Convective heat transfer in the thin film of an elongated bubble in the absence of phase change

The injection of an elongated bubble into a microchannel is a well-established strategy for enhancing convective heat transfer, with applications ranging from electronics cooling to miniaturized heat exchangers. So far, a rigorous characterization of the heat removal capabilities of elongated bubbles (also known as Bretherton's or Taylor's bubbles) is still lacking. To address this gap, we investigate the forced convection problem in the thin film formed by an elongated bubble under uniform wall heat flux, examining the competition between advection, diffusion, viscous dissipation, and an imposed heat flux at the channel walls. By means of two-scale asymptotic analysis, we derive a one-dimensional advectiondiffusion-heat-transfer equation with shape-dependent effective coefficients. This model generalizes the classical Graetz problem to capillary-driven flows and extends the Aris-Taylor dispersion to the case of an elongated bubble, clarifying the interplay between the imposed heat flux and the recirculating flow patterns at both front and rear menisci. Interestingly, the model recovers the P & eacute;clet-squared scaling of the effective diffusion coefficient and can be used to determine the heat transfer coefficient in the film region. In fact, we derive a closed-form scaling law for the Nusselt number, revealing the influence of the problem parameters (i.e., the bubble profile and the axial temperature gradient) and the dimensionless groups (P & eacute;clet, Brinkman, and capillary numbers) on forced convection. Grounded in first principles, our analysis contributes to a deeper understanding of capillary-driven forced convection in confined environments.