A Reciprocal Theorem for Convective Heat and Mass Transfer from a Particle in Stokes and Potential Flows

In the study of convective heat and mass transfer from a particle, key quantities of interest are usually the average rate of transfer and mean distribution of the scalar (i.e. temperature or concentration) at the particle surface. Calculating these quantities using conventional equations requires the 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 non-uniform boundary conditions in Stokes flow at small Peclet numbers, and (ii) extension of Brenner's theorem for the invariance of heat transfer rate to flow reversal.