the experimental observations by Radha et al (2016 Nature 538 222–5) and MD simulation results by Neek-Amal et al (2018 Appl. Phys. Lett. 113 083101) with our deformable-wall model. We find that for channel-height H o < 4 Å, the flow-rate prediction by the deformable-wall model is 5%–7% more compared to Neek-Amal et al (2018 Appl. Phys. Lett. 113 083101) well-fitted rigid-wall model. These predictions are within the errorbar of the experimental data as shown by Radha et al (2016 Nature 538 222–5), which indicates that the derived deformable-wall model could be more accurate to model Radha et al (2016 Nature 538 222–5) experiments as compared to the rigid-wall model. Using the model, we study the effect of the flexibility of graphene sheets on the flow rate. As the flexibility α increases (or corresponding thickness T and elastic modulus E of the wall decreases), the flow rate also increases. We find that the flow rate scales as m ˙ flexible ∼ α 0 for ( α Δ p W / E H o ) ≪ 1 ; m ˙ flexible ∼ α for ( α Δ p W / E H o ) ∼ O ( 10 − 1 ) ; and m ˙ flexible ∼ α 3 for ( α Δ p W / E H o ) ∼ O ( 1 ) , respectively. We also find that, for a given thickness T , the percentage change in flow rate in the smaller height of the channel is more than the larger height of the channels. As the channel height decreases for the given reservoir pressure and thickness, the Δ m ˙ / m ˙ increases with H o − 1 followed by H o − 3 after a height-threshold. Further, we investigate how the applied pulsating pressure influences the flow rate. We find that due to the oscillatory pressure field, there is no change in the averaged mass flow rate in the rigid-wall channel, whereas the flow rate increases in the flexible channels with the increasing magnitude of the oscillatory pressure field. Also, in flexible channels, depending on the magnitude of the pressure field, either of the steady or oscillatory or both kinds of pressure field, the averaged mass flow rate dependence varies from Δ p to Δ p 4 as the pressure field increases. The flow rate in the rigid-wall channel scales as m ˙ rigid ∼ Δ p , whereas for the deformable-wall channel it scale as m ˙ flexible ∼ Δ p for ( α Δ p W / E H o ) ≊ 0 , m ˙ flexible ∼ Δ p 2 for ( α Δ p W / E H o ) ∼ O ( 10 − 1 ) , and m ˙ flexible ∼ Δ p 4 for ( α Δ p W / E H o ) ∼ O ( 1 ) . We find that both the flexibility of the graphene sheet and the pulsating pressure fields to these flexible channels intensify the rapid flow rate through nano/Angstrom-size graphene capillaries.

Many researchers observed enhanced water flow through carbon nanotubes (CNTs) and attributed the reason to large slips. Even after taking significant slip effects into account, there remain unaddressed observations of significant improvements in flow rates. As CNTS are deformable, we represent nanotubes with a deformable-wall using a linear pressure–area relationship. We assume lubrication assumption, and using the properties of nanoconfined water, we derive the model for deformable-nanotubes. We validated our derived model in its limiting cases with the previously reported results in the literature. We compare the predictions by our deformable-wall and rigid-wall model with the experimental results and the MD-simulation predictions by multiple literature studies. Many studies were well-predicted by the rigid-wall model with slips. However, we find that there are many studies with high porosity and thin wall tubes, where elasticity or deformability of the tube is essential in modeling, which is well-predicted by our deformable-wall model with slips. In our study, we focus on investigating the impact of two key factors: the deformability, and the slip length on the flow rate. We find that the flow rate inside the tube increases as the deformability increases or the thickness T and elastic modulus E of the tube-wall decrease). We also find that the flow rate in deformable tubes scales as m˙deformable∼1/α0 for (Δp/αAo)≪1, m˙deformable∼1/α for (Δp/αAo)∼O(10−1) and m˙deformable∼α2 for (Δp/αAo)∼O(1). Further, for a given deformability, the percentage change in flow rate in the smaller diameter of the tube is much larger than the larger diameter. As the tube diameter decreases for the given pressure, Δm˙/m˙ increases. We find that for rigid-tube, the flow rate varies m˙rigid∼Δp, whereas for the deformable-tubes, the flow rate scales as m˙deformable∼Δp2 for (Δp/αAo)∼O(10−1), and finally to m˙deformable∼Δp3 for (Δp/αAo)∼O(1). We further find that slip also significantly increases flow rate, but, deformability has more substantial effect.

The power-law fluid flow in tree-like self-similar branching networks is prevalent throughout the natural world and also finds numerous applications in technology such as oil recovery and microfluidic devices. We investigate analysis of optimal power-law fluid flow conditions and the optimal structures within tree-like branching networks, focusing on maximizing flow conductance under the constraint of the network tube’s volume and the surface area. The study considered fully developed laminar power-law fluid flow regimes without considering any losses in the network system. A key observation was the sensitivity of the dimensionless effective flow conductance to the network’s geometrical parameters. We found that the maximum flow conductance occurs when a dimensionless radius ratio β∗ satisfies the equation β∗=N−1/3 and β∗=N−(n+1)/(3n+2) under constrained tube-volume and surface-area, respectively. Here, N represents the bifurcation number of branches splitting at each junction, and n is the fluid power-law index. We further find that this optimal condition occurs when pressure drops are equipartition across each branching level. We validated our results with various experimental results and theories under limiting conditions. Further, Hess–Murray’s law is justified and extended for the shear-thinning and shear-thickening fluid flows for an arbitrary number of branches N. Further, in this study, we also derive the relationships between the geometrical and flow characteristics of the parent and daughter tubes as well as the generalized scaling laws at the optimal conditions for the other essential parameters such as tube-wall stresses, length ratios, mean velocities, tube-volume, and surface-area of the tube distributing within the networks. We find that the fluid power-law index n does not influence the constrained tube-volume scaling at the optimal conditions; however, the scaling laws vary with n under the constrained tube’s surface area. These findings offer valuable design principles for developing efficient transport and flow systems.

Yield stress shear thinning/thickening fluids flow through flexible channels, tubes are widespread in the natural world with many technological applications. In this paper, analytical formulae for the velocity profiles and flow rate are derived using the Herschel–Bulkley rheological model in both rigid and deformable shallow channels, employing the lubrication approximation. To account for deformable walls, the approach outlined by Gervais et al (2006, Lab on a Chip 6 500-7) and Christov et al (2018 J. Fluid Mech. 841 267-86) is utilized, applying small displacement structural mechanics and perturbation theory, respectively. The newly derived formulae also enable the analysis of flow dynamics in Newtonian fluids, power-law fluids, and Bingham fluids as their limiting cases, all of which have been previously described in the literature and also serves as the validation cases. It is observed that deformability increases the effective channel height and the flow rate within the channel. Multiple scaling relationships for the flow rate are identified under different applied pressure regimes and deformability parameters. Additionally, it is noted that increasing the yield stress results in decreased velocity in both the plug flow and non-plug flow regions. Higher yield stress also corresponds to an increase in the yield surface height and the solid plug within the central region, leading to a reduction in the flow rate. Furthermore, the shear thinning/thickening index is found to have no impact on plug height, although an increase in this index causes a reduction in the flow rate due to the corresponding increase in shear thickening of the material.