Modelling of enhanced water flow in deformable carbon nanotubes using a linear pressure-diameter relationship

Numerous researchers have documented a notable enhancement in water flow through nanotubes While modelling, these researchers typically treated the CNTs with rigid walls. The flow rates of water within carbon nanotubes (CNTs) are significantly influenced by the nanoconfined density, viscosity and the slip length. Despite considering substantial slip effects, there are unresolved findings of massive enhancements in flow rates. Recently, using a linear pressure-area relationship for the deformable tube walls, Garg (2023) derived a model for the flow rates. In contrast to that, this paper takes a different approach, utilizing a small displacement structural mechanics framework with a linear pressure-diameter relationship, to incorporate the deformable nature of carbon nanotubes and derive another deformable model. We compare predicted flow rates with previous findings. The rigid-wall model with slips accurately predicted the outcomes of numerous studies. Nonetheless, we observed that in many studies featuring high porosity and thin-walled tubes, the inclusion of tube elasticity or deformability is crucial for accurate modeling. In such cases, our deformable-wall model with slips performed exceptionally well in predictions. We also compare and contrast the flow physics and flow rate scaling of the current model with the predictions from the Garg (2023) deformable model. We also find that as the deformability $1/\beta$ increases, the flow rate also increases. Although the scaling for how the flow rate and flow physics varies are different than reported by Garg (2023) with pressure-area model. We find that the flow rate in deformable tubes scales as $\dot{m}_{\text{deformable}}\sim 1/\beta^0 $ for $\Big ( \Delta p/\beta \sqrt{A_o} \Big ) \ll 1$, $\dot{m}_{\text{deformable}}\sim 1/\beta $ for $\Big ( \Delta p/\beta \sqrt{A_o} \Big ) \sim O(10^{-1})$ and $\dot{m}_{\text{deformable}}\sim 1/ \beta^4 $ for $\Big ( \Delta p/\beta \sqrt{A_o} \Big ) \sim O(1)$. Further, for a given deformability factor $\beta$, the flow rate in the smaller diameter of the tube is much larger than the larger diameter where the flow rate increases with $D_o^{-1}$ followed by $D_o^{-4}$ as diameter decreases. We also find that for the rigid tube, the mass flow rate varies linearly with pressure, whereas for the deformable tubes, the flow rate scales as $\dot{m}_{\text{deformable}}\sim \Delta p^2 $ for $ \Big ( \Delta p/\beta \sqrt{A_o} \Big ) \sim O(10^{-1})$ during transition from $\dot{m}_{\text{rigid}} \sim \Delta p $ to $\sim \Delta p^5 $, and finally to $\dot{m}_{\text{deformable}}\sim \Delta p^5 $ for $ \Big ( \Delta p/\beta \sqrt{A_o} \Big ) \sim O(1)$. On the otherhand, the scaling reported by Garg (2023) was $\dot{m}_{\text{deformable}}\sim \Delta p^3 $ for $ \Big ( \Delta p/\alpha A_o \Big ) \sim O(1)$.