The Graetz–Nusselt problem extended to continuum flows with finite slip

Abstract

Graetz and Nusselt studied heat transfer between a developed laminar fluid flow and a tube at constant wall temperature. Here, we extend the Graetz–Nusselt problem to dense fluid flows with partial wall slip. Its limits correspond to the classical problems for no-slip and no-shear flow. The amount of heat transfer is expressed by the local Nusselt number $\mathit{Nu}_{x}$, which is defined as the ratio of convective to conductive radial heat transfer. In the thermally developing regime, $\mathit{Nu}_{x}$ scales with the ratio of position $\tilde{x}=x/L$ to Graetz number $\mathit{Gz}$, i.e. $\mathit{Nu}_{x}\propto (\tilde{x}/\mathit{Gz})^{-{\it\beta}}$. Here, $L$ is the length of the heated or cooled tube section. The Graetz number $\mathit{Gz}$ corresponds to the ratio of axial advective to radial diffusive heat transport. In the case of no slip, the scaling exponent ${\it\beta}$ equals $1/3$. For no-shear flow, ${\it\beta}=1/2$. The results show that for partial slip, where the ratio of slip length $b$ to tube radius $R$ ranges from zero to infinity, ${\it\beta}$ transitions from $1/3$ to $1/2$ when $10^{-4}<b/R<10^{0}$. For partial slip, ${\it\beta}$ is a function of both position and slip length. The developed Nusselt number $\mathit{Nu}_{\infty }$ for $\tilde{x}/\mathit{Gz}>0.1$ transitions from 3.66 to 5.78, the classical limits, when $10^{-2}<b/R<10^{2}$. A mathematical and physical explanation is provided for the distinct transition points for ${\it\beta}$ and $\mathit{Nu}_{\infty }$.

Image-Based Modeling of Blood Flow and Oxygen Transfer in Feto-Placental Capillaries

“…The approximation Eq (12) is expected to hold for sufficiently short capillaries with Pe eff ) 1, as long as Pe eff is not large enough for oxygen transfer to be limited by diffusion through the villous volume. However, for the images in this study, if the ratio of capillary length L to radius R 0 plus villous thickness d is sufficiently large (Pe eff ( LR 0 =ðd þ R 0 Þ 2 ), oxygen is expected to diffuse to the centreline of the downstream region of the capillary, suppressing concentration gradients across the capillary cross-section [46] (if the simplifying assumption is made that capillary branches do not affect each other, the longest path between the inflow boundary and outflow boundary through a branched capillary could be used as L in this inequality). In this case the concentration is said to be fully developed, or equilibrated, in the downstream region of the capillary.…”

“…, oxygen is expected to diffuse to the centreline of the downstream region of the capillary, suppressing concentration gradients across the capillary cross-section [46] (if the simplifying assumption is made that capillary branches do not affect each other, the longest path between the inflow boundary and outflow boundary through a branched capillary could be used as L in this inequality). In this case the concentration is said to be fully developed, or equilibrated, in the downstream region of the capillary.…”

Tuning of salt separation efficiency by flow rate control in microfluidic dynamic dialysis

“…As an idealization, it is assumed that the solute is consumed at the channel wall, i.e., the membrane in a dialyzer, that the time scale for solute diffusion is longer than the time scale of axial convection (~ flow rate), and that there is no radial convection. This enables calculation of the diffusive solute flux to this boundary wall and its scaling with the flow rate as a function of the distance from the feed channel inlet, which takes the form of a 1/3 power law for the parabolic velocity profile of a fully developed laminar flow (Kirtland 2010;Haase et al 2015). Here, we follow the model derived by Yeh et al (Yeh and Hsu 1999;Yeh and Chang 2005), where the total mass transfer rate M (mol/s) in a planar counter-flow dialyzer is given by Eq.…”

An asymptotic theory for the high-Reynolds-number flow past a shear-free circular cylinder

“…The reduction in vorticity generation and flow separation offers additional advantages such as slip-enhanced transport (Haase et al. 2015; Haase & Lammertink 2016; Rehman, Kumar & Shukla 2017) and slip-induced flow stabilization (Legendre et al. 2009; Muralidhar et al.…”

“…This exceptional feature of a shear-free surface is targeted in devising patterned superhydrophobic surfaces that effectively reduce drag by inducing significant slip over underwater bodies (Ou, Perot & Rothstein 2004;You & Moin 2007;Rothstein 2010;Bocquet & Lauga 2011;Muralidhar et al 2011;Karatay et al 2013). The reduction in vorticity generation and flow separation offers additional advantages such as slip-enhanced transport (Haase et al 2015;Haase & Lammertink 2016;Rehman, Kumar & Shukla 2017) and slip-induced flow stabilization (Legendre et al 2009;Muralidhar et al 2011;Seo & Song 2012;Li et al 2014;Xiong & Yang 2017;Sooraj et al 2020) as well. Advances in theoretical analysis and fundamental understanding of flow past shear-free surfaces is of significant technological importance and paramount for an effective realization of the full range of their drag and dissipation reducing, and transport enhancing capabilities.…”