Enhanced flow in deformable carbon nanotubes

Water transport through nanopores is widespread in the natural world and holds significant implications in various technological applications. Several researchers observed a significant increase in water flow through graphene-based nanotubes. Graphene sheets are deformable, so we represent nano/Angstrom-size tubes with a deformable wall model using the small displacement structural mechanics with a linear pressure-area relationship. We assume the lubrication assumption in the shallow tubes, and using the microstructure of confined water along with slip at the capillary boundaries, we derive the model for deformable nanotubes. Our derived model also facilitates the flow dynamics of Newtonian fluids under different conditions as its limiting cases, which have been previously reported 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 literatures. 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 modelling, 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 of the nanotubes and the slip length on the flow rate. We find that the flow rate inside the tube increases as the deformability $1/\alpha$ increases (or corresponding thickness $\mathcal{T}$ and elastic modulus $E$ of the wall decreases). We find that the flow rate in deformable tubes scales as $\dot{m}_{\text{deformable}}\sim 1/\alpha^0 $ for $\Big ( \Delta p/\alpha A_o \Big ) \ll 1$, $\dot{m}_{\text{deformable}}\sim 1/\alpha $ for $\Big ( \Delta p/\alpha A_o \Big ) \sim O(10^{-1})$ and $\dot{m}_{\text{deformable}}\sim \alpha^2 $ for $\Big ( \Delta p/\alpha A_o \Big ) \sim O(1)$. We also find that, for a given deformability factor $\alpha$, 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 reservoir pressure, $\Delta \dot{m}/\dot{m}$ increases $A_o^{-1}$ followed by $A_o^{-2}$ after a threshold with the tube diameter. We find that for the rigid tube, where the deformability parameter $1/\alpha=0$, the mass flow rate varies linearly, i.e., $\dot{m}_{\text{rigid}} \sim \Delta p $, whereas for the deformable tubes, the flow rate scales as $\dot{m}_{\text{deformable}}\sim \Delta p^2 $ for $ \Big ( \Delta p/\alpha A_o \Big ) \sim O(10^{-1})$ during transition from $\dot{m}_{\text{rigid}} \sim \Delta p $ to $\sim \Delta p^3 $, and finally to $\dot{m}_{\text{deformable}}\sim \Delta p^3 $ for $ \Big ( \Delta p/\alpha A_o \Big ) \sim O(1)$. We further find that the slip also significantly increases the mass flow rate in the nanotubes. Still, the deformability has, in comparison, a more substantial effect in increasing the mass flow rate to several orders than the slips