- Rapid Communication
- Access by Xinjiang University
Reciprocal theorem for convective heat and mass transfer from a particle in Stokes and potential flows
Phys. Rev. Fluids 1, 022001(R) – Published 21 June, 2016
DOI: https://doi.org/10.1103/PhysRevFluids.1.022001
Abstract
In the study of convective heat and mass transfer from a particle, key quantities of interest are usually the average rate of transfer and the mean distribution of the scalar (i.e., temperature or concentration) at the particle surface. Calculating these quantities using conventional equations requires detailed knowledge of the scalar field, which is available predominantly for problems involving uniform scalar and flux boundary conditions. Here we derive a reciprocal relation between two diffusing scalars that are advected by oppositely driven Stokes or potential flows whose streamline configurations are identical. This relation leads to alternative expressions for the aforementioned average quantities based on the solution of the scalar field for uniform surface conditions. We exemplify our results via two applications: (i) heat transfer from a sphere with nonuniform boundary conditions in Stokes flow at small Péclet numbers and (ii) extension of Brenner's theorem for the invariance of heat transfer rate to flow reversal.
Physics Subject Headings (PhySH)
Article Text
References (24)
- A. Acrivos and T. D Taylor, Heat and mass transfer from single spheres in Stokes flow, Phys. Fluids 5, 387 (1962).
- A. Acrivos and J. D. Goddard, Asymptotic expansions for laminar forced-convection heat and mass transfer part 1. Low speed flows, J. Fluid Mech. 23, 273 (1965).
- J. D. Goddard and A. Acrivos, Asymptotic expansions for laminar forced-convection heat and mass transfer part 2. Boundary-layer flows, J. Fluid Mech. 24, 339 (1966).
- N. A. Frankel and A. Acrivos, Heat and mass transfer from small spheres and cylinders freely suspended in shear flow, Phys. Fluids 11, 1913 (1968).
- H. Brenner, Forced convection heat and mass transfer at small Peclet numbers from a particle of arbitrary shape, Chem. Eng. Sci. 18, 109 (1963).
- H. Brenner, On the invariance of the heat-transfer coefficient to flow reversal in Stokes and potential streaming flows past particles of arbitrary shape, J. Math. Phys. Sci. 1, 173 (1967).
- H. Brenner, Invariance of the overall mass transfer coefficient to flow reversal during Stokes flow past one or more particles of arbitrary shape, Chem. Eng. Prog. Symp. Ser. 66, 123 (1970).
- L. G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes (Cambridge University Press, Cambridge, 2007).
- J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, with Special Applications to Particulate Media (Nijhoff, Leiden, 1983).
- L. G. Leal, Particle motions in a viscous fluid, Annu. Rev. Fluid Mech. 12, 435 (1980).
- H. A. Stone and A. D. T. Samuel, Propulsion of Microorganisms by Surface Distortions, Phys. Rev. Lett. 77, 4102 (1996).
- H. Masoud and H. A. Stone, A reciprocal theorem for Marangoni propulsion, J. Fluid. Mech. 741, R4 (2014).
- H. A. Stone and H. Masoud, Mobility of membrane-trapped particles, J. Fluid. Mech. 781, 494 (2015).
- S. Michelin and E. Lauga, A reciprocal theorem for boundary-driven channel flows, Phys. Fluids 27, 111701 (2015).
- A. Dörr, S. Hardt, H. Masoud, and H. A. Stone, Drag and diffusion coefficients of a spherical particle attached to a fluid-fluid interface, J. Fluid. Mech. 790, 607 (2016).
- H. Brenner and S. Haber, Symbolic operator solutions of Laplace's and Stokes' equations part 1. Laplace's equation, Chem. Eng. Commun. 27, 283 (1984).
- E. E. Michaelides and Z. Feng, Heat transfer from a rigid sphere in a nonuniform flow and temperature field, Int. J. Heat Mass Transfer 37, 2069 (1994).
- C. G. Bell, H. M. Byrne, J. P. Whiteley, and S. L. Waters, Heat or mass transfer at low Péclet number for Brinkman and Darcy flow round a sphere, Int. J. Heat Mass Transfer 68, 247 (2014).
- F. A. Morrison and S. K. Griffiths, On the transient convective transport from a body of arbitrary shape, J. Heat Transfer 103, 92 (1981).
- W. E. Olmstead, Reciprocal relationships in viscous hydrodynamics, Acta Mech. 21, 289 (1975).
- M. Roper and M. P. Brenner, A nonperturbative approximation for the moderate Reynolds number Navier-Stokes equations, Proc. Natl. Acad. Sci. USA 106, 2977 (2009).
- A. Acrivos, Heat transfer at high Péclet number from a small sphere freely rotating in a simple shear field, J. Fluid Mech. 46, 233 (1971).
- G. K. Batchelor, Mass transfer from a particle suspended in fluid with a steady linear ambient velocity distribution, J. Fluid Mech. 95, 369 (1979).
- T. D. Taylor, Heat transfer from single spheres in a low Reynolds number slip flow, Phys. Fluids 6, 987 (1963).