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We compute steady planar incompressible flows and wall shapes that maximize the rate of heat transfer ($$Nu$$) between hot and cold walls, for a given rate of viscous dissipation by the flow ($$Pe^2$$), with no-slip boundary conditions at the walls. In the case of no flow, we show theoretically that the optimal walls are flat and horizontal, at the minimum separation distance. We use a decoupled approximation to show that flat walls remain optimal up to a critical non-zero flow magnitude. Beyond this value, our computed optimal flows and wall shapes converge to a set of forms that are invariant except for a$$Pe^{-1/3}$$scaling of horizontal lengths. The corresponding rate of heat transfer$$Nu \sim Pe^{2/3}$$. We show that these scalings result from flows at the interface between the diffusion-dominated and convection-dominated regimes. We also show that the separation distance of the walls remains at its minimum value at large$$Pe$$.
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This content will become publicly available on October 10, 2026
Shape dynamics of nearly spherical, multicomponent vesicles under shear flow
In biology, cells undergo deformations under the action of flow caused by the fluid surrounding them. These flows lead to shape changes and instabilities that have been explored in detail for single component vesicles. However, cell membranes are often multicomponent in nature, made up of multiple phospholipids and cholesterol mixtures that give rise to interesting thermodynamics and fluid mechanics. Our work analyses shear flow around a multicomponent vesicle using a small-deformation theory based on vector and scalar spherical harmonics. We set up the problem by laying out the governing momentum equations and the traction balance arising from the phase separation and bending. These equations are solved along with a Cahn–Hilliard equation that governs the coarsening dynamics of the phospholipid–cholesterol mixture. We provide a detailed analysis of the vesicle dynamics (e.g. tumbling, breathing, tank-treading and swinging/phase-treading) in two regimes – when flow is faster than coarsening dynamics (Péclet number$${\textit{Pe}} \gg 1$$) and when the two time scales are comparable ($$\textit{Pe} \sim O(1)$$) – and provide a discussion on when these behaviours occur. The analysis aims to provide an experimentalist with important insights pertaining to the phase separation dynamics and their effect on the deformation dynamics of a vesicle.
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- Award ID(s):
- 2147559
- PAR ID:
- 10647835
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1020
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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