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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 $\mathit{Nu}_{\infty }$ .

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References

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Barron, R. F., Wang, X., Ameel, T. A. & Warrington, R. O. 1997 The Graetz problem extended to slip-flow. Intl J. Heat Mass Transfer 40 (8), 18171823.
Barrow, H. & Humphreys, J. F. 1970 The effect of velocity distribution on forced convection laminar flow heat transfer in a pipe at constant wall temperature. Wärme Stoffübertrag. 3 (4), 227231.
Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2007 Transport Phenomena, 2nd edn. John Wiley & Sons.
Bocquet, L. & Barrat, J.-L. 2007 Flow boundary conditions from nano- to micro-scales. Soft Matt. 3, 685693.
Colin, S. 2011 Gas microflows in the slip flow regime: a critical review on convective heat transfer. J. Heat Transfer 134 (2), 020908.
Eckert, E. R. G. & Drake, R. M. 1972 Analysis of Heat and Mass Transfer. McGraw-Hill.
Enright, R., Hodes, M., Salamon, T. & Muzychka, Y. 2013 Isoflux Nusselt number and slip length formulae for superhydrophobic microchannels. J. Heat Transfer 136 (1), 012402. doi:10.1115/1.4024837.
Ezquerra Larrodé, F., Housiadas, C. & Drossinos, Y. 2000 Slip-flow heat transfer in circular tubes. Intl J. Heat Mass Transfer 43 (15), 26692680.
Graetz, L. 1882 Über die Wärmeleitungsfähigkeit von Flüssigkeiten. Ann. Phys. 254 (1), 7994.
Graetz, L. 1885 Über die Wärmeleitungsfähigkeit von Flüssigkeiten. Ann. Phys. 261 (7), 337357.
Jakob, M. 1949 Heat Transfer, vol. 1. John Wiley & Sons.
Karniadakis, G., Beskok, A. & Aluru, N. 2005 Microflows and Nanoflows: Fundamentals and Simulation. Springer.
Lafuma, A. & Quéré, D. 2011 Slippery pre-suffused surfaces. Europhys. Lett. 96 (5), 56001.
Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Microfluidics: the no-slip boundary condition. In Springer Handbook of Experimental Fluid Mechanics (ed. Tropea, C., Yarin, A. L. & Foss, J. F.), pp. 12191240. Springer.
Lévêque, M. A. 1928 Les lois de la transmission de chaleur par convection. Ann. Mines, Mem., Ser. 13 (12), 201–299, 305–362, 381–415.
Majumder, M., Chopra, N. & Hinds, B. J. 2011 Mass transport through carbon nanotube membranes in three different regimes: ionic diffusion and gas and liquid flow. ACS Nano 5 (5), 38673877.
Maynes, D., Webb, B. W. & Davies, J. 2008 Thermal transport in a microchannel exhibiting ultrahydrophobic microribs maintained at constant temperature. J. Heat Transfer 130 (2), 022402.
Maynes, D., Webb, B. W., Crockett, J. & Solovjov, V. 2012 Analysis of laminar slip-flow thermal transport in microchannels with transverse rib and cavity structured superhydrophobic walls at constant heat flux. J. Heat Transfer 135 (2), 021701. doi:10.1115/1.4007429.
Navier, C. L. M. H. 1823 Mémoire sur les lois du mouvement des fluids. Mem. Acad. Sci. Inst. Fr. 6, 389–416, 432–436.
Nusselt, W. 1910 Die Abhängigkeit der Wärmeübergangszahl von der Rohrlänge. Z. Verein. Deutsch. Ing. 54 (28), 11541158.
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42 (1), 89109.
Shah, R. K. & London, A. L. 1978 Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. Academic.
Sparrow, E. M. & Lin, S. H. 1962 Laminar heat transfer in tubes under slip-flow conditions. J. Heat Transfer 84 (4), 363369.
Whitby, M. & Quirke, N. 2007 Fluid flow in carbon nanotubes and nanopipes. Nat. Nano 2 (2), 8794.
Wong, T.-S., Kang, S. H., Tang, S. K. Y., Smythe, E. J., Hatton, B. D., Grinthal, A. & Aizenberg, J. 2011 Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity. Nature 477 (7365), 443447.
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The Graetz–Nusselt problem extended to continuum flows with finite slip

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